3 Linear Equations

p. Here, we take the radius p of the sphere greater than / x - I. Tn contrast with (2.10), we now place the center of the sphere at the point x rather thanx. Performing the same transformations a s in the preceding case, we obtain the formula

x

x

Let u s subtract Eq. (2.11) from (2.10) and let us transform the result as follows:

X"[

1

dyi dyi

Ix

-y

In-'

-

I x-

y

128

LINEAR EQUATI ONS

The t e r m s in the braces cancel each other out, as is easily seen.

(2.12)

Here, p is an arbitrary number greater thanlx - 21. With the aid of this representation, we can bound the quantity 1D2wl(m), En in t e r m s

of En just as was done above for vxiXn(x'. xn)-vXlXn(x'.%,). We shall not repeat the details. F r o m Lemmas 2.1 and 2.2, we derive l e m m a 2.3. I f the functions f(x) and Q ( X ) are equal to 0 outside the sphere 1x1c k , then, for the solution U ( X ; of the Dirichlet problem Au =f (XI. u Ixn=0 = 'p (x'). (2.13)

in the half-space x,, >0, which converges to 0 as I x I + 00, the following inequality i s valid:

where the constant c depends only on n. Let u s represent the solution u ( x ) a s a sum of potentials. To do this, we extend f ( x ) onto the half-space x , ,< 0 a s an even function, so that / ( x ' , - x , , ) = f ( x ' . xn). We keep our former notation f o r this extended function. Obviously, Ifl(@), ( X n > o l - If[(=), En. Fromf(x), let u s construct the Newtonian potential

129

SCHAUDER'S A PRIOR1 E S T I M A T E

The function v ( x ) = u ( x ) + w (x) will satisfy Laplace's equation and, for x n = 0, it becomes the function p ( x ' ) w ( x ' , 0). It approaches 0 as 1x1 +me Therefore,

+

21

(4= -

JK

(Jd

-Y) I Y (Y')

Yn= 0

+

'w (Y'l

0)l dY'1

and the solution u (x) is v ( x ) -w (x). On the basis of Lemmas 2.1 and 2.2, inequality (2.14) is valid for u ( x ) . This completes the proof of Lemma 2.3. An arbitrary function u ( x ) of compact support belonging to C2,(En)can be represented in the form of a Newtonian potential (I

with finite density Au. Therefore (cf. Lemma 2.2), for this function, (2.15)

Let u s now turn to the derivation of inequality (1.11). Suppose that u ( x ) E C,,a(a), that S E C,,., that the coefficients L belong to Co, and that 'p (s)= u Is E C,, (S). Let us take a finite number of indefinitely differentiable nonnegative functions (x), ., ,C ( x ) of compact support such that, for every x E

(a),

(L

a,

N

[,

..

xEG.

xcC,(x)=l.

k=l

(2.16)

with the property that the diameters of their supports do not exceed some small number 6. The value of 6 will be made more precise below. It is determined only by quantities that we know. h accordance with (2.16), the function u ( x ) can be represented in the form N

2 U h ( x ) = u (x)s k=1

where

uy ( x ) =sL u ( x ) Ck (x).

Let us denote LU by f(x). Let us multiply the equation Lu = f by C, and then represent it in the form MU

where Fk

=f c k

+

a i j (x) ~

a i j (2uxibX,+

k

I /

=~F , (x). . ~

uC&xiX,)

- aiuxiCk - a d , .

(2.17)

130

LINEAR EQUATIONS

If the support Q, of the function C,(x) is contained in M, and if ' b ( x ) to all En by assigning it the value 0 outside Q k , we can regard u k (x) a s a function of compact support in C2,~ ( E , that ) satisfies Eq. (2.17) with inhomogeneous t e r m F , ? ( x ) . On the other hand, if the carrier 9 , is only partly contained in Q , we set up in it n, in such a way new regular coordinates y, = y, ( x ) , for i = 1 , that the portion S , of the boundary S that belongs to '2, will be represented by the equation y, = 0 and the region Q , n Q will belong to the half-space y, 0. The functions y, ( x ) must be elements of C2,(2,). In the new coordinates, the function (y) = uk ( x ( y ) ) will satisfy an equation of the same form (2.17), namely,

we extend

...,

>

uk

M u , =&/ (Y)

'kYiY,

= F&(Y),

(2.18)

where &,](y) and F , ( y ) are calculated from a l j ( x ) and F , ( x ) by a familiar procedure.Let u s extend u k (y) to the entire half-space y, > 0 by setting ' k (y) = 0 for values of y in (y, 0) that do not belong to the image of the region Q k n 8. We keep the same notation as before for the function i k ( y ) thus extended. In the region y,> 0, it satisfies Eq. (2.18) and, for y, = 0, it satisfies the boundary condition

>

(2.19)

(y) lV,,=O = G k (y')*

where ' 9 k (s) = '9 (s) [ k (s) and .9, (YO = y k (S (Y) 1. Let u s represent Eqs. (2.17) and (2.18) in different forms: Mo', = atj ( x " )U k x i x , = Fk ( X I

[ a i j (x')

-

( X ) ] Uhxix,

(2.17 ')

and (2.18')

where xu and y" are arbitrary points in M k and G k respectively. Instead of x and y, let us introduce new rectangular coordin?tes z with ?rigin at the points x') andybin such a way that Eqs. (2.17 ) and (2.18 ) will be transformed respectively into a;j (o)ukZi~,~4'&=F;(z)+[n~j

(2.17")

(o)-a~j(z)]'Lkz,z 1 /

and

> >

and in such a way that the half-space (y, O } in the second case will be transformed into the half-space ( z, 0).

131

SCHAUDER‘S A PRIOR1 ESTIMATE

Let us apply inequality (2.15) to the solution of Eq. (2.17”). Then,

Recalling the form of F ; ( z ) , the nature of the changes from x to and from y t o z , and the rules for evaluating the HGlder constant for the product of two functions, we obtain

y(x)

The maximum of the quantity Ia;,(O) - a ; , ( % ) ] is over all z in the support of u k . Let us assume the diameter of the support to be sufficiently small so that

This is the only restriction that we make on the choice of the diameter 6 of the interior regions Q k . For such values of 6, we obtain from (2.20) n

2 I I1kzlZl I(=), i.I=1

En

4 2 c ~I

f

( x ) 12,0, P~ -k 2 c ~1

1,

Q ~ *

(2.22)

For the solutions u k ( z )of Eqs. (2.18”) that satisfy the boundary condition (2.19) (or, more precisely, the condition obtained from it by making the linear transformation from y to z ) , we apply Lemma 2.3. From Eq. (2.14) with condition (2.21) [with constant c taken from (2.14)], we obtain by a procedure analogous to that followed above

(2.23)

I

Tf we use inequality (2.1), we can replace ~ u ~ z i z j ! , z )and ,E,z

“k?.Z.

I ‘k x

~

I1(1).

l2,

01,

En

(Zn”O)

in the left sides of (2.22) and (2.23) by the norms

and 1 U k 12. n, ( Z n ” O ) ’

Returning now to the original variables

and summing these inequalities over all k, we obtain

132

L I N E A R EQUATIONS

I u ( x ) 12. a, 3 -<%(I u 12,0, 3

+I f

In.

3

+I

'p 12,

LI,

s).

(2.24)

From this and from (2.1) we obtain the desired inequality (1.11). It is easy to show that the constant c in (1.11) canbe chosen independently of the dimensions of the region 3. To do this, one should construct the functions Ck ( x ) in such a way that their supports 8, will have diameters not exceeding some fixed number c ( n ) that is independent of the dimensions of 8. Finally, let us prove inequalities (1.12) and (1.13). Let S, be the portion of the boundary S on which the function 'p(s) belongs to C2,a(Sl). A s special cases, S , can coincide with all S or it may even be the empty set. Let u s denote by d , the distance from the point x to S \ S, and let u s denote min ( d x , d y ) by dXy. We introduce the following norms:

We need these norms with 1 = 0, 1, 2. The following relationships are valid for them: lemma 2.4. F o r arbitrary positive E < 1 and an arbitrary func-

tion u ( x ) E c, (a),

MI [ul

< E M , [ill + CE

M , [ u ] ,< E M , + , [ u ]

+

CE

_ _ _I m-'

--

M,,[ u ] , 1

< tn,

(2.25)

l
(2.26)

I

m+'l-l

M o [ u 1.

where the constants c depends on 1, m , and a but not on u ( x ) or E . The proof is elementary but long (cf. [31, 501). We shall not go through it here. We introduce yet one other notation: 2

Let us prove that, for an arbitrary function

N z + ~Q ,Iul,

N2+o,3 Iul


"a.

3

ILnl+ M, Iul

u ( x ) with

+ 1'9 12,

u,

finite norm (2.27)

The constant c in this inequality is determined by the constant of ellipticity v in (1.2), the norms of the coefficients L in Co,II (s), and the norm of the portion S, in C2, The function y ( s ) is u ( x ) IxZs Es,'

..

133

SCHAUDER'S A PRIOR1 E S T I M A T E

Inequality ( 2 . 2 7 ) is a somewhat more precise form of inequality (1.13).

On the basis of ( 2 . 2 6 ) , instead of the norm Nz+a,[ u ] , it will be sufficient for us to find a bound for the quantity M 2 + . , s [ u ] , which, by definition, is equal to

Let us choose two points x and y in Q at which

(2.28)

Two possibilities may arise:

Tn the first case,

Then, inequality ( 2 . 2 7 ) will be a simple consequence of ( 2 . 2 6 ) with I = rn = 2.

In the second case, that is, with I x -yl


d x ,y,

let us

suppose for definiteness that dx,y=d,. Consider the sphere K of radius 3d,/4 with center at the point x. For this sphere, we introduce the function C ( z ) E Cz, (s) defined to be equal to 1 for Ix - z 1 < d x / 2 and equal to 0 for I x - zI >/ 3 d x / 4 . The function v ( z )= u (z)C(z) satisfies the following equation in K n 8 :

where F = Ln . C

+ 2ai~uzlCz,+

aijiiCziz,

+w L i .

Let u s extend w(x) to the entire setG by setting it equal to 0 outside n M. We still use the notation v ( z ) for this extended function.

K

134

LINEAR EQUATIONS

Inequality (l.ll),which we proved above, is applicable to it. For all possible arrangements of K with respect to S,(that i s , whether K does or does not intersect Sl),this inequality canbe written in the following somewhat cruder form:

Let u s assume the function C(z) chosen in such a way that JCt, I

< ..c, d,

lCl(a) < &, and ICzizj( ,< 4 (which, obviously, is possible). After dX

dx

some simple computations, we see that

and

I CY 12, =, K n S , .S 2 +.I 'P 12. , K n S , dx C

If we substitute these inequalities into (2.30), multiply the result by

d?", and keep (2.28) in mind, we obtain

and, on the basis of (2.26), this inequality yields (2.27). This completes the proof of all the assertions made at the beginning of this section for 1 = 2. For 1 > 2, they can be derived in an elementary manner from the case 1 = 2.

3. THE SOLVABILITY IN cz,,(a) OF OTHER BOUNDARY -VALU E PROBLEMS The solvability of the second and third boundary-value problems and the oblique-derivative problem for sufficiently smooth given

THE SOLVABILITY O F OTHER BOUNDARY-VALUE

PROBLEMS

135

conditions is investigated basically in the same way in which this was done in the preceding section for the first boundary-value problem. The analytical basis consists of a priori bounds of the Schauder type, that i s , inequalities of the type (1.11). For the second and third boundary conditions, such inequalities were proven in 1955 by Miranda [51]. These conditions for the operator

take the form Au = a l l ( x ) U X l cos (a. x,)

+b ( x ) u = p ( x ) .

(3.2)

+

For b ( x ) E 0, we have the second boundary condition; for b ( x ) 0, we have the third. However, in contrast with the first boundary condition, we need to impose more stringent restrictions on the functions a , / ( x ) close to S than was necessary for the interior estimates. Specifically, if the function u ( x , belongs to C2,I then it is natural to require that the coefficients in L belong to Co, (G). In this case, every t e r m in Lu, and hence the entire expression L u , will be a function in Co,a(32). The boundary condition (3.2), on the other hand, contains the first derivatives of u ( x ) ; these belong to C,,.(Q). Therefore, the function Q ( X ) must, in the general case, be an element of Cl,m(a)(since, in particular, when the ail are constants and b ( x ) = 0, the function qa is a linear combination of the first derivatives of u ) . But then, we need to require that the a,, be elements of C,,%(S). Thus, there is a difference in the requirements on the a I i ( x ) within 8 and on S. But it is unavoidable. Therefore, when we investigate the second and third boundary-value problems in the nonlinear case, a knowledge only of Miranda's inequalities regarding the linear problems is insufficient. (Miranda found a bound for the norm I u 12, =, for an arbitrary function u in C2, (H)in t e r m s of

(a),

l q n ' m p l ,

and lA~l,,,&s.)

To study the nonlinear problems, a m o r e general result, proven in 1959 by Fiorenza [52] was needed: Theorem 3.1. Suppose that the boundary S of a region 5! i s a class C2. surface, that the coefficients u i i ,a,, and a in the operator L belong to Co,a(G),and that the ellipticity condition

> v z E.: n

a,, ( x ) E,E/

r=1

v = const

>0

(3.3)

is satisfied. Suppose that the coefficients b , ( x ) and b ( x ) in the

136

LINEAR EQUATIONS

boundary operator

are elements of Cl, ( S ) and that n

2 bi (n)cos (n,X i )Is > i= 1

V",

V"

= const

> 0.

(3.4)

Then, f o r an arbitrary finction u ( x ) in C2, (g), (1

and

where the constants c depend only on n, the constants v and v,, in conditions (3.3) and (3,4), and the boundary S , which is assumed to belong to the class C Z s a .

T H E S O L V A B I L I T Y OF OTHER BOUNDARY-VALUE

137

PROBLEMS

The second of these inequalities has the following structure:

where the constant c, is independent of u ( x ) and is determined only by S and the coefficients in the operators L and B . This dependence of c1 on the a l l , the a,, a , the bi, b , and S is clearly shown in (3.5) and (3.6). Knowledge of it is essential for study of the second and third boundary-value problems for quasilinear equations (cf. Chapter 9) but is not significant in the linear case. With the aid of inequality (3.7), we can easily show that, for the problem L u - ~ u = f(X).

(3.8)

Bu(,=~(s),

where h is a complex parameter, the following theorem is valid: Theorem 3.2. Suppose that the conditions of Theovem 3.1

regarding S and the coefficients L and B are satisfied, that f ( x ) E C o .m(%), and that y ( s ) E C,,=(S). Then, .the problem (3.8) has a unique solution in C , , ( B ) for arbitrary f and 'p in the classes mentioned for all h except f o r a countable number of values A,. X,, , . . that constitute the spectrum of the problem (3.8). For these exthe absolute value I l k ( ceptional values k=h,, for k = 1, 2 , approaches infinity (or, more precisely, Re hk + - 00) as k -+00. Each of the kk i s of finite multiplicity, that is, the homogeneous problem corresponding to it

...,

LU -

h k ~

(3.9)

= 0 , BU 1s = 0 ,

has only finitely many linearly independent nontrivial solutions in Cz,a(G). These same h,, for k = 1, 2, , with the same multiplicity, constitute the spectrum of the problem adjoint to (3.9). The inhomogeneous problem (3.8) i s not solvable for all f and 'p; the conditions of orthogonality to the solutions of the homogeneous problem adjoint to (3.9) must be satisfied.

. .,

uk(x)

This theorem is proven in basically the same way as the corresponding assertions were proven for the f i r s t boundary-value problem in Section 1. This time, however, we need continuation along the parameter not only for the operator L but also for the boundary operator B . The overall procedure is a s follows: We denote by M , the operator that assigns to each element v ( x ) in C,,.(a) the pair of functions { L v ; B v ) , which we treat as an element of the space C,,,LI (s) X C!, (S). Let u s consider the family of operators M,v=(L,v; B,v),

where

11,

138

L I N E A R EQUATIONS

and

into For every T in [0, 11, the operator M, maps the space Cz, the space C,,,(n)n Cl, I (S). Inequality (3.7) or, more precisely, the inequality I

I42,01, P

4

c1(

I L u la, P + lB7u 11, a. S)'

which is valid if the coefficient a ( x ) of u satisfies the condition a ( x ) < 0 andb(x) >0, ensures uniform boundedness of the operators M;'(M,-MM,) in the space C z , a ( a )provided we know that M;' exists and is defined on the entire space Co,.@) x C1,"(S). This, together with the fact that the operator M;' is defined for T = 0 on a set that is dense in Co, (n) x Cl, ( S ) , ensures the existence of operators M;' that a r e defined on the entire space C",, bounded (9)x C,, .(Sj for all T i n [0, 11 and, in particular, it ensures a unique solution of the problem (I

BU IS ='p (s)

Lu = f.

(3.10)

in C2,I (G) for arbitrary f in C", ( G ) and 'p in C,,(S). In the general case of arbitrary a ( x ) in Co,=(Q) and b ( x ) in Cl, (S), we have Fredholm's alternative, formulated in Theorem 3.2. The existence of IM,;' on a set that is dense in C, (G) x Cl, (S) can be proven either directly on the basis of the theorems on electrostatic potentials or in the manner in which this was done in Section 1. The inequalities of the type (1.13) with function C ( x ) that a r e necessary for the second procedure hold even in the case that we a r e considering the boundary condtion Bn Is =9. '1

(I

~

4. GENERALIZED SOLUTIONS IN W : ( Q , . THE FIRST FUNDAME NT A L INEQUALlTY

Let us now study equations with unbounded coefficients. Consider equations of the form

under the assumption that the leading coefficients

a,,

a r e bounded

GENERALIZED SOLUTIONS

139

and that Eq. (4.1) is strictly elliptic; that i s , assume that

In Section 2 of Chapter 1,we showed that, to consider generalized solutions of such equations in W: (Q), we need to assume* that

If at least one of these conditions fails to be satisfied, the theory of generalized solutions in W: (Q) is not applicable to such equations since in that case, the theorem on uniqueness “in the small” is violated. We shall say that a function IL ( x , belonging to Wi (Q) is a generalized solution of Eq. (4.1) in W i ( Q ) if it satisfies the integral identity

for an arbitrary function q(x) in W : ( Q ) (cf. Section 1, Chapter 1). It is easy to see that if the functions f iand f and the coefficients in Eq. (4.1) are smooth and if the function u ( x ) is its classical solution or belongs to Wi(Q) and satisfies Eq. (4.1) almost everywhere, then it also satisfies the identity (4.4). To see this, let us multiply (4.1) by q, integrate over 9 , and transform the series of t e r m s by integrating by parts. This leads us to the relation (4.4). The converse is also true: if the function u(x),,belongs to W;(Q) and satisfies the identity (4.4) for arbitrary q E W:(Q) [or at least for arbitrary q E W : (Q)]and if the other functions appearing in (4.4)a r e smooth, then u (x) satisfies Eq. (4.1)almost everywhere. Both these phenomena exemplify the fact that the concept of a generalized solution of Eq. (4.1) is indeed an extension of the old concept of a classical solution. For the identity (4.4) to be meaningful for arbitrary q E W ; (Q), we impose the following restrictions on f l and f:

*For the case q

n a; 3, cf. end of this section.

:

140

LINEAR EQUATIONS

It is easy to verify that, when conditions (4.2), (4.3), and (4.5) a r e satisfied, the integrals of all terms in the identity (4.4) a r e finite for arbitrary u and rl in W : ( Q ) . Therefore, in that case, we may treat q in the definition given above of a generalized solution

a s an arbitrary function in lbl. This broadening of the concept of a solution is convenient from several points of view. In the first place, it makes it possible to study equations of the form (4.1) with nondifferentiable and even discontinuous and unbounded coefficients such that the left side of Eq. (4.1) may not be meaningful. Such equations a r e encountered in a number of problems, for example, in diffraction problems. In the second place, such solutions correspond to one of the principal approaches in functional analysis to the study of linear operators. Specifically, the identity (4.4) defines a bilinear form for the Operator L:

where the elements u a r e to be determined and 7 i:. an arbitrary number of W:(8). A s will be shown in the following section, it is then comparatively simple to determine the u if we also know their boundary values. In the third place, generalized solutions in W: (8) correspond to the physical nature of problems that lead to equations of the form (4.1). We shall clarify this with an example 3f the Dirichlet problem for Poisson’s equation Au== f,

u(,=O.

(4.7)

A s we know, this problem is encountered when we seek the equilibrium position u of an elastic homogeneous membrane that i s fastened along its boundary S and that is subjected to external forces f. The potential energy of such a membrane is given by the integral

I (u) =

J(

IVU12+

2uf)dx.

e

and its equilibrium state, according to Hamilton’s principle, is

GENERAL1 ZED SOLUTIONS

141

determined from the condition that the functional I ( u ) attain its smallest possible value in comparison with all ' ~ in&;@). t From this it follows that, for the equilibrium state u ( x ) , Sl(u)= 2

j-( V u V q + f 7 ) P

d x =0

(4.8)

for arbitrary q in W;(Q). Thus, in accordance with Hamilton's principle, the problem of finding the equilibrium position of the membrane reduces to finding the function u in the class W : (a) that satisfies the integral identity (4.8). The practice that has developed over the years of solving the problem of the minimum for the integral I ( u ) with the aid of the solution to the boundary-value problem for the Euler equation corresponding to it has considerably complicated the problem by introducing an extra requirement not inherent in the physical nature of the problem, namely, the requirement that the solution possess derivatives of twice as high order as the derivatives appearing in the energy integral I ( u ) . In many ways, this departure from the physical situation caused difficulties in solving the variational problems and the Euler equations associated with them. It turns out that the problem of finding the equilibrium position of a membrane is considerably simpler than problem (4.7) for finding the classical solution of the Euler equation corresponding to it. The broadening of the concept of a solution to Eqs. (4.1) proposed above in connection with the problem (4.7) consists in dropping the Euler equation and returning to the identity (4.8), which is simply the integral Eq. (4.4) for Poisson's equation. Let us now make clear what we mean by a generalized solution in W : ( 8 ) of the first boundary-value problem for Eq. (4.1). Obviously, for such solutions u ( x ) to exist, it is necessary that the function 'p (s) giving the boundary values of u ( x ) admits an extension 'p ( x ) to the entire region 8 that belongs to W2 (Q). Thus, suppose that 'p ( x ) E W: (51) and that the boundary condition for the desired solution is (4.9) must belong to W : ( Q ) , it i s sensible to write uls='pls.

Since our solution

u

condition (4.9) in the form u ( x ) -'p ( x ) E lb: (Q). From what has been said, we can see the naturality of the following definition. A generalized solution in W : ( Q ) of the problem(4.1), (4.9) is defined as a function u belonging to W:(Q) that satisfies the identity (4.4) or, what amounts to the same thing, the identity (4.10)

I42

for arbitrary

LINEAR EQUATIONS

(a) and that satisfies the condition

u ( x )-

p ( x ) E &: (Q). Since we shall not be studying generalized solutions in other functional classes in the next two sections, let u s agree for the time being to call them simply generalized solutions of an equation o r of a first boundary-value problem. The boundary-value problem with condition (4.9) is easily reduced to a boundary-value problem with homogeneous boundary conditions. Specifically, instead of u , we introduce the function

(4.11)

w ( x ) = I1 ( x ) - 'p ( x ) .

For this function, we obtain from (4.10) the condition

and from (4.9) the condition (4.13)

v l s =0 ,

which we understand in the sense that w E lb: (a). Instead of the function u , let us seek the function

'u

as a function

in $1 (52) that satisfies the identity (4.12). When we find it, we shall also have the function u =v+'p, which is a generalized solution of the problem (4.1), (4.9). Suppose that conditions (4.2), (4.3), and (4.5) are satisfied. Let u s show that, under these conditions and for ~ ( x in ) W : ( Q ) , the expression

is a linear functional over the space W : ( Q ) and that

(4.15)

GENERAL1Z E D SOLUTIONS

where f = (fl.

..., f,)

and /

and where

143

n

c ( 4 , Q) and c ( 4 , Q) a r e constants in the inequalities

which are valid for arbitrary functions u,E W: (Q) and 7 E ki (8) [cf. formulas (2.19) for E = 1, (2.13), and (2.17 } in Chapter 21. F r o m our assumptions (4.2), (4.3), and (4.5), let us find a bound for Jl(?)(with the aid of inequalities (4.16), Holder's inequality, and the generalized Cauchy inequality [cf. (1.4) and (1.2) in Chapter 21 a s follows:

Inequality (4.15) follows from these inequalities.

I44

L I N E A R EQUATIONS

Let u s turn now to the derivation of the first fundamental inequality (also called the energy inequality) for elliptic operators. We recall that, for an arbitrary function [cf. (2.15), Chapter 21

'u ( x )

E &; (Q),

the inequality

where E is an arbitrary positive number, is satisfied. Let u s represent the coefficient a ( x ) in the form of a difference: a ( x ) = a t ( x ) - a - ( x ) , where a + ( x ) = inax { a ( x ) - a , ; O), a - ( x ) = - a, max [ - a ( x ) + a,; 0) , and

+

I-

d

a ( x )dx.

We define (4.20)

Let u s show that the quadratic form L(w, w ) possesses the following property: l e m m a 4.1. Suppose that conditions (4.2) and (4.3) are satis-

fied, Then, f o r an arbitrary function w E W i (Q),

I [I

VWl2

Q

.++a-+x

4

4 ;(9%v ) f

c , (4)llv 112,,(9)r

where

Proof: By virtue of the ellipticity condition (4.2), we have

1I (v

Q

vw12

+a-wZ) d x Q L (u. w ) + n

(4.21)

GEN ERALl Z E D SOLUTl O N S

145

where E is an arbitrary positive number. Using HGlder's inequality to obtain a bound for the last two t e r m s and using the notation (4.20), we obtain

From this inequality and from (4.22) for

E

= v/2, we obtain

If we bound the last t e r m with the aid of inequality (4.19) and take E=

2M (2v

v24

+ 1)

C*

(4)R '

then, by collecting similar t e r m s , we obtain (4.21). This completes the proof of Lemma 4.1. Let us use inequality (4.21) to find a bound for the generalized solution u of the problem (4.1), (4.9). From the relations (4.12) and (4.13) for the function w = u --'p and from inequality (4.21), we obtain "IVulZ+yo

4

-

4

W q d n ,<,~(~)+~1(4)Il~Il~,ce).

9

where 1(u) is given by Eq. (4.14). (4.15), we have

Then, on the basis of inequality

146

LINEAR EQUATIONS

1

If we transpose the t e r m

I I V V ~ ~ ~t o, ~the ( ~ )left side, collect t e r m s of

like powers, and multiply the result by 2, we get

/

(4.24) 16 c 2 (9. QI 'p* f . vz

f)+

2~*(~~l141;2(Q).

Of special interest are the c a s e s in which we can discard the t e r m ~ ) the right side of this equation. To find out what with l j ~ 1 1 2 ~ (on

these cases a r e , let us use the inequality [cf. (2.14), Chapter 21 I -

(4.25)

Q11vvlI:2(Q) From this it is clear that, if

l l v l l L 2 ( QComes" )~

and the fact that min 51

a- ( x ) = - uo.

2

(4.26)

then it follows from (4.24) that [jVOl2rlX

i

16 <----c2(q. (1 - 6) v2

Q,

'p,

f, f ) .

(4.27)

lf 'p- 0, then v is a solution 11 of Eq. (4.1), and inequalities (4.24) and (4.27) for it take the forms

and

respectively. In the general case for 'pf 0, inequalities (4.24) and (4.27) yield the following inequalities for generalized solutions u = v + ' p of the problem (4.1), (4.9):

I47

GENERAL1 ZED SOLUTIONS

'C&.

P, '9. f. f).

In these inequalities, the constant c ( q , 'p. P, f . f) is takenfrom (4.15) and c l ( q ) from (4.21). Inequality (4.30) is the first fundamental inequality for the solutions of the problem (4.1), (4.9). For ip= 0, instead of this inequality, we can use inequality (4.28); also, when (4.26) is satisfied, we can use inequalities (4.29) and (4.31). We know that inequality (4.26) is satisfied in the follwoing two cases. In the first case, it is satisfied for regions 51 of sufficiently small area since

In the second case, it is satisfied for arbitrary regions provided

(4.32)

It is quite important that this inequality be satisfied for the operators Lu- Au for certain values of 1, for example, for all sufficiently large positive )i. This follows from the fact that the coefficient of u ( x ) in the expression Lu-Au is equal to a ( x ) - ) i and (a ( x ) -A)" = a, - A. In what follows, we shall consider the question of the solvability of the first boundary-value problemfor the entire set of operators L - E with arbitrary complex A. However, it will be convenient to begin this study with values of A such that condition (4.32) is satisfied. Without loss of generality, we may assume that this )i is 0, that is, that inequality (4.32) is satisfied for the operator L itself. The following section is devoted to a study of this case. In Section 2 of Chapter 1, we proved the necessity of the assumption q > n in (4.3) for the propositions proven in the present chapter to be valid. However, in that section, we pointed out that, for n >3, the limiting case q = n retains some of the features of the Dirichlet problem in its classical form. Let u s show this. Suppose that condition (4.2) is satisfied for L and that (4.33)

Let us represent the functions bi ( x ) - a, ( x ) and sums b,(x)-a,(x)=c;(x)+c;(x),

a+ ( x ) in

a+ (x)=c")+c"(x).

the form of (4.34)

148

LINEAR EQUATlONS

where c; ( x ) and c’ ( x ) a r e bounded functions and c; ( x ) and cN ( x ) a r e respectively elements of L,(Q) and L, (Q) with small norms. Suppose that 2

Obviously, in the general case, M., increases without bound a s E’ approaches 0 and the nature of this dependence of Ms.on E’ is determined not only by the size of M in (4.33) but also by L. For L we have an inequality analogous to (4.21). Its derivation i s the same a s for (4.21): We need to findabound for the right-hand member of inequality (4.22) by using Eqs. (4.34). As one can easily see, this leads us to the inequality J’(lvv12+~a-v2)dx,
+ +

e

2 (v 1) --y2-

M 2 r

+

V)+

2(v+

(4.36)

1)

l14!;2(9)yz E’ lrvlr;

2n

n -4y

(Q) *

On the basis of (4.16), the last term on the right does not exceed the quantity

Therefore, if

E‘

i s chosen so that

(4.37) then, we have from (4.36) the desired inequality:

Just a s we derive (4.24) from (4.21), wederive from this inequality the first fundamental inequality for the generalized solutions of Eqs. (4.1) in $;(Q):

SOLVABILITY OF THE F I R S T BOUNDARY-VALUE

PROBLEM

149

In carrying out this derivation, we note that inequality (4.15) is also meaningful for q=n. F r o m this we easily get the corresponding inequality for an aribtrary generalized solution of Eq. (4.1) in W:(Q)that is analogous to inequality (4.30). If the relation I';'

5 [' ;ti --

M,,-rnin 9,

1

2

a - ( x ) cirnes;i-&l

>a,>

0.

(4.40)

holds between the numerical parameters characterizing Eq. (4.1) for Q, c Q , it follows from (4.39) that

for generalized solutions v ( n ) of Eq. (4.1) that belong to $i(Qd. We know that inequality (4.40), like inequality (4.26), is satisfied in the following two cases: (1)for regions Q, of sufficiently small measure and (2) for operators Lu - hu with sufficiently large h >/)lo. However, in the present case, the smallness of m e s '2, and the size of A,, are determined not only by I.( in (4.5) and (4.33) but also b y d and M,. in (4.35) and (4.37). 5. SOLVABILITY OF THE FIRST BOUNDARY-VALUE PROBLEM IN W:(Q) Inequalities (4.29) and (4.41) enable us to a s s e r t the following uniqueness theorem for the problem (4.1), (4.9): Theorem 5.1. The problem (4.1), (4.9) has no move than one generalized solution in W: (9)i f conditions (4,2), (4.3), and (4.26) aye satisfied. For n > 3, these conditions can be replaced by the conditions (4.2), (4.33), and (4.40). Proof: Inequalities (4.29) and (4.41), the right sides of which are zero (since, the inhomogeneous t e r m and the boundary condition are homogeneous), are valid for the difference u = uf -u" of two possible generalized solutions in W i ( Q ) of this problem. Therefore, u ( x ) is equal to 0.

1 50

LINEAR EQUATIONS

It follows from this theorem, in particular, that, for an arbitrary differential operator L that is defined in s1 and that satisfies conditions (4.2) and (4.3), the theorem on the uniqueness of the Dirichlet problem is valid in any sufficiently small (as regards measure) subregion 9,of the region 9,withmes 9,determined only by the constants q, v , and I.* When na3,this is also true for arbitrary L , the coefficients of which satisfy conditions (4.2) and (4.33), but, in this case, mes 8,depends on the choice of L. In the regions Q, c Q of arbitrary size, the uniqueness theorem for the Dirichlet problem is valid for the operators L -hE with h > h,, if t satisfies conditions (4.2) and (4.3) or (4.2) and (4.33) and if ho is sufficiently great. In the first case, h, is determined only by q, v, and p . In the second case, it is determined by the number v in (4.2) and the numbers E' and Me, in (4.37) and (4.40), more precisely, by inequalities (4.37) and the inequality 1-B,

7 -

* Mt,-rninu-(x)-h,, u [Y 0,

1

2 -

cirnes" Q,>8,>0.

Let us now investigate the solvability of the Dirichlet problem for (4.1). We shall do this for the entire set of operators L - E , where )i is an arbitrary complex number, with no restrictions on the smallness of the region 9. Let us suppose that conditions (4.2), (4.3), and (4.5) a r e satisfied for (4.1). Without loss of generality, we may assume that the uniqueness theorem, or, more precisely, condition (4.32), is satisfied for the operator L itself (that is, with A = 0). Then, for a possible solution u of the problem (4.1), (4.9) [throughout this section, we a r e speaking only of generalized solutions in W:(Q)],inequality (4.31) is valid, and, for an arbitrary function 7 ( x ) in

i2(Q),we have

*This is true because

where

I a,, I

mes

--2

'

Q II a II L q , 2 ( Q y

a,= mes-Q

a (x) d x Q

and therefore condition (4.26) i s satisfied for those P the measures of which satisfy the inequality

SOLVABILITY

OF THE FIRST BOUNDARY-VALUE

PROBLEM

151

which follows from (4.21) and (4.32). Let us prove Theorem 5.2. When conditions (4,2), ( 4 3 , and (4.32) a r e satisfied, the problem (4.1), (4.9) has a generalized solution in Wi(S1) f o r arbitrary f(x) in L*; ( Q ) , f(x) i n L,(sZ), and g ( x ) in W:(&).

-

n+2

Proof: We define a new scalar product

in the space $$(sZ). On the basis of our assumptions regarding all and a , the norm 1. I corresponding to this scalar product is equivalent to the norm in the space

fii(L-2).

This is ture because, on the

basis of (4.2) and (4.32), if v c $ $ ( Q ) ,

where

On the other hand, it follows from (4.2) and (4.3) that

from which, on the basis of (4.19), we have

Instead of finding the solution u of the problem (4.1), (4.9), it will be sufficient t o find the function v = u--(g from conditions (4.12) and (4.13). The identity (4.12) can be written in the form

152

LINEAR EQUATIONS

where the expression l ( 7 ) i s defined by Eq. (4.14). Let u s show that the integral

defines, for fixed a in ii(Q), a linear functional on the space $;(Q). That I , ( v , q) is linear inq is obvious; the boundedness of this integral is obvious from inequalities (4.18), which lead to the inequality

and hence, on the basis of (4.19), t o the inequality 111

?)I S~c,(q)ll~ll w;(e)IIT)IIW ; ( Q )

(5.6)

+

with constant c2 (q)= pc (q) I2 c (q)]. According to a theorem of Riesz on linear functionals (cf. [2], p. 396), the functional I,(v, q) can be uniquely represented in the form of the scalar product I , (a.7 )= [Av, ql

(5.7)

Equation (5.7) defines an operator A on an arbitrary element v

G;(Q).

This operator is bounded in and (5.61,

from which we get

Of

i:(Q) since, by virtue of (5.2)

S O L V A B I L I T Y O F THE F I R S T BOUNDARY-VALUE

153

PROBLEM

Let us show that the operator A is completely continuous in i : ( Q ) . Let [ V , ( X ) ) , for m = 1, 2, denote a sequence of elements in the space $i(Q) with uniformly bounded norms

...,

m=l.

2,

... .

...,

Then, the norms of the elements Av,, for m = 1, 2, in @;(Q) a r e also uniformly bounded. Since the operator for the injection of 2n is completely continuous the space Gi(Q) into L,(Q), for p < n--2, (cf. Theorem 2.1, Chapter 2), there are subsequencesof (v,] and [Av,] that converge strongly in the space L,(Q), where p = 24/(4 -2) (we recall that q > n everywhere). Without loss of generality, we may assume that the sequences (v,] and (Av,] themselves converge strongly in L (Q). From the equation

q-2

[ Avi

- Av,,

Avi

-Av,] =1, ( ~ i v,,

Av,

- AV,).

the uniform boundedness of {v,) and ( A v , ) (for m

(5.9)

..

= 1, 2, .) in the norms of W ; ( Q ) and L2q (Q), and from inequality (5.5) a s applied q-2 to the right side of (5.9), it is clear that

Consequently,

(Av,]

converges strongly in the space

i:(Q).

This

proves the complete continuity of A in @:(Q). Under our assumptions on 'p. f , a ndf , the right side of (5.4) is, as was shown in the preceding section [cf. inequality (4.15)] , a known linear functional in $I:@). Therefore, we can represent it as the scalar product of a well-defined element F in W : ( Q ) and r):

L(s)= [ F , sl.

(5.10)

By virtue of (5.7) and (5.10), the identity (5.4) can be written in the form [v+ Av,

4 = [ F , q].

Since this identity must be satisfied for arbitrary 7 in

(5.11)

$/1(Q),

it

154

LINEAR EQUATIONS

follows that (5.11) is equivalent to the operator equation in W i ( Q ) : (5.12)

Av= F.

V+

Since A is a linear completely continuous operator in W i ( Q ) , Fredholm's theorems a r e valid for (5.12). The first of these asserts that Eq. (5.12) has a solution v for arbitrary F in homogeneous equation

$i(Q) if the

(5.13)

w +Aw =O

has only the trivial solution w r O . But any solution of Eq. (5.13) in @:(Q) is nothing other than a generalized solution in W : ( Q ) of the problem (4.1), (4.9) with p = f r f = O since (5.13) is equivalent to the integral identity [w+ Aw, 71 = 0.

and this is simply the identity L(w. ?)=Or

which defines a generalized solution of Eq. (4.1). On the basis of Theorem 5.1, we have w=O, so that Eq. (5.12) does indeed have a solution v for arbitrary F . This solution i s the desired function v since (5.12) i s equivalent to (5.11) and (5.11) is simply the identity (4.12). It determines the solution u of the problem (4.1), (4.9): u ( x ) =v ( x ) p (x). This completes the proof of Theorem 5.2. Now consider the problem

+

Lu-ku=-+

d f1 axi

f.

UIS"'p

(5.14)

for arbitrary complex h. The assumptions on the known quantities in (5.14) remain a s before. In the present section, we shall assume that the Hilbert spaces Lz(Q), W:(Q), and $:(Q) a r e complex-valued spaces and we shall keep the old notations for the scalar products and norms in them. Thus (u. v )

J u ( x ) v ( x )d x .

D

and [ u , v ] denotes

SOLVABILITY OF THE F I R S T BOUNDARY-VALUE

PROBLEM

1.55

In accordance with the definition given in Section 4, we shall refer t o a function u in W&2) that satisfies the identity L(u. i)+h(u.

q)=(f,.

rlx,)-(f.

(5.15)

?)

for arbitrary q in $':(S) with the property that u - Y E i : ( Q ) as a generalized solution of the problem (5.14) in W i ( Q ) . The expression L ( u , 7 ) is the same a s in (4.6). For the function v = u -Q, the identity (5.15) generates the identity (5.16)

L(v. i)+h(v3 7)==4(7)*

where

Just as above, the identity (5.16) can be replaced by an equivalent operator equation in the space c:(S): v

+ AV +hBv

(5.18)

= FA.

where F , is the element in W ; ( Q ) defined by the identity

4 (7) = IF,,

(5.19)

71,

and B is the completely continuous operator in $'A@) identity

defined by the (5.20)

[Bv. T ] = ( v , 7 ) .

It is easy t o see that B is symmetric and positive-definite and hence has an inverse defined on the range R ( B ) of its values. That B is completely continuous is proven in the same way as the complete continuity of A was proven above. A s is shown in Theorem 5.2, the operator E+ A has a bounded inverse in Eq. (5.18) is equivalent to the equation v+~(E+A)-'Bv=(E+A)-~F,.

i;(a).

Therefore, (5.21)

156

LINEAR EQUATIONS

The operator (E+ A)-'& being the product of a bounded operator and a completely continuous one, is completely continuous. Therefore, all of Fredholm's theorems a r e applicable for Eq. (5.21). The homogeneous equation v+A (E

+ A)-'Bv

(5.22)

=0 ,

corresponding to (5.21) is equivalent to the identity

It is natural to refer to the function v in $;(Q) that satisfies (5.23) for arbitrary q in $;(Q) a s a generalized eigenfunction in W:(Q)of the problem Lv =hv,

(5.24)

v l s =0 ,

and to A a s the eigenvalue corresponding to it. According to the second of Fredholm's theorems, all nontrivial (that is, not identically zero) linearly independent solutions of Eqs. (5.22) can be numbered in increasing order of the absolute for k = 1, 2 , , corresponding values of the eigenvalues A=A,, to them. It is easy to show that all the h, a r e located in the interior of some parabola of the form ReA=-cc( ImA(C1,where c and c, a r e determined by the constants n, q, and v in (4.2) and (4.3) with c > 0 and 0 < c, < 1. Let us state these facts in the form of a theorem. Theorem 5.3. Suppose that the conditions of Theorem 5.2 are satisfied. Then, the problem (5.14) has a unique generalized solution in W:(sZ) f o r arbitrary 'Q E Wi(Q),f E L, (Q) and f E L2; (8)

. .,

c

n+l

throughout the complex plane except at a countable set of values A=A, ( f o r k = 1 , 2, ...) such that ~ , I + C O U S k + w . To every A = 1, there corresponds a finite number of linearly independent ) what amounts to the same thing, solutions of Eq, (5.24) in ~ ; ( Q or, of identity (5.23). These exceptional values h,, for k = 1, 2, constitute the spectrum of the problem (5.14). ForA=A,, where k = 1, 2, the problem (5.14) has a solution i f and only i f

...,

...,

lhR( w ~ ) - O ,

where the

w;

I =1,

. . ., N , ,

(5.25)

are all solutions in $;(Q) of the equation (E+ A') w + ~ , B W = 0,

(5.26)

SOLVABILITY

OF T H E F I R S T BOUNDARY-VALUE

PROBLEM

157

that is adjoint t o Eq, (5.22). The number of conditions (5.25) coincides with the number of linearly independent solutions of Eq. (5.22)

for A = A,. We have already proven all the assertions in this theorem except that the conditions for solvability of the problem (5.14) for )i =A, are of the form (5.25). It follows from the third of Fredholm’s theorems that these conditions [for (5.21)l are formulated a s

follows:

[ ( E f A)-’Fi,. z:] = 0,

where the

1 = 1,

. .., N,.

(5.27)

z i are solutions of the equation z +IkB [ ( E

+A)-’].z

= 0,

(5.28)

which i s adjoint to (5.22). Since the operator B has an inverse defined on R ( B ) and since all the k k a r e nonzero, we can, by virtue of (5.19), transform (5.27) to the form

But this requirement coincides with (5.25) if we remember that, on the basis of (5.28), the functions w,6=B-’z: satisfy (5.26). A s one can easily see, Eq. (5.26) is equivalent to the identity pijW.r,Txi Q

+aiwxi7--biWT.ri - awy+xkw<) d x

TE

= 0.

t:(a

which formally corresponds to the equation d L’w -Akw = --(o,,wx -biw) - aiwxi d.Yl

1

+ ow - -Akw = 0.

This completes the proof of Theorem 5.3. It answers the basic question posed at the beginning of the chapter regarding the solvability of the first boundary-value problem for second-order linear elliptic equations. We note that the dependence of f A ( T )on A is caused by the nonhomogeneity of the boundary condition in (5.14). If y r 0 , then l A ( q )= I (T) in (4.15). If, in addition, f zs 0, condition (5.25) denotes ordinary orthogonality o f f to all the w;:

LA, (WL) = I (WL) =-(f. w ; ) = 0.

158

LINEAR EQUATIONS

Theorem 5.4. Suppose that the conditions of Theorem 5.2 are of the form satisfied f o r all the operators L,, for rn = 1, 2, (4.1) by the same constants. Suppose that the sequence {a$ ( x ) } are uniformly bounded and converges almost everywhere to a,] and that the sequences of the functions a?. b y , a", f". f'", and ye"'converge to a,, b,, a , f , f , and 'p in the norms of the spaces L, ( Q ) , L, (Q), Lqn(S), L2(51), L 2; (Q) and Wi(C2) respectively. Then, the sequence of gen__

...,

-

n+2

eralized solutions uNLin W: (Q) of the problems (5.29)

converges strowly in W: ( 8 )to a generalized solution in W: (52) of the limit problem (4.11, (4.9).

The proof is extremely simple and the reader himself can carry it out, using the following outline. On the basis of Theorem 5.2, each of the problems (5.29) has a unique solution u". For vm=pvrn, an identity of the type (4.12) is valid. The same i s true for v = u -'p, where u is a solution of the limiting problem. It follows from these identities that

If we set 7 = v" -v , after several elementary transformations and inequalities of the same type a s in Section 4, we arrive at the desired conclusion that 11 v" -w 11 w ; approaches 0.

Let us consider also the limiting case in which q = n >3. Suppose that the assumptions (4.2), (4.5), and (4.33) a r e satisfied for (4.1) and that, with the aid of some inconsequential simplifications, we have 'p ( x )= 0, that is, that the desired solution u ( x ) vanishes on the contour S. Let us represent the coefficients of the nonleading terms in the forms of sums: ui ( x ) = a; ( x )

+ a;

(X).

+ by ( x ) ,

bi ( x ) = b; ( x )

a ( x ) = a+' ( x )

+ a+" ( x ) .

where a;, b; and a+'' have a small norm in L, (Q), L,, (a), and L,,, (a) respectively and a;. b; and a+' a r e bounded functions, so that

S O L V A B I L I T Y OF T H E F I R S T BOUNDARY-VALUE

PROBLEM

159

Just as above, let us transform the identity (5.15) corresponding to the problem (5.14) t o anequation of the form (5.18), more precisely, to the equation u

+ A‘u $- A”u +ABu = F ,

(5.31)

assuming, without loss of generality, that a - (x) >/ 0. Here, the operator B and the element F are determined by the identities (5.20) and (5.10) respectively and the operators A’ and AN are determined by the identity (5.7) except that we need to replace the functions a,, bi and a+ in I , ( u , 7) with the functions a ; , bj, and a+’ in the case of A’ and with a;. b;, and a+’ in the case of A”. It follows from what we proved above that the operator B is symmetric, positive, definite and completely continuous, that the operator A’ is completely continuous, that A“ is bounded, and that its norm approaches 0 a s P approaches 0. Furthermore, on the basis of Theorem 5.2, the operators E+ A’+A’B have inverses for all sufficiently large A’ (where A‘ >/A;) and (quite important) that the norms of their inverse operators do not exceed some number p2 that depends only on n, v, and 51 and is independent of M,.. This last fact can be seen from inequality (4.39) and the facts that the role of a- in that inequality is now played by aA’ and that 6, can be taken equal to 0 (since, for A’, the functions c; and C” a r e equal to 0). The number Ah, on the other hand, is determined not only by v but also by M,”. Thus,

+

Let us apply the operator (E+ A’+h’B)-’ to both sides of Eq. (5.31) and let u s write the result in the following form: u+(E+

A‘+A”)-’A‘’u$-(h--’)(E+ =( E

A’+A’B)-’Bu

+ A‘ +A’B)-’F.

=

(5.32)

We choose E” sufficiently small so that the norm of the operator A” will be less than l / p 2 . Then, the operator E+ (E+ A’ k’B)-’A’’ will have an inverse, and Eq. (5.32) can be transformed into an equivalent equation of the form

+

u

+ (A - A’)

[E+ ( E

= [ E + (E+ A’

+ A’ +A’B)-’A‘’]-’ + A’B)-’A’’]-’

(E+A’+h’B)-’Bu

(E+ A’

+A‘B)-’F.

=

(5.33)

Tn this equation, the operator multiplied by ( h - h’) is completely continuous (since B is completely continuous and the other operators

1 60

LINEAR EQUATIONS

a r e bounded). Therefore, all the Fredholm theorems a r e applicable to Eq. (5.33). Rewording these theorems in terms of the original Dirichlet problem, we obtain Theorem 5.5. Suppose that conditions (4,2),(4,5), and (4.33) are satisfied f o r Eq. (5.14) and suppose that cp(x)E Wi(Q). Then, all the conclusions of Theorem 5.3 are valid for the problem (5.14). 6. THE SECOND AND THIRD BOUNDARY-VALUE PROBLEMS

The solvability of the second and third boundary-value problems for the equation

in the space W k Q ) is investigated in essentially the same way a s the solvability of the first boundary-value problem was investigated in Sections 4 and 5. We only need to define what it means to solve these problems “properly” in the space W i ( Q ) . For an arbitrary function in W : ( Q ) , it makes no sense to say that it satisfies the condition

du where = a l p x jcos (n. x ~ ) ,where n is the outer normal to S, and dN

where a (s) and cp (s) a r e given functions defined on S, because its first derivatives a r e defined only almost everywhere on an ndimensional region 52 and may fail to be defined (or may be equal to 00) on the entire (n - 1)-dimensional surface S. Therefore, we need to put the condition (6.2) a s well a s Eq. (6.1) in a different form, one that will be suitable for anarbitrary function W i ( Q ) . This can be done a s follows. We shall say that afunction u ( x ) in W i (8) that satisfies the identity

THE SECOND AND THIRD BOUNDARY-VALUE

PROBLEMS

161

zf all the functions in (6.3) were sufficiently smooth, it would not be difficult to verify that when we integrate the first integral in (6.3) by parts, we a r r i v e at the identity

from which, by virtue of the sufficient arbitrariness of 7 , Eqs. (6.1) and (6.2) follow. Thus, the definition that we have given for a generalized solution is indeed an extension of the classical concept of a solution of the problem (6.1), (6.2). Solvability of the problem (6.1), (6.2) in the class k'/:(n) is established in an extremely simple manner. Suppose that the assumptions of Theorem 5.2 a r e satisfied for the coefficients in L and the functions f i and f. Let us first prove the first fundamental inequality. If we set 7 = " in (6.3), we can derive inequality (4.23) just. as was done in Section 4. Then, by using inequalities (2.19) of Chapter 2, we get

where E, is a small positive number and c r , ( a , , b,, a + ) i s a known positive constant that approaches rn a s --t 0. Tf the functions a, (p, and a, were identically equal to 0, the integral over S would vanish and (6.5) would become an inequality of the type (4.24). Here, we do not need to impose any smoothness conditions on S. In the general case, the conditions on a, '9, the a , , and S reduce to the requirement that, for an arbitrary function u in w ~ ( Q ) [-

and

a

+ a, cos (n,

4

xi)] U ? ds I

[E

e

I Fu l2

+

ct(a,

a,) 4 d x

(6.6)

162

LINEAR EQUATIONS

with arbitrarily small

E and finite c,(a, a,) and c(cp) [where the constant c, (a, a,) can increase arbitrarily a s E -+ 01. If the conditions on a, 'p, and the ai a r e formulated in terms of their membership in the spaces L,(S), then, a s the injection theorems show (cf. Section 2, Chapter 2 ) , we need to assume for this that the function a =- a al cos (n, xi)belongs to the space Lp(S) with p > n 1 and '9 EL, (S), where r >/ 2(n - l ) / n when n > 2 and r > 1 when n = 2. We need to assume that the boundary S is piecewise-smooth. When these conditions a r e satisfied, we obtain from the relations (6.5)-(6.7) after a number of simple bound-finding operations analogous to those in Section 4

+

-

This inequality is of the same nature a s inequality (4.24). 'It is used to prove that, for the problem (6.1), (6.2), the three Fredholm theorems hold for the space W : ( Q ) and, in particulr, that, for all sufficiently large )., the boundary-value problem (6.2) for the equations Lu-).u=-+f

dfl

an-,

has a unique solution for arbitrary f, f,, 'p, and a with the properties indicated above. We shall not give proofs of these propositions here since they a r e quite analogous to those given in Section 5 for the first boundary-value problems.

7. INTERIOR ESTIMATES IN L2 OF THE SECOND DERIVATIVES OF A N ARBITRARY FUNCTION IN TERMS OF THE VALUES OF A N ELLIPTIC OPERATOR APPLIED TO IT 'In this and the following section, we shall prove the second fundamental inequality for elliptic operators. 'It enables us to find a bound for the norm in L, ofthe second derivatives of an arbitrary function u in terms of the norms in L, of the function 11 itself and of the values of the elliptic operator applied to it and the norms of the boundary values of u . This inequality was established independently in [55, 10, and 791. The derivation that we give below,

163

INTERIOR ESTIMATES

like all our further investigations on the solvability of boundaryvalue problems in the space Wi(Q),a r e taken from the articles [lo, 11, 121 of one of the present authors. We begin by finding interior bounds that are independent of the values of u on the boundary S. This time, the functions in question belong to the spaces Wi, and the operator L [cf. (4.1)] on them must be calculated directly and must yield functions in L,. To ensure this, in addition to the restrictions (4.2), (4.3), we also require that the generalized derivatives dail/axk and dai/dx, exist , that the dai,/axRbelong to Lq(L2), and that the da,/dx, + a belong to L T n ( 8 ) , where = max (4. 4 ) and q > n.* These conditions ensure that every term in the expression

Lu where a, =%+a, ox,

( x )uXiK/

+ a, ( x ) + i(x)u,

(7.1)

UKi

aai + b, and a = -+ dx,

a

yields, for arbitrary u in

so that L is a bounded operator from W:(Q),a function in b(Q), Wi(Q)into Lz(S) and

II L41'2(Q)scII"Ilw;(Q)

(7.2)

with coeificient c depending only on the constant /.iin (4.2) and the nOrmS II a, II L q ( Y ) and II ^a II L?, (Q). Thus, we write the operator L in the form (7.1) and assume, in addition to conditions (4.2) and (4.3), that

l.1, k = l ,

..., n,

.

q > n , q = m a x ( q ; 4).

Let us first consider the operator L a s applied to functions in that is, to functions Wi(Q)that a r e of compact support in 9. Let us prove the validity of the following proposition: lemma 7.1. If conditions (4,2), (4,3), and (7.3) are satisfied f o r the coefficients of the operator L , then, for an arbitrary function

W:(Q),

u

EWQ),

with constant c depending only on the size of q, ditions (4,2), (4,3), and (7.3) and not on u or&. 'Regarding the possibility of replacing q by n for n situation i s the same here.

v,

and

I-(

in con-

>3, cf. end of Section 4.

The

164

LINEAR EQUATIONS

Proof: The set C, (Q) of all infinitely differentiab!e functions that a r e of compact support in Q is dense in the set W i ( Q ) [in the sense of convergence in W i ( Q ) ] . This is true since the usual averages with indefinitly differentiable kernel and sufficiently small radii of averaging yield an approximation from below for a function in W i (a). Therefore, it will be sufficient to prove inequality (7.4) for functions u inCm(51). For arbitrary u E W i ( Q ) ,this inequality

i s obtained by closure in the norm W i ( Q ) . Here, we need only remember that the operator L, like the operator from W:6?) into L2(Q),is bounded. Thus, suppose that

u f Cm (9). Let

u s consider the integral

+(,^,,,*, + a^u)2]fix. Let u s transform the first term on the right by twice integrating by parts a s follows:

J' (Lu)2 d x = f [SE

3

UX,

d

(a,j a k / " x k x * )

+ . . .] d x =

Let us show that (7.6)

To do this, we choose an arbitrary point xOEQ and set up new rectangular coordinates in a neighborhood of it: yk = a k , ( x l - x;). We choose an orthogonal matrix (an*! in such a way that it will reduce a quadratic form a,,(xO)ErEj to diagonal form, that is, in such a way that

where k1( x ) .

. . . , I,, ( x )

a r e the eigenvalues of the form ail ( x ) ti€,.

165

INTERIOR ESTIMATES

Then, a s one can easily see, n

I, ( x o )=s,2 As ( x o )ht (x") U: s t (xo). t=1 On the basis of the ellipticity condition (4.2) or, what amounts t o the same thing, on the basis of the assumption hi(X)>V,

[=I,

..., n,

we obtain

But

so that inequality (7.6) is established. Therefore, from (7.5), it follows that

The remainder of the proof consists in showing that, for arbitrary E > 0, (7.8)

where the constant c, depends only on E (here, c, 00 as E. --f 0) and the constants q , v , and 1 in conditions (4.3) and (7.3). To prove inequality (7.8), we shall use H61der's inequality, inequality (4.19), and inequality (2.22) of Chapter 2. With the aid of these inequalities, we obtain a bound for the integral I, by the standard procedure. For example, let u s find bounds for certain of the t e r m s in the expression for 12. --f

166

LINEAR EQUATIONS

Let us define

(Here, we are not using a summation convention.) It is easy to see that

To find a bound for the third factor in the second term on the right side of the last inequality, let us use inequality (4.19). Then, we obtain

The bound obtained for the integral I, is an inequality of the type (7.8) since E and a r e arbitrary positive numbers. A s an example, let us consider the expression

for arbitrary values of the indices 1 and j . Obviously,

Let us suppose first that n = 2, 3. Then, by using inequality (2.22) of Chapter 2, we obtain

INTERIOR E S T I M A T E S

167

Suppose now that n >4. Again using inequality (2.22) of Chapter 2, we have

Bounds a r e found 'for the remaining terms in I2 in an analogous manner. Here, in finding a bound for the integral

we need to consider the cases n = 2, 3 and n > 4 separately, just a s we did above. Thus, inequality (7.8) is proven. It follows from (7.7) and (7.8) that

Then, if we take, for example,

E

= +/2, we obtain

This inequality together with inequality (4.24) gives u s the second fundamental inequality (7.4). Let u s draw some conclusions from inequality (7.4). First of all, let us show that the operator L defined above a s the differential operator (7.1) on the set D ( L ) = W ; @ ) has a closure in L2(Q) (cf. [2]). Let { u k ) r Z 1denote a sequence of elements in W i ( Q )that

168

LINEAR EQUATIONS

converges in L, ( 8 )to 0 with the property that [ Lu,} converges in L, (Q) to some element f. We need to show that thenf=O. To do this, note that, from inequality (7.4) a s applied to the function u , -ul, for k , 1 =1,2,...,it follows that the sequence ( u k ]itself converges in the norm of Wi(S1) and, consequently, converges to its limit in the norm of W ; ( 8 ) . But the limit of ( u k ]in L2(&) is u=O. Consequently, ( u k ) converges to 0 in the norm of W i (52) and hence, a s one can easily see, ( L u k )converges to 0; that is, f0. Thus, the operator L admits closure. Furthermore, from what we have just shown, it follows that D ( 0 , that is, the domain of definition of the closure L, coincides with $i(Q), that is, with the closure of the set W ; ( O ) or, what amounts to the same thing, with the closure of C,(U) in the norm of Wi((.?). It is worth noting that on an arbitrary smooth portion of the boundary of the region 9, functions in 1% (8)vanish in mean together with their first derivatives (cf. Section 2, Chapter 2). A s we know, the possibility of closure of the operator L is equivalent to having its adjoint operator L* defined on a dense set in L,(Q). The adjoining operator to L is easily defined in explicit form if', in addition to our other assumptions, we assume that the daij daij that coefficients a,] have derivatives - and dxl,d*ai' that -CL,(P), dx1

dX/

d2aU

d 3 d

EL-(Q), 9

that the

a,

and b, have derivatives 5%

dxi

2

db

and& in

L; (Q), and that the coefficient a belongs to LG (8). Specifically, on

-2

2

'u ( x )

E W i ( Q ) c D (L*), it is defined by

In fact, the conjugate operator L* and its domain of definition D ( L * ) a r e characterized by the fact that, for an arbitrary element u

in D ( L ) = W z ( Q ) , the identity (Lu,u ) = ( u ,

L*V)

(7.11)

holds. But this identity is valid for arbitraryuEWi(Q)if L* is defined by Eq. (7.10) and if v E W:(Q). From all this, it is easy to show that, under certain regularity assumptions regarding S,D ( L ) coincides with W i (Q).

THE SECOND FUNDAMENTAL INEQUALITY

1 69

8. THE SECOND FUNDAMENTAL INEQUALITY FOR ELLIPTIC OPERATORS One of the principal questions associated with an elliptic operator L is that of the solvability of various boundary-value problems for it. These problems consist in finding solutions u ( x ) of the equation

+

Lu = ( x ) (8.1) in a region 9 that satisfy some specified boundary condition on the boundary S of the region 2. The following boundary conditions are considered the basic ones: (1) the first boundary condition

(2) the second boundary condition (8.3) (3) the third boundary condition

where Q (s) and a (s) a r e functions defined on S. From the point of view of the methods that we a r e using, it is convenient to reduce nonhomogeneous boundary conditions (8.2)(8.4) to homogeneous ones. To do this, we replace the unknown function u with a new unknown function z, by setting fJ

(4=2, ( x )

+u1

(x).

where a,(%) belongs to the domain of definition of the differential operator L and satisfies one of the conditions (8.2)-(8.4). Then, v must satisfy the equation Lv =J, -Lu,,

that is, an equation of the same form a s Eq. (8.1) and the corresponding homogeneous boundary condition. Thus, we assume that u satisfies Eq. (8.1) and one of the homogeneous boundary conditions, for example, the condition uJs=0.

(8.2 ')

1 70

LINEAR EQUATl O N S

Trom the point of view of functional analysis, the problem (8.1), (8.2 ) consists in finding the operator L-' inverse to the operator L , yhere L is defined on the set of functions satisfying condition (8.2 ). In the preceding section, we defined the operator L on the set W i ( Q ) . Since functions in W l ( Q ) a r e equal to 0 on the boyndary S, any one of them is a solution of the problem (8.1), (8.2 ) with corresponding right-hand member (namely, q~ = Lu). However, such functions $ [where belongs to R ( L ) , that is, to the range of values of the operator L or even to R ( L ) ] do not fill the entire space L,(M). For example, if we take any twice continuously differentiable 1)

function v ( x ) that vanishes on S such that problem of finding

11

from the conditions Lu=Lv.

uI,=O,

-dn

aw Is

f 0,

and pose the (8.5)

then one of the solutions of this problem is v. Tf we assume that condition (4.26) or condition (4.32) i s satisfied for the operator L. then the function I( = v will be the only possible solution to the

(& L8 (Q) and therefore, problem (8.5). But, obviously, v 4 .!I= Lv @ R Thus, R (L) does not coincide with L, (Q). Thus, if we wish to solve the problem (8.1), (8.2') for all qI E L, ( Q ) or even for all members of a set that is dense in L, (a),we need to extend the operator L. A s we know, even for the simplest elliptic operator, namely, the Laplacian operator, the operator

(z).

on $:(a) has infinite defective and hence it admits an infinite set of distinct extensions. The boundary-value problems for the operator L generate various extensions of the operator L from its original domain of definition D (L)= W i (8)and, in a definite sense, vice versa. Here, we shall consider only one side of this question; specifically, we shall study those extensions of the operator L from W i ( Q ) that a r e associated with the fundamental boundary conditions (8.2)-(8.4). It i s desirable to extend the operator L in such a way that the range of values of the extended operator L^ (or L+AE for some h ) will fill the entire space L,(Q), the functions in the domain of definitionof the operator L^ will satisfy in a certain sense the boundary condition of the boundary-value problem in question, and the operator (or L^+AE) will have a bounded inverse on L,(M). Clearly, for any one of the three fundamental boundary-value problems (8.2)-(8.4), we need to place in the domain of definition D ( i ) of the operator

E,

first of all, all sufficiently smooth functions

THE SECOND FUNDAMENTAL INEQUALITY

171

that satisfy the corresponding homogeneous boundary condition since any such function is a solution of the boundary-value problem in question. For definiteness, let us take the first boundary-value problem. We denote by Wz,o(Q),the closure in the norm of Wi(51) of the set of all functions in C2@) that vanish on the boundary S. Suppose that the region 51 is such that W i , o ( Q ) c ~ ~ ( 5 This 1 ) . will be the case for the classes of regions that we a r e considering, namely, regions with piecewise-smooth boundaries (cf. Section 1, Chapter 1 and Section 2 of Chapter 2). We denote by L^ the extension of the operator L obtained by extending D ( L ) to the set W;,,(Q). A s before, the operator

L^ defined on

W i , , ( O ) is of the form (7.1).

We shall later show that the operator 2 is closed with respect to the set W:,o(Q). Here, we shall show that the operator L^ admits closure and that all the functions in the domain of definition D of the operator ?, have generalized second derivatives that a r e square-summable over an arbitrary strictly interior subregion 52' of the region 8. Both these assertions a r e easily derived from the inequality

(z)

which is valid for arbitrary u in W;(Q). Here, C(x) is any positive indefinitely differentiable function of compact support on the region 51. The constant c(C) in inequality (8.6) depends only on the constant of ellipticity v, the numbers q and p in conditions (4.2), (4.3) and (7.3), and max(K(. Inequality (3.6) is proven in the same way a s inequality (7.4). We only need to consider the expression

instead of

and to transform the highest-order terms a s follows:

172

LI N EAR

--ax, d

( @ l / a k h 2 ) lExkxluxl

E QUATI ONS

$-

a

1d x .

(aijakic2)~ E Xx u x 1 I 1,

On the basis of what was said in Section 7 [cf. (7.6)], the principal positive term

has a lower bound given by

A l l the other terms a r e bounded from above. Most of them differ from the corresponding terms in the expression for

only by the factor C2. Exceptions a r e the terms containing the derivatives of C, namely, the terms

(these disappear for C r 1). We must give a bound for them as follows:

where E is an arbitrary positive number and the constant c depends only on maxlal,l and maxIVC(. lf we go through the remaining reasoning and find the necessary inequalities just exactly a s we did above in deriving the

173

THE S E C O N D F U N D A M E N T A L INEQUALITY

fundamental inequality (7.4), we obtain the inequality

where Q’ is the carrier of C. It is then easy to find a bound for the integral

in t e r m s of

J [ ( L U ) * +u2] d x . P

To do this, consider the integral

J’ Lu . uq2 d x , 8

where ~ ( xis) a smooth function that is equal to unity in to 0 on S , and transform it to the form

L2’

and equal

from which it follows that

The right-hand member of this inequality does not exceed the quantity J[(L1l)2+51VUI~7~+CI(T)U21dX, 8

where c,(ri) is a known constant that depends on

E,

the quantities

174

LINEAR EQUATIONS

max lVql, M , and q in (4.2), (4.3), and (7.3) and where E i s an arbitr a r y positive number. This is shown by means of the same line of reasoning and the same types of bound a s in the first energy inequality (4.24). We only need to apply inequalities (4.19) and (4.16) not to the function u but to the function uq. This leads us to the inequality

which, together with (8.8), yields (8.6). It follows from inequality (8.6) that the operator admits closure (this is proven just a s the assertion that the operator L admits closure was proven in Section 7) and that the functions in D ( z ) will have generalized second derivatives that a r e squaresummable over an arbitrary strictly interior subregion 8' of the region 8 (because, for an arbitrary subregion 0' of this type, we can choose C ( x ) so that C ( x )= 1 when x 6 W). These assertions

a r e valid for all three boundary-value problems (8.2)-(8.4) since inequality (8.6) is valid for an arbitrary function u in Wi(2). $et us turn again to the first boundary-value problem (8.1), (8.2 ) and let us show that, if the boundary S of the region 0 is sufficiently smooth, then D

(z) = W i , (8);that is, the operator

is

closed with respect to W;,,(Q). To do this, we shall extend the second fundamental inequality for elliptic opeators to arbitrary functions in w;. (Q). Let us suppose that the region 52 and its boundary S possess the following properties: (1) a s always, the boundary S is a piecewise-smooth surface with nonzero interior angles (cf. Section 1, Chapter 1); (2) for almost all (in the sense of the measure on S) pointsx" in the surface S , there exists a tangent plane to S and the equation for a portion of the surface S in a neighborhood of the point xu in a local Cartesian coordinate system (with the y,-axis directed along the outer normal to S that passes through xo and the yl-, Y,+~axes lying in the plane tangent to S at the point x o ) is of the form

...,

y, = (Yl.

*

-

9

Y,,-Ib

Here, the function w is twice differentiable and the eigenvalues p, ( x O ) .. . ., p,,-, ( x o ) of the quadratic form

175

THE SECOND F U N D A M E N T A L I N E Q U A L I T Y

at the point xo a r e bounded from above by a nonnegative constant, which we denote by k: (8.10) We shall say that surfaces S possessing these properties are piecewise-smooth surfaces with curvature bounded from below by the number K. For example, for an n-dimensional sphere of radius R, we have pk =- 1 / R for k = 1 , ., n 1. Therefore, for K we may immediately take 0 in the case of all spheres. We note that the principal curvatures of a sphere of radius R a r e equal to 1 / R and thus a r e indeed bounded below by 0. As a second example of such surfaces, we may take the surface of a nondegenerate ndimensional polyhedron (not necessarily convex) or the surface obtained from the surface of such a polyhedron by a topologically twice differentiable transformation with bounded second derivatives and positive jacobian. We shall show now that, if the boundary S of the region Q is a piecewise-smooth surface with curvature bounded below by K , then inequality (7.4) is valid for all functions in C,(8) that vanish on the surface S. lemma 8.1. Suppose that S i s a piecewise-smooth surface with curvature bounded b e l m by a number K. Then, f o r an arbitrary that twice continuously differentiable function u defined on vanishes on S ,

.. -

(8.11)

where the constant c depends only on the constant of ellipticity v of the operator L, on the numbers q and p in conditions (4.2), (4.3), and (7.3)*, and on the surface S (it i s independent both of the function u and the size of the region8). The proof of this lemma is in two parts. The first part coincides with the proof of Lemma 7.1 and reduces to obtaining inequalities (7.6) and (7.8). Specifically, we transform the integral S ( L u ) ?d x D

just a s we did in the proof of Lemma 7.1. Since the function

u

is

*With regard to the possibility of replacing q by n for n > 3, cf. end of Section 4. The situation is the same here.

176

LINEAR EQUATIONS

not of compact support in the region 51 this time, in making the two integrations by parts, we handle the boundary integrals and, therefore, in the right-hand member of Eq. (7.5), we have the following integr a1: /I, d s E S

/

U i j akl[UxkxlUxiCOS(fl, %j)--xlxlUxiCOS(fl.

XR)]~S.

S

Repeating the procedure indicated in the proof of Lemma (7.1), we obtain [cf. inequalities (7.6) and (7.8)]

where E is an arbitrary positive number and c, is a known constant depending on E. We note that Eq. (7.5) with the correction of the boundary integral

and hence inequality (8.12), is valid for anarbitrary function u E C,(Q) although third derivatives of u appeared in an intermediary stage of the derivation. This is true because, by virtue of the piecewise smoothness of the boundary S, an arbitrary function in C2@) can be approximated in the norm of C,@) by functions in the class C,(a) (cf. Section 2, Chapter 2). Therefore, the relationships in which we are interested a r e valid for arbitrary u E C,(Q). Let us now consider Is. Let x3 denote an arbitrary point on the surface S a t which the derivativesdh/dyidyj, for i, j = 1, n 1, exist. Let us take an orthogonal matrix (Ckl) and let us use it to shift from the coordinates ( x l , . . ., x,,) to a local coordinate system

...,

(Yl.

*..)

-

YIJ:

y k = ckl ( x l - xy).

k = 1.

. . ., n ,

(8.13)

where the direction of y,, coincides with the direction of the outer normal at the point x(’. By virtue of the orthogonality of the matrix ( c k l ) , we have x1 - x;=

Ck1Jlk.

1 = 1,

. . .,

n.

(8.14)

THE SECOND FUNDAMENTAL INEQUALITY

177

It follows from (8.14) that cos(n, x l ) = c n 1 ,

Thus, at the point

xo,

1=

1,

. . . , n.

we have

where bpq== a k i C p k C q l . x0

p . 4 = 1,

..

,

n.

Let us now use the boundary condition #Is= 0. Close to the point with coordinates y I = . . . = y , = 0, this condition takes the form u(y,,

...

1

y , l - ] , o(y,.

...,

y,l-J=O.

Close to no, it is identically satisfied in y,. . . ., yn-l. Let US differentiate this identity with respect to y, and y j , for i, j = 1, n 1, remembering that, at the point xlJ,

...,

-

_ do -0, dYl

1=1,

..., n-

1

This yields

at the point xo. With the aid of these relations, we can simplify the expression (8.15) for Is (nu): (8.17) For p = n and arbitrary q or for arbitrary p but q = n, the terms in the square brackets in (8.17) cancel each other out. Therefore, in view of (8.16), Eq. (8.17) takes the form

178

LINEAR EQUATIONS

..

We shall assume that the coordinates y,, . , y n - l in the tangent plane a r e chosen in such a way that all the mixed derivatives a20/aypdy,, for p , q, = 1, ., n 1 vanish at the point x o (which obviously we can always arrange by means of an orthogonal transformation of the coordinates y,, . . ., y n - ] ) . Then,

..

-

(8.19) Therefore, by virtue of property (2) of the surface S and the fact that 0 < bnnbpp - b;,, p2, the inequality

<

where M i s the constant in condition (4.2), is valid for Is(xo). From this inequality and inequality (8.12), it follows that

Since the surface S is piecewise-smooth, it follows [cf. Section 2, Chapter 2, inequality (2.25)] that

where the constant c, depends only on the surface Sand is an arbitrary positive number. When we substitute this into (8.21), we obtain

Then, by choosing E=El=+l

V2

+K(n-

l)p2c1]-',

THE SECOND FUNDAMENTAL INEQUALITY

179

we have

This, together with inequality (2.23) of Chapter 2, yields inequality (8.11). Remark: lt is easy to see that, if 8 is a convex region, then K = 0, I , > O , and the boundary integral

in inequality (8.12) can simply be discarded, and we can derive (8.11) directly from the resulting inequality. In particular, for convex Q and constant a,] that satisfy the condition

where

v

> 0, the inequality (8.22)

is valid for an arbitrary function u ( x ) c Wg.0 (55).

Thus, we have proven the second fundamental inequality for elliptic operators. This was done for an arbitrary function u in the class C,@) that vanishes on the boundary S. But, by definition, such functions are dense in W i , ,,(Q). Therefore, inequality (8.11) is valid for an arbitrary function u in the class W;,o(Q). But then, from inequality (8.11), we get the following theorem: Theorem 8.1. The differential operator i s closed with respect

to the set W i , O(Q). Thus, D ( I )= XI(2)= W:, (Q). It is natural t o expect L^ on W;, o ( Q ) to be that extension of the operator L from the set W i ( Q ) that corresponds t o the first boundary-value problem for the operator L. We a r e interested in those cases in which the second fundamental inequality for the elliptic operator L can be written in the form (8.23)

180

L I N E A R EQUATIONS

that is, without the term IIuIIL2(9)in the right-hand member. This is possible, for example, when the region 8 is sufficiently small or, more generally, when condition (4.26) is satisfied. This is true because, in such a case, inequality (4.29) is satisfied and from it we obtain the following inequality for the function u and the operator

L:

IIUII w ; ( 9 )
IIu II w; (9)4 c IIAfd I1L3(&)'

(8.24)

lf the boundary S of the region !! satisfies the conditions of Lemma 8.1, then, just a s above, we obtain from inequality (8.11) 2

II~ll;;(p) But since u E W:,

< c (IlAullr,(n,+

(8.25)

l I 4 ~2> ( 9 ) ) *

(a),we have

and, on the basis of inequality (4.25), this yields

--2

Let us take E = c02 mes n 52. Then, from inequality (8.25), we obtain inequality (8.24). From Lemma 8.1, we obtain the Corollary. Suppose that uEWi(53) and that the hypotheses of Lemma 8.1 regardirg the region 53 and the coefficients in the operator L are satisfied. Then,

It /I2w; (9)< c [ II Lu II

2:

+ II u /I2, 3 +

(9)

(9)

c1

2

II'9 II w;@)*

(8.26)

where 'Q ( x ) i s an arbitrary function f o r which II ( x ) - '4 ( x ) E W i , o ( Q ) and the constants c and c1 are determined by the same quantities as c in (8.11). Inequality (8.26) is easily derived from inequality (8.11) a s applied to the function u - '9 and from inequality (7.2) for 9.

T H E SECOND F U N D A M E N T A L I N E Q U A L I T Y

181

Zn this section and in the precedingone, we have given two types of inequalities bounding the norms in W i of an arbitrary function u ( x ) : interior bounds (with no assumptions regarding the smoothness of the boundary of S o r the boundary conditions of u ) and bounds that apply to an entire region (in the case of these, assumptions of such a type are necessary). It is also useful to have a bound for the norm of u in W i for a region Q only a portion of the boundary of which satisfies the conditions of Lemma 8.1. For this, we have lemma 8.2. Suppose that a portion S , of the boundary of a of Lemma 8.1 and that C ( x ) is a region 52 satisfies the conditions function that is smooth in 9 , that assumes values in the interval [ 0 , 11, and that vanishes outside S \ S,. Then, f o r an arbitrary ficnction u ( x ) in W: ( 9 )that vanishes on S , ,

(8.27)

Here, the constad c depends only on the quantities v, p , and q in (4.21, (4.31, and (7.3),on max IVCI, and on S , . This Lemma is proven in essentially the same way as Lemma 8.1 and inequality (8.6). Specifically, we take the integral

1C ( L U ) ~

dx

e

and transform it just as in the derivationof inequality (8.11). Since C is nonzero on S,, when we integrate by parts, we isolate the integral

J’ C*uija/zluxi[uxkxl (COS n, x j ) - u x r r j

cos (n, xk)]d ~ .

S,

We can find a bound for it just as we did for the integral

in the proof of Lemma 8.1, remembering that C = 0 on S \ S,. This yields inequality (8.27). Lemmas analogous to Lemmas 8.1 and 8.2 (and even more general inequalities for the derivatives of u of arbitrary 1 > 2 ) are

182

LINEAR EQUATIONS

valid also for other boundary conditions: For conditions (8.3) and (8.4) and for conditions with an “oblique derivative” (in this connection, see [lo-121 of one of the authors). The method explained here is suitable also for these cases. A slight modification is necessary only in the case of the bound for the boundary integral

which is transformed to approximately the same form by using different boundary conditions. Here, the boundary S must belong to c2 Remark: In investigating the convergence of the approximate solutions to boundary-value-problems that are calculated according t o Galerkin’s procedure, the following inequality has proven useful: r .

.

(8.28)

This is a generalization of inequality (8.23) to the case of two distinct elliptic operators. This inequality is valid for an arbitrary function IL in W;, ,,(Q and for any two elliptic operators if the coefficients in these operators satisfy conditions (4.2), (4.3), and (7.3) and if the coefficients a (x) and a (x) in these operators do not, for the function u , exceed negative numbers of sufficiently great absolute value. In the general case, if this last condition is not satisfied, instead of inequality (8.28), we have the inequality (8.29)

These inequalities are proven in the same way as inequalities (8.11) and (8.23). We only need to consider the integral

instead of the integral

J Lu . Lu d x ,

e

transform i t s principal t e r m s by twice integrating by p a r t s just as

ON T H E S O L V A B I L I T Y O F F I R S T BOUNDARY-VALUE

PROBLEM

183

was done above, and use the familiar proposition on the possibility of simultaneously reducing two positve-definite quadratic forms to the sum of squares (cf. 171, 721).

9 . ON THE SOLVABILITY O F THE FIRST BOUNDARY-VALUE PROBLEM IN THE SPACE W;,o(Q) The second fundamental inequality for elliptic operators L enables us to investigate in a comparatively simple manner the solvability of the Dirichlet problem in the space W i ( Q ) . Regarding the boundary of the region Q and the operator i., suppose that the hypotheses under which Lemma 8.1 was proven are satisfied. Let u s consider the following Dirichlet problem in 8: Lu=qJ(x),

nl,=0

(9.1)

f o r q~( x ) E L, (Q). We can assume without loss of generality that the boundary condition is homogeneous. Let us suppose first that the problem (9.1) has no more than one solution in @i(L?); more precisely, let u s suppose to begin with that condition (4.32) is satisfied. In this case, the second fundamental inequality can be written in the form where v is an arbitraryfunction W-i,,,(Q). Onthe other hand, we know [see (7.2)] that Inequalities (9.2) and (9.3) show that the differential operator L s e t s up a one-to-one correspondence between its domain of definition Ws, (8) (we have agreed to denote such an operator by L^) and its range R ( L ) c L&). Let u s show that, when the boundary S possesses a certain degree of regularity, the set H ( L ) coincides with the entire set L,(Q). We shall say that aregionMpossessesproperty :I1 if the problem Au=$,

ul,=O

(9.4)

has a solution in Ws,o(Q) for every set 91 of functions $ ( x ) that is dense in L,(Q). We have the following proposition: Lemma 9.1. Su#pose that the conditions of Lemma 8.1 and inequality (4.32) aye satisfied for L and s. Suppose that 9 possesses

184

L I N E A R EQUATIONS

property $1. Then, the problem (9.1) has a unique solution in W;,,(Q) f o r arbitrary J, ( x ) in L, (8). This lemma is proven in essentially the same way a s Lemma 1.1 of Chapter 3 except that, instead of the correspondence C2,=(G) C,, =@), we now take the correspondence W i , o ( c 2 ) t t L,(Q) and, instead of Schauder's inequality (l.ll),we need to use inequality (9.2). Tn addition, we need to remember that the conditions imposed on L a r e satisfied for the entire set of operators L,u= Au +T ( L -A) u that were introduced in the proof of Lemmas 1.1 and 9.1 and that the constants appearing in these conditions can be taken a s general for all 7 in [0, 11. This method of continuity along a parameter and inequalities (8.11) and (9.3) enable us to assert the validity of the following proposit ion: lemma 9.2. Suppose that the conditions of Lemma 8.1 are satisfied f o r L and S. Then, the operators L,, where 0 ,< T ,< 1, are closed on w ~ , ~ ( S ; )and their ranges L T ( W i , o ( 8 ) )coincide with each other. Just a s in Section 1 of Chapter 3, it is easy to show that condition 'Jz is satisfied for spheres, parallelepipeds, and regions that can be topologically mapped onto a sphere or parallelepiped by a function y = y ( x ) in W i ( & ) , where q > n , with nonzero jacobian. Therefore, from Lemma 9.1, we have Theorem 9.1. Szlppose that the conditions of Lemma 8.1 and inequality (4.32) are satisfied f o r L. Szlppose also that the region 8 either belongs to the class w:, where q > n, o r can be topologically mapped into a parallelepiped by the function y = y (x) in W i (Q), where q > n, with nonzero jacobian. Then, the problem (9.1) has a unique solution in W i ,,(9)f o r arbitrary q~ ( x ) in L, (8). We shall not give a proof of Theorem 9.1 (or Lemma 9.1) here since the proof is completely analogous to the proof of Theorem 1.3 given in Section 1. Instead of inequalities (1.9) and (l.ll), one needs to use inequalities (4.29) and (9.2). Let us say a few words about condition 9. For R ( L ) and L 2 ( Q ) to coincide for the entire set of elliptic operators L with wellbehaved coefficients, this condition is obviously necessary and, if the conditions of Lemma 8.1 and inequality (4.32) a r e satisfied, it is also sufficient (cf. Lemma 9.1). Condition !It is by no means satisfied for all regions. Its satisfaction or nonsatisfaction for regions with angular points, for example, depends on the size of the angles at these points. Let us illustrate this with an example. For 8, we take the circular sector ( 0 -<. r << 1 , 0 ,< 0 0,) in the x , x,-plane. The eigenfunctions of the Laplacian operator for this sector with homogeneous boundary condition have singularities at the point r = 0. For Oo
CONDITIONS

UNDER WHICH GENERALIZED SOLUTIONS

BELONG

I85

singularities are such that the eigenfunctions are elements of Wi(Q), but, for x < I), < 2x, theydonotbelongto Wj(62). Consequently, the solutions of the problem (9.4) in the sector with angle 8, E ( n , 2 ~ ) are not elements of W g , o ( 8 ) for arbitrary + in L , ( Q ) . On the other hand, inequality (8.24) does hold even for such a sector. From all this, it follows that the closure of the operator A from C,(G) leads to the operator A, which is defined on W i , o ( Q )and which maps iVS,o(Q) into the proper subspace L,(8). This operator admits a further extension with preservation of symmetry, and such an extension is necessary if we wish to solve the problem (9.4) for arbitrary ,:t in L,(Q). It is easy to show that this extension is unique and is obtained by adding to the set W i , o ( Y )all elements of the n

form crKj;f(r)sis --, where xo

00

c

is an arbitrary constant and f ( r ) is

any twice continuously differentiable function that is equal to 1 close to r = 0. On the other hand, if fJ,& r , then A (and hence any other symmetric elliptic operator L ) is self-adjoint even on the set W i , o ( Q ) . ln the following sections, we shall show that the differentiability properties of the solutions of elliptic equations are local properties, that is, that they depend on the corresponding characteristics L ,+ , and S only in aneighborhood of the point in question. By using this fact together with an example of Guseva and inequalities (8.11), (8.27), and (8.28), and also by using certain facts in the theory of the extension of symmetric operators, Birman and Skvortsov have shown [53] that the defective number of the operator I. is exactly the same as the number of angles exceeding n. This makes it possible to describe all selfadjoint extensions of the operator L.

10. CONDITIONS UNDER WHICH GENERALIZED SOLUTIONS IN W:(S)BELONG TO Wi(S2) Suppose that the coefficients of the operator

satisfy conditions (4.2) and (4.3) and suppose that u is a generalized solution in Wi(i2) of the equation

for

fl

E L, (8) and f E L *; (a).

-

n+2

186

LINEAR EQUATIONS

We do not make any assumptions of the type of inequalities (4.32) or, more generally, any assumptions a s to whether h = 0 i s or is not a point of the spectrum for L under some boundary condition. Thus, in particular, II can be an eigenfunction for I , under some boundary condition. The purpose of the present section is to prove that, if in some subregion Q' of the region Q , the coeffieients of the operator L also satisfy conditions (7.3) and if the function (i, belongs to Lz(Q'), then the solution u has generalized .second derivatives in that subregion that are square-summable over &''c Q' and it satisfies Eq. (10.1) for almost all x in M'. On the other hand, if Q' is adjacent to a sufficiently smooth portion S, of the boundary of s1 and if the values of u on that portion coincide with the values of some function Q ( X ) € Wg(Q),then u ( x ) will have square-summable second derivatives close to S , also. In particular, if 8' coincides with Q , if the boundary S possesses a specified degree of smoothness and if the generalized solution u of Eq. (10.1) in the class W : ( Q ) coincides on S with the function Q ( x ) Wg(Q), ~ then u must belong to W:(Q). Let us suppose first that 51' i s an interior subregion of Q . Let u s take an arbitrary sphere K, that belongs to 8' and that has a radius p sufficiently small for condition (4.26) to be satisfied for the operator L in K,, so that

4-ai??xi - b,v7xl - a?2)d x

Lp (71 7)

(aij7xi?x, KP

(10.2)

zz cllsllf;(,P)

with constant c > 0 for arbitrary 7 E W i (K,). Let us construct a sequence of infinitely differentiable functions u,(x), for m = 1, 2, that converges to the solution II in the norm of W: (K,) and let us consider the boundary-value problems

...,

Lv=+,

m=1,

2. ...,

(10.3)

in the sphere K,, where S, is the boundary of the sphere K;. On the basis of Theorem 9.1, each such problem has a unique solution v, in the space W:(K ). Inequality (4.31) holds for the functions w m ; also, the right side of this inequality is uniformly bounded for all tri = 1 , 2, that is, l / v m ~ l w ; ( K p ) 4m c ,= l .

2.

... .

...;

(10.4)

Furthermore, inequality (8.6) is also valid for w,,, and its right side is, on the basis of (10.4), also uniformly bounded for all m = 1, 2, that is,

...;

(10.5)

CONDITIONS UNDER WHICH G E N E R A L I Z E D SOLUTIONS BELONG

187

Here, C (x) is an arbitrary twice continuously differentiable nonnegative function that is equal t o 0 close to the boundary S,. A s a consequence of (10.4) and (10.5), we can extract from the sequence [ v m ]a subsequence ( v , ~that ~ ~ converges ) weakly in the norms

i 4with arbitrary

of W:(K,) and W i K v 'u

T

= 2, 3,

...to some function

[I.

that also satisfies inequalities (10.4) and (10.5). Let u s show that the equations coincides with For the functions u and

a r e valid for arbitrary 7 in Wi(K,). Therefore, L, (u - V / f I k . T ) = 0.

Let u s now set 7 = umk-v , , ~ . Then, L, (urnk -'Urnk.

u7ik

- vm,)

+L, (u -urnk. umk

Therefore, in view of The second t e r m approaches 0 as k -+a. + 0 also. This shows that ii =v in K,. (10.2)s /I u,nk - vmk "w:(Kp) Thus, for interior regions Q', the assertion made at the beginning of this section is proven. Suppose nowthat the subregion M, is adjacent t o the boundary of the region s1. We may assume without loss of generality that 51, is sufficiently small that inequality (4.26), and hence inequality (4.31), is satisfied for the operator L in Q,. Furthermore, let u s suppose that 8, satisfies the conditions imposed on the region in Theorem 9.1. This assumption imposes a restriction on that portion S, of the boundary S that is common to the boundaries of 52 and C,. Let u s assume that u I s , = 0 (otherwise, we would only need t o subtract from u ( x ) a function p ( x ) that coincides on S, with IL and belongs to Wz ( 2 , )and then c a r r y out our calculations for the function 11 (x) - 9 (x)). We introduce a sequence of functions [I,, for m = 1, 2, that vanish on S,, that belong to W ; ( Q , ) , and that converge to u in the norm of Wi(8,).Let u s consider the problem

...,

L"J= y' .

vls,=u,,,Is"

m=1.

2,

....

(10.6)

in the region Q , , where S' is the boundary of B,. On the basis Of Theorem 9.1, the problem (10.6) has a unique solution v, in the class Wi(Ql). On the basis of inequalities (4.31) and (8.27), (10.7)

188

L I N E A R EQUATIONS

with constants independent of m. Here, C(x) is an arbitrary nonnegative twice continuously differentiable function that vanishes close to S'\S,, that is, t o that part of the boundary S' that does not belong to S. Then, just a s above, we conclude that the function representing the limit of the sequence (71,) satisfies inequalities (10.7) and coincides in 8, with the solution u. In this way, we investigate the differentiability properties of u close to the boundary. Together with the results of the investigation of u within P, this enables us to draw conclusions a s to whetherthe solution u belongs t o W i ( Q ) . Let u s summarize all this in the form of Theorem 10.1. Let U ( X ) be a generalized solution in W:(P)of Eq. (10.1). Let us suppose that the coefficients and inhomogeneous terms in (10.1) satisfy the conditions n

(10.8)

Suppose that, for some interior subregion 8' of the region 9, the coefficients of the operator L satisfy not only conditions (10.8) but also the conditions* (10.9)

where

4rmax (7,

4) and

(10.10)

Then, the solution u (x) has generalized second derivatives in 9;it satisfies Eq, (10.1) for almost all x in 9', and

Ill41 w g ( o , ) Q C ( V 9

P I

4.

~)p4f2(Qr)+ ll$Ilf2(Q,)]*

(10.11)

where C is a smooth nonnegative function of compact support on s?'. 'With regard to the permissibility of replacing The situation i s the same here.

11

by

II

for n

>3, cf. end of Section 4.

C O N D I T I O N S UNDER W H I C H G E N E R A L I Z E D SOLUTIONS BELONG

189

Suppose that the boundary of a region P, c 0 coincides in part with the boundary S of the region Q. Let us denote this common portion by S,. Suppose that 8 , satisfies the conditions imposed on ihe region in Theorem 9.1. Suppose also that the bounday, values of u on S , are given by a function q ( x ) belonging to W:(2,). Then, the assertions just made remain valid for the solution u in the subregion 9, in a strengthened f o r m inasmuch as the ficnction C ( x ) in (10.11) i s necessarily equal to 0 only in a neighborhood of that Portionof the boundary - of- 52,. that is not contained in S. (In this case, we need- to add Ily(lw 2 I to the expression in the square brackets in ( l O . l l ) . ) 2 ( I! Swpose that conditions (10.9) and (10.10) are valid f o r 8 , = % that Y i s a region of the type indicated in Theorem 9.1, and that ( x ) defining the boundary values of u belongs to W ; (9). Then, u E W i ( Q ) and (10.12)

For such regions 8 ,

The proof of this theorem follows from the considerations made above if we note that regions satisfying the conditions of Theorem 10.1 can be partitioned into regions of the same type but of sufficiently small size. Such a supplementary partition of 8, or 2.! in the general case must be performed because, in proving that u belongs t o W; @,), we assumed above that the size of 8 , was sufficiently small so that inequality (4.26) is satisfied for L in 8,. The restrictions that must be imposed on the coefficients in the operator L and on the function II) in order for an arbitrary generalized solution u of Eq. (10.1) to belong to W i ( Q ) are due to the nature of the problem. They are such that every t e r m in the expression for Lu, when written in the form (10.1) and in the form

for u W ; ( Q ) , belongs to L2(S2) and all the t e r m s other than the principal t e r m al j U x , r , determining the type of equation are subject to a l p x i x , in the sense that, for these terms,

1 90

LINEAR EO'JATIONS

with arbitrarily small positive E and with. c, increasing without bound a s E-+ 0. This requirement i s expressed in terms of the membership in the different sets L,(Q) of the coefficients of the unknown function u in the expression L , and in these terms it cannot be weakened. 11. OTHER WAYS OF PROVING THE SECOND FUNDAMENTAL INEQUALITY

Zn Sections 7 and 8, we derived the second fundamental inequality for elliptic operators of the form

For such operators, the requirement of differentiability of the coefficients a , , and u i i s caused by thenature of the situation. However, if the operator is of the form (11.1)

it is natural to seek to prove the second fundamental inequality (11.2)

for u E Wi, (51) without assuming differentiability of the coefficients in M. One might think that a sufficient assumption, in addition to the assumption of ellipticity, would be boundedness of maxJa,j(and of the norms

8

However, example (2.19) of Chapter 1 shows that, for n > 2 , boundedness of the u i j is insufficient. Let us show that inequality (11.2) holds for continuous ail ( x ) with the constant c depending on the modulus of continuity of the uti. We shall not give a complete proof of this proposition. The basic idea involved i s the same a s Schauder's idea for proving inequality (l.ll),which gives a

191

O T H E R W A Y S OF P R O V I N G T H E S E C O N D F U N D A M E N T A L I N E Q U A L I T Y

bound for 1u12,a,Q.* It consists in reducing the entire question to finding the corresponding bounds in small regions for elliptic operators in which the leading coefficients a i j a r e replaced by their values at some point in the small region chosen and to combining these bounds by using the continuity of thenij(x). ln connection with this idea, the general outline of the proof of inequality (11.2) is as follows: Suppose that II E IV;. (9). We denote a value of the operator M applied to u by + ( x ) . Let u s cover the region Q with small overlapping regions Q k c 8 , where k = 1, N , the boundaries of which a r e piecewise-smooth surfaces with curvature bounded below by a number K (cf. definitions in Section 8). By virtue of the continuity of the coefficients their values at points of the region52, differ only slightly from their values at some fixed point x t of that region. We write the equation Mu = 9 in the form

...,

M,u

=a,

(xi) uxi

i

- qJ - [ a i ( x ) - a, (xo")]U X i X j - niuxi - au = F -

.

We recall that the coefficient of uXixj on the right is sufficiently small. F o r operators Mo in Q,, we have inequalities of the type (8.27), specifically, llCu II w;( Q k )

s

c1

I1 Mou II L2 ( Q k )

+

CPll u

/I w; (Bk)

(11.3)

with constant c1 depending only on the constant v in the ellipticity conditions (4.2). The right-hand member of (11.3) does not exceed the quantity 1

t C 3

( ll~ll,;(,k)+

II+IIL2(sk)).

*In a footnote in [ 9 I], this idea is described by Schauder in connection with the derivation of inequality (11.2) for the c a s e of plane regions (i,e., n -1 2) and continuous coefficients ail. Schauder knew that such inequalities had been proven e a r l i e r by Bernstein in [221 (see also [54]) for equations of the form 2

in a c i r c l e with a r b i t r a r y measurable coefficients ail ( x ) satisfying the inequalities 2

L

Y

2 El i=l

al

p

2 ET i=1

and for general equations of the form

Mu = f with differen-

tiable coefficients aij ( x ) , ai ( x ) and a ( x ) , where a ( x ) 0 in convex regions. Schauder himself felt that his principal achievement was removing Rernstein's assumption that the region be convex. These r e s u l t s a r e omitted in Soviet literature up to the middle fifties and they do not appear in the surveys of Bernstein and Petrovskiy.

192

L I N E A R EQUATI ONS

...,

Let u s sum (11.3) over all regions Q k , for k = 1, N , remembering that the function L can be chosen nonzero on that portion of the boundary 3f 9, belonging to S (we assume that u vanishes on S). Using the inequalities that we have, we obtain the inequality

If we choose the regions Q, so small that c 4 m a x n i a x ( ~ , ~ ( x ) - - ~ ~ ( X4 ~ )2 I1 , i, j. k x E Q k

(11.5)

we obtain from (11.4) (11.6) Again using the inequality (11.7) with arbitrary E > 0, we obtain inequality (11.2) from inequality (11.6). From these remarks, the reader himself will be able to c a r r y out the complete proof of inequality (11.2). He need only note that in inequality (11.3) the constant c, can be chosen independently of the region Q k and that it is determined only by the smallest eigenvalue v of the quadratic form aljS,Ej. This is clear from the derivation of inequalities (7.4), (8.20), (8.22), and (8.27). Specifically, inequality (7.6) gives u s a lower bound for the principal positive term; bounds for all the other t e r m s are obtained in t e r m s of it and the derivatives of u of lower order. Let u s state the result that we have been describing in the form of l e m m a 11 .l. Suppose that the regionP satisfies the same conditions as in Lemma 8.1. If the coefficients a,! in the operator M defined by the expression (11.1) are continuous in C and i f the ( ~ ) , llailL ;(Q), where q^=max [ q , 41 f o r q > n , are n o m s I ~ U , J I ~ ~ and -

finite, then, f o r an arbitr&'jy function u in W i , o ( Q ) , inequality (11.2) is valid with constant c depending on the region 51, the values of v and u in the double inequality

193

OTHER WAYS O F PROVING THE SECOND FUNDAMENTAL INEQUALITY

n

n

the modulus of continuity ofa,,,undthe norms IlalllLo(Q)and Ilall,

T(n).

-_

A s will be shown in Section 17, for n = 2 we can discard the 'requirement that the a,,(") be continuous. For n >2, we cannot do this. Keeping the scheme for proving inequality (11.2) that we have been discussing, one can show that, for elliptic operators M, the following more general inequality is valid:

(11.8) where p is an arbitrary number greater than 1 and u ( x ) is an arbitrary function in Wyl,o(Q). With regard to thecoefficients in M, we need to assume that the a,, ( x ) are continuous and satisfy inequalities (4.2), that the i,( x ) are q-summable over Sq (where q > n) if p < n and p-summable if p > n , and that a ( x ) is (q/2)-summable (where q > n ) over 2 if p ,< n and (p/2)-summable if p > n. The constant c,,,, is determined by n, r , p , v, ,u, the modulus of continuity of the a l l @ ) , and the norms of a , and a in the spaces L , ( Q ) indicated. It also depends on the boundary S, which is assumed to be twice continuously differentiable (or at least belongs to W i , where q = max ( p ; n + E ) for E > 0). Proofs of inequality (11.8) for the case of bounded coefficients a, and a appear in the articles [86] and [87].

12. CONDITIONS UNDER WHICH GENERALIZED SOLUTIONS IN Wz BELONG TO Cl,a FOR C 2

>

Suppose that the function u ( x ) belongs to %It, and for almost all

x in Q , satisfies the equation Lu =a,/ ( x ) u x p j

+

a , (4u x ,

+a

( x ) =f

(

4

1

(12.1)

and inhomogeneous t e r m in which are are elements the coefficients of C,+ (Q), where 1 2. Suppose that

>

where Y > 0. Here, !Dl is the set of elements Wi(S2) with finite essential max. Let u s show that u ( x ) actually belongs to Cl,a[Q)a Q

We need only do this for an arbitrary sphere K, of small radius p.

1 94

LINEAR EQUATIONS

We choose the size of the radius p in accordance with the coefficients in L as follows: F i r s t , we take p sufficiently small that, for the operators Lou E a i f (x ) uXix,+ ai ( x ) uxi 0

+a ( x ) u , where x" E K,.

and an arbitrary function v ( x ) in W!,o(K,), the inequality (12.2) is satisfied.

In the second place, it is necessary that inequality (1.9) be satisfied for L" in K , and hence that the theorem on the uniqueness in the class C,, =(Kp)be valid. Finally, we need to take p such that cn

rnax ( a i j ( x f ' ) - a i , ( x ) l i. 1: xE Kp

.< -,21

(12.3)

where c is the constant in inequality (12.2). Obviously, we can satisfy all three of these requirements if we take p sufficiently small. of infinitely difLet u s take a sequence ( u r n ) ,for rn = 1, 2, ferentiable functions in the sphere K , that is uniformly bounded and that converges in the norm of W:(K,) to the function u (x). For each of the u,,~,let u s consider in Upthe problem

...,

Lvm = f *

Vmlsp

=~rnl~~*

(12.4)

where S, is the boundary of Up. From what was said above, the problem (12.4) has a unique solution v, in C l , (K,). The difference wnl= u, -v, satisfies the equation Lwn1=Lu, -j, which we write in the form Low, = (L" - t)w,,

+ Lu,

-f.

(12.5)

and the homogeneous boundary condition wrnlSp=O.

For this difference, inequality (12.2) yields

But (Lo- L) w, = [a,, ( x o )- ui/ (x)]w , , , ~ ~and, ~ ] , therefore,

From this inequality and inequality (12.6), we have by virtue of the

I95

T H E BOUNDEDNESS O F G E N E R A L I Z E D SOLUTIONS

assumption (12.3) ,

11% I1 w; (Kp) ,< 2c IILu, -f II Lq (K,)’

(12.7)

A s one can easily see, the right side of this inequality approaches zero a s m - m , so that {vm) converges in W i ( K , ) to u. But for wm, where rn = 1, 2, we have the uniform bounds with respect to rn [cf. (1.13) and (1.9)] IemLIi, a, Kp 4 c (1, a, C) ( I fie, K, max Iv m 1). (12.8)

...,

+

K?

where C ( x ) is any function in Cl, (K,) that is of compact support in K,. The constant c ( 1 , a. L) depends on this function and on 1 and a but is independent of m. It follows from (12.8) that the function u representing the limit of the sequence (vm)will belong to C,, ( K , \ s,) and will satisfy inequalities (12.8). The desired assertion is proven. Analogously, by using inequalities (1.11) and (1.12) instead of (12.8), we can show that u ( x ) belongs to Cl. (G)o r to Cl, (S2 US,) if we also assume that the boundary values of u on S (resp. on the portion S , of the boundary S ) belong to C l , aand if S itself (resp. S,) is a class Cl,a surface. Slight additions to our reasoning that a r e necessary for this are described at the end of Section 10. Let u s formulate these assertions in the form of Theorem 12.1. S q p o s e that u ( x ) E ID{ and satisfies almost everywhere in Q the elliptic equation (12.1), the coefficients and inhomogeneous term, in which are functions in C , - 2 , u ( E ) ,where l>, 2. Then, u ( x ) i s a function in C,p ( Q ) . If, in addition, the boundary S (resp. the portion of it S , ) i s a surface in the class C l , gand if the boundary values of u on S (resp. on S , ) are defined by a function in the class Cl, a, then u ( x ) belongs to C l , , [resp. to C,, (Q u SJ]. (L

‘I

11

(a)

~

13. THE BOUNDEDNESS OF GENERALIZED SOLUTIONS IN Wi(Q) Suppose that u ( x ) is a generalized solution in W:( Q ) of the equation (13.1)

and that the coefficients L satisfy the conditions*

(13.2) 2

‘For the c a s e 4 = n

> 3, cf. end of this section.

196

LINEAR EQUATIONS

In Sections 4 and 5, we established theorems on the solvability in W: (Q) of the Dirichlet problem for (13.1) under the assumptions that f E L2;i( Q ) ,that f rE L, (M), and that the boundary values of u are given ;+z

by a function ‘9 ( x ) E Wi (52). In this and in the following section, we shall show that under somewhat better properties of f and the f i , an arbitrary solution of this type belongs t o the Halder class Co,m(’sz). In Section 2 of Chapter 1 , we showed with the examples presented there that the following conditions a r e necessary for this: (13.3)

Let u s show that these are also sufficient conditions for u to belong to C,,(Q). Furthermore, we shall obtain bounds for the quantities max IuI and I u I , in t e r m s of the constants Y and 1 in conditions (13.2) and (13.3) and for I I u I I ~ , ( ~ ) . Of course, the bounds for these norms for all Q depend also on the corresponding characteristics of the behavior of the function u on S. Thus, let u s suppose that (13.2) and (13.3) a r e satisfied and that u is a generalized solution of Eq. (13.1) in W:(Q), that is, that u belongs to W : ( Q ) and satisfies the identity

for an arbitrary function 7 E &: (Q). Now, let u s take for 7 the function ~(x)=C*(x)max ( u ( x ) - k ; 01,

where k is an arbitrary positive number and C(x) is an arbitrary smooth nonnegative function of compact support on an arbitrary sphere K , c Q whose values lie between 0 and 1. Such an ~ ( x is) an admissible function since, on the basis of Lemma 3.3 of Chapter 2 , it belongs to the class lb:(Q).The function q is nonzero only on the set dk, of points x in K , , where u ( x ) > k. If we substitute this function into (13.4), we obtain after some simple calculations,

J ( a i i l f x i u1x C * + a i j ~ x , ( u - - ) . ‘k, p

+(up

-ffl)[Ux,C2+

XC~,+

2 ~ c x i ( u - k ) ] - ( ~ ~ u x ~ + a ux- ~

x ( u - k ) CZ} d x = 0.

(13.5)

197

T H E BOUNDEDNESS O F G E N E R A L I Z E D SOLUTIONS

On the basis of (13.2) the first t e r m is not less than v l V u ) 2 [ 2 . Let u s leave this t e r m on the left and find a bound for it from below. Let u s transpose the other t e r m s in (13.5) to the right and bound them from above, using inequalities (1.1) and (1.2) of Chapter 2 giving a small E to factors containing nXi. This leads to the inequality

(13.6)

Let us now set

E

= ~/6.Then,

(13.7)

From this, we get the inequality

(13.8)

where

and c is a constant depending only on v and 1 in (13.2). On the basis of the assumptions (13.2) and (13.3), the quantity l(DII,q(9)is finite and is determined only by the size of

p.

2

I98

LINEAR EQUATIONS

Let u s bound the last integral in (13.8) by using Halder's inequality:

where c, is determined only by v, p , and q in (13.2) and (13.3). By virtue of inequality (2.13) of Chapter 2, the first t e r m on the right in inequality (13.10) satisfies the inequality

-

= 2/n 2/q > 0 and the constant c p ( q )depends only on n where and q. If we assume the radius p is such that c2 (4) c1cp"eI

1

4y

(13.12)

I

then, on the basis of (13.10) and (13.11), it follows from (13.8) that JIVu12C2dx p

q

.f(u-kh)2IVCl~dn+

'k0

(13.13)

p

+(k2+

1)mcs

I--;

:A,,,].

The constant 7 is determined only by the quantities v , p , and y. Inequality (13.13) is proven for all k and p that satisfy condition (13.12). Let us set C equal to unity within the sphere KP-.IP,concentric with K,,where a is anarbitrarynumber in (0, 1) and otherwise choose C so that it satisfies the condition I V C J ,.< c/op. Then, inequalities (5.12) of Chapter 2 follow from (13.13), so that we conclude from Theorem 5.3 of Chapter 2 that, for arbitrary Q ' c Q , the quantity esssential max u is finite and is bounded above by an ?'

expression involving only 7, q , I/ u 11 L 2 ( Q ) , and the distance between !!' and S.

199

T H E B O U N D E D N E S S OF G E N E R A L I Z E D S O L U T I O N S

Analogous considerations for the function - u ( x ) lead to the bound from below for essential min u. Thus, we have bounds from P'

above and below for the expression essential rnaxlul. Q'

Inequalities (13.13) remain valid for spheres K, that intersect S provided the k satisfy the inequality k

> inax [ fu ( x ) ] . Kpns

Tn accordance with Theorem 5.3 of Chapter 2, this enables us to give a bound for essential max [ + u ( x ) ] close to those portions S , of the boundary S on which the quantity essential max [ k: u ( x ) ] is S,

finite. In particular, if M, = essential max/u (x)l is finite, we are in s

a position to give a bound for essential maxlu(x)l. We can do this 9

last not only by using Theorem 5.3 of Chapter 2 but also by using the simpler Theorem 5.1 of Chapter 2. .Specifically, if M,, < 00, then we set 7 (x)= inax ( u ( x )- k ; 01, where k > M,,, in (13.4). Such an After setting up a sequence of inequalities analogous to those above (these are somewhat simpler since all t e r m s containing derivatives of the function C, which in the present case is identically equal t o 1, drop out), we obtain the inequalities ~ ( x belongs ) to $:(Q).

with constant 7 depending only on q, v, and p . Here, k is an a r bitrary number greater thanM,and A, is the set of points x in 9 at which u ( x ) > k. Analogous inequalities can be established for the function - u ( x ) . On the basis of Theorem 5.1 of Chapter 2 , these inequalities together imply the boundedness of essential max Iu ( x ) l 2

and the possibility of giving a bound for this quantity in t e r m s of v. P. 4. and lluilL,cp,* Thus, we have proven Theorem 13.1. Suppose that u ( x ) i s a generalized solution in W: (9) of Eq, (13.1) and suppose that conditions (13.2) and (13.3) are , quantity essential max satisfied. Then, for an arbitrary V c Q the P'

( u ( x ) l is finite and on v, p q, (1 u /IL1(*,,

bounded from above by a constant depending only , and the distance from Q' to the boundary S of the region 9. If, in addition, essential max 1uix)l < 03 f o r some portion S,

S , of the boundary S , then essential maxlu(x)J,where Q, i s a sub%

region of Q that lies at a positive distance from S \ S , , is finite and

200

L I N E A R EQUATIONS

bounded f r o m above by a constant depending only on v, p , q, /Iu 11 L,(p,, and the distance from 51, to S \ S , . In particular, i f essential max lu(x)l

< 00, then

s

essential max I U ( x ) ~is finite and bounded above by 9

a constant depending only on v, H , q, and I/u II L!(a,. Remark 1: If, for n > 3 , wereplace conditlon (13.2) for the b, by

the assumption

-

(13.14)

2

then inequalities (13.13) will be satisfied for sufficiently small p, more precisely, for values of p that satisfy not only the condition (13.12) but the condition (13.15) This is true because, on the basis of H’dlder’s inequality and the inequality (13.11) with q = n, the inequality

is valid for the t e r m

However, in the present case, the quantity depend also on

and the choice of p

Thus, if we replace the assumption (13.2) regarding the B , ( x ) for n > 3 with the assumption (13.14), then the assertions of Theorem

CONDITIONS UNDER WHICH GENERALIZED S O L U T I O N S SELONG

201

13.1 remain in effect except that the values of essential maxlu(x)I PI

now depend also on A(p). A s the examples in Section 2 of Chapter 1 show, this dependence is a genuine dependence. Remark 2: If we relax the assumptions regarding the inhomogeneous terms f i ( x ) and f ( x ) in the hypotheses of Theorem 13.1, we can, by using (13.4) with y = uCmC2, where cm=2(*)m-

(for m = 0, 1, following:

...),

1,

and also inequality (13.13) with q= n, prove the

IffiEL,(Q)andfEL,(&),where 2
Iff

nr n-r

-};ns

n--2s

E L, ( Q ) and f E L -, ( Q ) for some p > 0 , the integral 2

i s finite. 14. CONDITIONS UNDER WHICH GENERALIZED SOLUTIONS IN W:(Q)BELONG TO C,,= Let us show that, when conditions (13.2) and (13.3) a r e satisfied, an arbitrary generalized solution u ( x ) in W i ( Q )of Eq. (13.1) belongs to the class C , for some a > 0. To do this, we note that, from the proof of Theorem 13.1, more precisely, from inequality (13J3), we get Lemma 14.1. Suppose that conditions (13.2) and (13.3) aye ~

satisfied and that u ( x ) i s a bounded generalized solution W : ( Q ) of Eq. (13.1). Then, u ( x ) belongs to the class B2 (a, M , y, m,l/q), where M = essential max I u ( w ) l and y i s a constant determined by P the quantities v, p , and q in conditions (13.2) and (13.3). From this lemma, Theorem 13.1, and Theorems 6.1 and 7.1 of Chapter 2 regarding functions in the classes d,,we can easily derive Theorem 14.1. Suppose that u ( x ) is a generalized solution in W:(Q)of Eq, (13.1), the coefficients and inhomogeneous terms f i and f of which satisfY conditions (13.2) and (13.3). Then, u ( x )

202

LINEAR

EQUATIONS

belongs to the class C,, (?), where Q' i s an arbitrary interior subregion of the region &, and the norm 1 ~ 1 o~ , .is bounded f r o m above by a constant depending only on the quantities v , p , and q in conditions (13.2) and (13.3), llu/l,,2(Q,,and the distance f r o m 52' to the boundary S of the region 8. The exponent a > 0 i s determined by these same quantities. If we assume that essential max)u (x)J= M < 03, then 11 (x) E Cq, (Q) for some a > 0 determined only by the quantities v, p , q, and hl. I f a portion S , of the boundary S satisfies condition ( A )and i f u IsI E C", (S,),then, for an arbitrary subregion 8 , of the region P that lies at a positive distance d from S \ S , , the function u (XIE C,,, @,), and the norm 1ul0, 9, is bounded from above by a constant depending only on v, p , q, I J U ( I ~ ~ , ~ , ,( u I . ~ , s,, B, d , and the constants 0, and a, in condition (A). These quantitzes also determine a > 0. In particular, i f the entire boundary S satisfies condition ( A ) and if I I ( , E C,,(S), then I I ( X ) ~ C ~ , ~and ( G )the norm ( L I ( ~is, ~bounded f r o m above by a ~ , , the constants constant depending only on V , p, q. ~ ~ U ~ JI U ~I ~ , , ~~ , pami 0, and a , in condition (A). Remark 1: Here, just a s with Theorem 13.1, condition (13.2) regarding the b , ( x ) can be replaced by the condition (13.14), but the values of the norms Iu 1 9, will then depend on A (p) a s well a s the quantities mentioned. Remark 2: The assertions proven in Sections 13 and 14 a r e consequences of the fact that the functions u ( x ) being investigated satisfy inequalities of the form (13.13). In these two sections, we have shown that functions u (x) that satisfy the identity (13.4) satisfy these inequalities also. However, if a function u ( x ) E W: (9)satisfies not (13.4) but, for example, the inequality 0

~

cL,

for all known positive ~ ( x in ) w:(Q),it also satisfies inequalities (13.13) and therefore its maximum can be bounded from above in terms of the quantities indicated in Section 13. In Chapter 5, we shall give another class of gLsubso1utions,99 that is, functions that satisfy not the equations or the identities substituted for them but some system of inequalities and we shall prove their HGlderness. A slight generalization of this class is the following class of ''subsolutions:" Suppose that u (x) E W i ( Q ) , that essential max I u (x)l,< M , and d that u ( x ) satisfies inequalities (14.1) for all q ( x ) that satisfy the conditions - M ,< u (x) + t q ( x ) < M for t E [ 0, 11. Ttturns out that all such u (x, belong to C,,a, where a > 0, and that their norms J u ) , satisfy the same bounding inequality a s do the norms of the

CONDITIONS F O R BOUNDEDNESS

203

generalized solutions of Eqs. (13.1). The proof of this is analogous to the proof of Theorem 4.2 of Chapter 5.

15. CONDITIONS FOR BOUNDEDNESS OF rnaxlVirl AND IuxiI FOR GENERALIZED SOLUTIONS IN Wi (52)

Let u s see what are sufficient conditions regarding the coefficients and inhomogeneous t e r m s in Eq. (13.1) for an arbitrary generalized solution of it in W:cQ!, to belong to Cl,a(Q). It follows from the results stated at the end of Section 11 that there exists an a priom' bound for llulj w;(p,, where q > n, in t e r m s of lluilL2(Q),the constants v and I-( in (13.2), the constant I-( in the inequality

and the constants characterizing the surface S c W i and the norm 11y111w;ce,of the function p ( x ) which determines the boundary values of u (XI. The bound for (lullwg,pxy with q > n, gives in turn a bound for

-

lull. =, *, where a = 1 n/q (cf. Theorem 2.1 of Chapter 2 on the inFrom these a priori bounds jection of W: (Q), where q > n. inC,, and the theorems proven above regarding the solvability of the first boundary-value problem in the spaces C2. and W i , it is easy to show the validity of Theorem 15.1. If the coefficients in Eq. (13.1) satisfy conditions (13.2) and (15.1), i f !c is a region of the class W ; , and i f '9 ( x ) E W i ( Q ) , then an arbitrary generalized solution u ( x ) of Eq, (13.1) in W i ( 2 ) n / q , and the quantities belongs to W i ( Q )nC,,=(G), where a = 1 II u 11 w;(,)and (11 1 aye bounded from above by a constant depending only on n , q. v. p. JjulJL,,p,, llpll w;(B),and the region c?. This theorem is proven by following the same procedure a s in the proof of Theorem 10.1. We can obtain an a priori bound for ~ u ~by, a, different ~ procedure, similar to the one used in Sections 13 and 14 to derive bounds for max IuI and 1u1(,,. Specifically, let u s differentiate Eq. (13.1) with respect toxR,for h = 1, n, and let u s write the result in the form o1

(a)).

-

,,

R,

...,

k=l,

where

.... n.

204

LINEAR EQUATIONS

and 8; denotes the Kronecker delta. Letustreat the set of relations (15.2) and (13.1) a s a system of equations in theu,, where k = 0, 1, n. Its characteristic feature is that the principal part, con, i s diagonal and is the sisting of terms of the form - a,,-dF dxl same in all equations of the system. We have succeeded in investigating such systems with a s great detail a s a single secondorder equation. Chapter 7 is devoted to these systems. There, we derive, in particular, a bound for the maximum of

...,

"(

")

which, when applied to the present case, yields the desired bound for max IVuJ, where Q'cQ.A bound for (Vul close to the boundary 9' does not follow immediately from the bound given in Chapter 7 for max IuI since the value of max lul in Chapter 7 is bounded in terms 9 of the value of max I u J , which we do not know for the vector

...,

I 1

S

du

only the quantity dn is unknown in S advance). To obtain such a bound, we need to examine further certain considerations regarding the boundary spheres. For lack of space, we have been obliged to omit them. However, they can be reestablished when we have read Section 16: The bound explained in that section for (Vul near the boundary between two media is very close to the bound for )Vu(close to S. Bounds for the quantities I uXi 9 , , where Q ' c P also follow from the results of Chapter 7. However, they can be derived from the theorems in Section 14 of the present chapter. ln fact, if the quantities M = niaxlul and M, = maxlVuI are already known, the reP 9 lationship (15.2) can be regarded a s the equation u = ( u , u,,,

u X n ) (actually,

(15.3)

...,

for u k for which all conditions of Section 14 a r e satisfied. Conn, the conclusions of sequently, for & = u x k y where k = 1, Section 14, namely, Lemma 14.1 and Theorem 14.1, a r e also valid. They provide a bound for I uxi where Q'cQ.To get a bound for

205

CONDITIONS F O R B O U N D E D N E S S

I uxr I(n)

close to any portion S , of the boundary S , we straighten that

portion out, introducing new coordinates. Without loss of generality, let u s assume that S, lies in the plane (x,, = 0) and that 8 is adjacent to it from above, that is, onthe side x , > 0. For the derivatives uxr, where T = 1, . . . , n- I, the boundary values on S , are known. Therefore, we can apply to uxTthe second portion of Theorem 14.1, dealing with a bound for lux I where T = I , . . . , n - 1, close to I

(a), QP'

S,. (Here, QP is the half of the sphere K, with center on S , belonging

to

n.)

for

To get a bound for

T = 1,

. . ..

n - 1, in

I ux, I n , , we note that P . . .) implies

?$(Qp.

membership of

u.~-,

n

F r o m this and Eq. (13.1), we get

A s was shown in Lemma 4.1 of Chapter 2 , these inequalities imply that u,, belongs to C,),~ ( 9and ~ they ) enable u s to get the desired bound f o r I u l'n lo,9. Thus, we have a second procedure for deriving bounds

for max lbul and I n ,

I

i (a)

for solutions of Eqs. (13.1). On the basis of

it we have proven Supplement to Theorem 15.1. Suppose that conditions (13.2) and (15.1) a r e satisfied for Eq. (13.1) and suppose that only a portion S, of the boundary S i s a s u r f a c e i n the class W i . Then, an arbitrary generalized solution u ( x ) in W : ( Q ) of Eq. (13.1) will belong to C,, @,), where 9 , is a subregion of Q lying at a positive distance from S \ S, if it coincides on S , with the function 'p ( x ) E W i (9).lts norm ( u (,, P, 11,

is bounded from above by a constant depending only on

quantities n,

v,

p,

the

and q in conditions (13.2) and (15.1), on the

norm llQll W ' ~ ( Q ) ' on the distance from 8, to S \ S,, and on the properties of the surface S,. 16. DIFFRACTION PROBLEMS

Ordinarily, the steady-state diffraction problems that we are considering are special cases of the problem of determininga

206

function

LINEAR EQUATIONS

u (x) in

a region Q that satisfies the equation

some condition on the boundary S of the region 8, and the following conditions on a surface 'l of discontinuity of the first kind for the coefficients ui,: (16.2)

Here, p ( x ) i s any positive function with a possible discontinuity of

I

dU the first kind on ,'I dN = u i j u x .cos (n, xi),where n is the normal to I

,'l and the symbol [u]denotes the jump in the function v a s it crosses .'l The surface 'l is contained in s1. A l l the given functions a r e assumed to be sufficiently smooth outside it. The functions 'p, and 'p2 a r e known functions. The unknown function u undergoes a discontinuity determined by conditions (16.2) on crossing I?. Outside I', it must satisfy Eq. (16.1). Most commonly, we encounter the case in which the surface I' partitions the region Q into several sub62,, where S2 = Q, U 8, . . . u 9,. This corresponds regions Q , , Q2, to physical problems in which there a r e several media of different substances [in each of which the equilibrium state is described by some equation of the form (16.l)l applicable to it and in which u obeys the relations (16.2) upon crossing from one medium into the other. Let us consider this case. Here, a s throughout the book, we consider only the case in which the region 8 is bounded. However, the greater part (including the fine points) of the reasoning that we follow with regard to the investigation of the differentiability properties of generalized solutions of diffraction problems a r e independent of whether the region Q is bounded or not. Therefore, it is immediately applicable to the case of arbitrary unbounded regions. ln order to make certain inconsequential simplifications in our presentation, let us assume that a in (16.2) is equal to 0. Suppose that the function n ( x ) satisfies the first boundary condition on the boundary S of the region &. (The cases of the other boundary conditions can be considered in a similar manner.) This boundary condition and condition (16.2) can be reduced to homogeneous boundary conditions by making the substitutions v (n) = u ( x ) - 'p ( x ) , where 'i. (x) is such that it coincides on S with u (x>l and undergoes jumps on' I that a r e determined by conditions (16.2). Suppose that we have made this reduction, and that u ( x ) satisfies conditions (16.1) in the regions Q,, . . ., 9,and the conditions

...,

DI F F R A C T I O N P R O 6 LE M S

20 7

(16.3)

on the surfaces S and.'I The problem (16.1), (16.3) can be reduced to finding a generalized solution in the space Wi(S2) of the first boundary-value problem for #la single" equation of the type (16.1) though with, generally speaking, coefficients that a r e not differentiable throughout the entire region P, that is, to the problem considered in Sections 4-5. This equation i s the following:

It is obtained by multiplying (16.1) by p and formally carrying out the operations of differentiation and regrouping of the terms. For the moment, these operations must be regarded a s purely formal ones. Let u s suppose that p ( x ) is a bounded function exceeding some positive constant p 0 and possessing generalized first derivatives in each of the regions Q,, for I = 1, t, that are g-summable, where q > n, over 9,. The coefficients u,,, b,, and a and the function f a r e assumed to satisfy the same conditions a s in Section 13, namely, conditions (13.2) and (13.3). We shall refer to a generalized solution in W:(Q) of the first boundary-value problem with homogeneous boundary condition for Eq. (16.4) a s a generalized solution in W:(Q) of the diffraction problem (16.1), (16.3) or, more briefly, a s a generalized solution of the problem (16.1), (16.3). In other words, we shall say that a

...,

function

u ( x ) belonging

to lb: (Q) and satisfying the integral identity

@:

for arbitrary E ( x ) E ( Q ) is a generalized solution in W: (Q) of the problem (16.1), (16.3). Suppose that the function u ( x ) satisfies all conditions of the form (16.1), (16.3). More precisely, suppose that u ( x ) belongs to Wi(52,) for each of the regions Q, (where 1 = 1, t ) , that, in each Q,, the coefficients a,, ( x ) a r e differentiable and belong, for example, to L,(Q,), where q > n, and that 11( x ) satisfies Eq. (16.1) for almost all x in Pi (where 1 = 1, t ) and conditions (16.3) for almost all points in r and S (in the sense of (n 1)-dimensional measure). Then, it is easy to verify that under these Conditions u ( x ) also be-

...,

...,

-

longs to the class Wi (&)andsatisfies the identity (16.5). Specifically,

208

L I N E A R EQUATIONS

...,

since u ( x ) belongs to Wi(Qi), for I = 1, t , and since [ u l l r = 0, it follows that u ( x ) E W: (Q). This, together with the condition u Is = 0 ensures that

u(x)

will belong to W:(9). Now, let us multiply Eq.

(16.1) by pE, where E E Wl (II) and , integrate over 8,. If we then integrate the first term by parts, taking p i d x a s “dv,” we obtain the identity (16.5). All the boundary integrals, which appear explicitly when we carry out the integration by parts, disappear: on I’ be-

and on S because El, 0. The converse is also CEII, true; specifically, if the given conditions of the problem possess the

[

cause p - -

=O

=

differentiability properties just indicated, and if the function on u (x)

belongs to

lb: (Q) and

W i (Qi),for I = 1,

...,t, and satisfies the

identity (16.5) with E E Wi(Q), then u ( x ) satisfies Eq. (16.1) (for almost all x in Q) and the conditions (16.3) (for almost all points on I’ and S). These considerations indicate that it is possible to replace the classical formulation of the diffraction problem in the form (16.1), (16.3) with another generalized formulation in the integral form (16.5). But when the diffraction problems a r e posed in this last way, they a r e reduced to finding a generalized solution of Eq. (16.4) in the class @:(Q). In the preceding sections, we investigated such a problem and proved its Fredholm solvability. We now formulate a portion of this result a s Theorem 16.1. Suppose that the following two conditions are satisfied: n

n

~ Z E : , < a r j E i E j , < P ~ E V~>,o , i=l t=l

I=1,

o

(16.6)

... t , q > n .

Then, the existence of a generalized solution w:(Q)of the diffraction problem (16.1))(16.3) is implied by the uniqueness of the problem. If I l f l l Lq (O)<

03,

2

(16.8)

then every generalized solution in w:@) of the problem (16.1), (16.3) belongs to C,,,(8)f o r some a > 0 . If, in addition, the boundary S satisfies condition (A), then u E C“,

(a).

20 9

DIFFRACTION PRO6 L E M S

This theorem is a direct consequence of the theorems in Sections

4, 5, and 14. In those sections, we gave sufficient conditions for the Dirichlet problem to have no more than one solution in W:(Q). Let

us find the conditions under which the generalized solution u ( x ) of the problem (16.1), (16.3) is a classical solution, more precisely, under which it will belong t o Cz, (Q,) n C,, (a,)for each 8,and hence will satisfy all the conditions of the problem in the form (16.1) and (16.3) directly. As always when we a r e studying elliptic equations, the investigation of the differentiability properties of their solutions can be carried out locally, over small regions. It follows from Theorem 12.1 that, if the functions al,, dai,/dxk, bi, a, f, p , anddp/dxi belong to the classes Co, ( Q l ) for 1 = 1, . . ., t , then the solution u ( x ) belongs to the class Cz,B ( Q l ) for 1 = 1, . . ., t. Furthermore, if the function u ( x ) vanishes (or coincides with a function in C2, on some portion S' of the boundary Q , that is contained in S but has no points in common with I' and if s' Cz, then u ( x ) E C2, (Q, u S'). One thing that the theorems proven above fail to give us is information regarding the behavior of the derivatives of u ( x ) in a neighborhood of .'I Let us examine this question for the first derivatives of u ( x ) , that is, for those derivatives of u ( x ) that appear in condition (16.3). In order to make certain insignificant simplifications in the presentation, let us assume that all the coefficients in the operator L and the derivatives dal,/dxk and dp/dxk a r e bounded, that is, that @)

e

@,

(16.9) Let u s also assume that conditions (16.6) a r e satisfied. Let u s take some small portion r,cl' that belongs to Cl, and straighten it out introducing new nondegenerate coordinates y = y ( x ) possessing bounded first and second derivatives with respect to x. In doing this, we do not destroy any of the properties of Eq. (16.1) that we need, and the equation keeps the same form. Therefore, we may assume without loss of generality that, even in the x coordinates, Pl is a portion of the plane x,= 0. Suppose tht the center of the sphere K , is on PI, that i t s left half (forx, < 0) belongs t o Q , , and that its right half belongs to Q,. Let us denote by C ( x ) any twice continuously differentiable function with values between 0 and 1 that is of compact support on K,. Let us show that u ( x )C ( x ) E W i ( K , f l Q l ) , for 1 = 1 , 2. The proof is essentially the same a s the proof of the analogous assertion for generalized solutions in the case in which the leading coefficients have no surface of discontinuity l'. The corresponding a priori bound constitutes the basis of the proof. In obtaining it, we need t o remember that, for the

210

LINEAR EQUATIONS

coefficients q j , there a r e no generalized derivatives of the form throughout the entire region & (they exist only in 52, and 8, separately) but that we still have the remaining derivatives d / d x f , for 1 = 1, n 1. Therefore, we need to obtain an apriom' bound such that the constant in it will be independent of d a f l / d x n . If we look at the derivation of inequality (8.6) in Chapter 3, it is easy to see that this derivation involved the existence of d a f j / d x , , i n & and the fact that these derivatives belong t o L, (Q), f o r q > n. To avoid this, let us consider not

d/dxn

...,

-

but

o r , what is essentially the same, let us consider identity (16.5) with n-1

where C ( x ) is a twice continuously differentiable function with values between 0 and 1 that is of compact support on Up. Let us integrate the first t e r m in this equation over each of the Q,,for 1 = 1, 2, once by parts, transferring the derivative d/L1xk with (uXkC*),, to the remaining factors. Since x k , for k n - 1, is a tan-

gential coordinate with respect to r,, we do not get a boundary integral over r,. There will also be no integral over the boundary of the sphere K, since C vanishes on it. The principal t e r m will be

It is not less than n-1

n

A s we can easily verify, none of the remaining t e r m s exceeds in

DIFFRACTION P R O B L E M S

21 1

absolute value the quantity

where E is an arbitrary positive number and c. ( I V C J ) is a known constant depending on max 1 VCI. Therefore, from our equation we get the inequality

KP

(16.10)

We derived this inequality under the assumption that the derivatives uxkxl are square-summable over K,. We could have derived the inequality without making this assumption by considering not the second derivatives of u but the difference quotients A/Axk of the first derivatives uxf and then finding a bound for the integral

To do this, we need to take E(x) in (16.5) in the form I.

.

)z1

k=l

(-&

c2).

When we have found a bound f b (uniform over the sphere A x , ) , be can then conclude that the derivatives uxkxi, for k ,< n - 1 and 1 4 n are square-summable over Up,,where p’ < p. It is possible to prove that the integral in the left-hand member of inequality (16.10) is finite in another way. This proof r e s t s on the existence of an a priori bound of the form (16.10) for sufficiently smooth functions u ( x ) and it uses the fact that the constant c ( I VCI) in (16.10) is independent of the magnitude of the derivatives dal,/dx,. The general procedure is the same as the procedure used in proving Theorem 10.1: the coefficients in Eq. (16.1) and the inhomogeneous t e r m f (x) a r e approximated by functions sufficiently smooth that all solutions of the approximating equations Lmu=fm,

m=I.

2,

....

(16.11)

21 2

LINEAR EQUATIO N S

are functions belonging to C2,*. Out of all the solutions of Eqs. (16.11), let us take those that coincide on the boundary of K, with the solution u ( x ) of the diffraction problem (16.1), (16.3) that we are studying. For spheres K, of sufficiently small radius, this requirement defines for each Eq. (16.11) a unique solution urn( x ) . We c a r r y out successive approximations such that the sequence p}converges to f in L 2 ( K p ) ,such that the sequences of the coefficients u?,, I!$, and urn,which are bounded in absolute value by some constant, converge almost everywhere to u l j ,b,, and a , respectively, such that the lowest characteristic value of the form a;€,€, will be

1 =I,

da;

no less than vo > 0, and such that the quantities f o r k ,< n - 1 , will remain uniformly bounded. For uYi, by. a'" and f"',we can, for example, take the averages of u,,, b,, a, and f, respectively, withaveraging radius equal to l / m . One can prove, just as in Section 10, that the functions u m ( x ) are uniformly bounded in the norm W:(K,) and that the sequence of these functions converges to u (x). Furthermore, as inequality (16.10) shows, the quantities /,, c(urn)will also be uniformly bounded. Therefore, the quantity I,, ( u ) is finite. Thus, we have shown that the derivatives uxkxi, for k < n - 1 and 1 ,< n are square-summable close to rl. It follows from Eq. (16.1) and inequality (16.10) that the quantity u; is summable in n n

each of the regions K, n 51, and K, n Q,, more precisely, that (16.12) where c ( I VC I ) is, in general, different from the c ( I VC I ) in (16.10) but is still a known constant. Note that, although we write the norm uC in W : ( K p ) in the expression on the left, the function u (and hence the function uC) does not belong to the space W;(K,) since u does not have a generalized derivative uxnxn everywhere in K,. Such a derivative exists only in each of the subregions K , n M, and K , n 9,. Here, the symbol denotes the quantity Inequality (16.12), together with the identity (16.5) ensures that the solution u ( x ) will satisfy Eq. (16.1) for almost all x in K, n Q k ) in each of the regions K , n Q k , for k = 1, 2 and that it will satisfy conditions (16.3) on for almost all x on I', in the sense of ( n 1)dimensional measure.

-

213

DIFFRACTION P R O B L E M S

Let us show now that if, in addition to conditions (16.6) and (16.9), we assume that max 1 f 1 is bounded, then the product uC will 9

belong t o C,,,(K,n Qk), for k = 1, 2. We first show that 1 V u l i s bounded, assuming that we know M = essential max j u 1. KP

For the function E in (16.5), we take the function d?,v/d.w,, for s < n - 1, where ~ ( x is) at the moment a sufficiently smooth function that is of compact support in K,. When we substitute it into (16.5) and integrate the first t e r m by parts, we get

Let u s take for K , a sphere that is concentric with the sphere K , chosen above [for which inequality (16.12) has been proven] but with a sufficiently small radius that the derivatives uxix, are squaresummable in the new sphere K,. Then, wemay take for qs in (16.13) an arbitrary function in &;(K,). In (16.13), let us set ys=uxsbrC2, where s , < r t - 1 and C(x)is a smooth function with values between 0 and 1 that is of compact support on K,, and

where

6 is a positive number not exceeding unity (which we shall choose

sufficiently small in what follows), and Nis a large positive number, which we shall let approach a. On the basis of the conditions (16.13), we have [uxs]I = 0 for s ,< n.- 1 and r,

This, together with the fact that u ( x ) belongs to W i ( K ,n 8,) for k = 0 1

1, 2, ensures that -qsbelongs to W 2(K,).Let us substitute this function

214

LINEAR EQUATIONS

q,, into (16.13) and sum the resulting equations with respect to s from s = 1 to s = t z 1. The principal terms generate the following two terms:

-

Let us represent the expression

in the form

1 2

x n

paijrbr-'bxf6x,C2-

In the equation that we a r e considering, let us transpose to the right side all terms except

Let us find a bound for this expressionfrom below and let us find a bound for the terms on the rightfromabove. In this way, we arrive at the inequality

n-1

n

(16.14)

DIFFRACTION P R O B L E M S

21 5

Let us use Eq. (16.1) to express the derivative u ~ , in . ~t e~r m ~ s of the derivatives uxsxf, where s ,< n - 1 and i ,
in the t e r m s containing I Vb I since, as follows from the definition of b ( x ) (we recall that 6 ,< l),the inequality

is valid, where v and 1-1 a r e the v and I-( in (16.6). By virtue of all this, from (16.14) and (16.1), we derive the fol1owi:ig chain of inequalities:

n-1

n

216

L I N E A R EQUATIONS

Here, E is an arbitrary positive number. For future use, we take only r , < r o = [ n / 2 ] numbers 6 and E sufficiently small so that

-1.

We choose the

Then, after collecting similar t e r m s in (16.15), we have

Using Eq. (16.1) for the derivatives uxnxn,we see that this representation of uXnxnand inequalities (16.17) yield the inequalities that we need:

21 7

DIFFRACTION P R O B L E M S

The constants c and cI that appear in inequalities (16.14)-(16.18) a r e constants determined by quantities known to u s and a r e independent of the number N that appears in the definition of the function b ( x ) . In addition to (16.18), we need yet another inequality, one that will be valid for an arbitrary function u ( x ) E W : (K,n Q*), for k = 1, 2, such that essential max 1 u I = M that satisfies on

rl conditions

<

03

KP

(16.3):

1

(16.19)

To prove inequality (16.19), consider the integral

where n- 1

n

and, a s before, b ( x ) is equal to

Let u s regard J as the sum of integrals over the regions K, r l Q, and I<, n S!, in each of which we integrate by parts, transferring the

218

derivative

LINEAR EQUATIONS

d/dxi

with uX1to the remaining factors. This yields

The surface integrals over ,'I that result from this cancel each other out. The integral over the boundary Upvanishesby virtue of the factor 1. Let u s use Cauchy's inequality in the form (1.2) of Chapter 2, the boundedness of

and the equivalence of the quantities

that is, the validity of the inequalities

to find a bound for the right-hand member of Eq. (16.20). In using Cauchy's inequality (12), Chapter 2 , to bound the right-hand member of Eq. (16.20), we need to assign small E to t e r m s of the form I V u I'vb'Cz and t e r m s equivalent to them. h this way, we obtain from (16.20)

21 9

D I F F R A C T I O N PROBLEMS

+El

Vb12i2+

vu126'+'C2+;b~-lI

+ 1 v u pvb'c2+ E

1

; vb'

I

I vc (2

dx.

If we choose E sufficiently small and collect similar t e r m s , we obtain (16.19). Inequalities (16.18) and (16.19) make it possible to determine that the integrals in the left-hand members of (16.18) and (16.19) a r e uniformly bounded both with respect to the number N in the definition of b ( x ) and with respect to u ( x ) if we know that II u II w;(Kp) ,= PI < 03. To see this, suppose that

..,

Let us construct a sequence of spheres UP,, where P , ~= p/2 + p/2"', for rn = 1, 2, that a r e concentric with Up. For r = 0, inequality (16.19) yields a bound for the integral

and hence for

in t e r m s of

that is, in t e r m s of p1 and

p.

2 20

LINEAR EQUATIONS

For r = 1 , inequaltiy (16.18) gives a bound for the integral

in t e r m s of

but, for arbitrarily large N , this last expression does not exceed

which is a quantity that is known to us. Thus, in particular, the integral

is finite and bounded from above by a known constant. By considering in succession the inequalities (16.18) and (16.19) for r = 1, 2, ro and letting N approachco, we can prove that the integrals

...,

a r e bounded. The constant c ( r o , l / p ) depends on r(, and l / p , the constants in (16.6) and (16.9), max I f 1 and IIUII~;(~,). K.

Let us now prove the boundLdness and Holderness of the tangential derivatives ux, for s ,< n - 1. In the identity (16.13), we set qs (x) = C2 (x) . max ( u X s(x) - k . 01 where k is an arbitrary number and :(x)is a smooth functionwithvaluesbetween0 and 1 that is of compact support on K,. We now take the sphere K,arbitrary though situated within that sphere for which the boundedness of the integrals (1.21) is established. Let us denote by A k , the set of points x in K,

22 1

DIFFRACTION PROBLEMS

at which U , , ~ ( X ) > k. Of course, this set depends on s but, for brevity, we omit the subscript s. If we substitute the function qs into (16.13) and make the elementary estimates, a s we have done several times before, we obtain

+ c ( J (1 + I vu

12)”

dx)’

rnes

,-I p

dk,p.

Ak,p

If we choose p so that 1/p <2/n, that is, so that 2p > nand remember that inequality (16.21) implies 1-summability of I V u I, where 1 = 2r, = 4 = 2 [ n/2] = 2 > n, it follows from inequality (16.22) that A

/I

(uxs - k)* I VC 1’ d x -t

Vuxs (‘C2 d x .
k, P

(16.23)

k. P

Analogous inequalities hold for the s e t s Bk,?, where ux, < k . On the basis of Lemma 5.4 and Theorem 6.1of Chapter 2, we conclude from these inequalities that max Iux,,I is bounded and uXs satisfies a Halder condition in an arbitrary sphere Up,, situated in the interior of the sphere Kp, - for which inequality (16.21) has been estab2

lished. It remains to show that lux,( i s bounded and that u,, satisfies a Halder condition in K,. n &, and K,. n 51,. Let us take an arbitrary sphere K,, in which the quantities max I uxs[ and 1 uxs. 1 Kp:

for s < n

- 1 , are known.

K,

If we set k = min uxs in (16.23) and assume KP

that the spheres K, to which (16.23) applies belong toK,,, we conclude from (16.23) on the basis of the Halderness of uxs that

II

vuxs12 C2dX 4 c2 [P

n-2+2a +

PR

(1

1

--p)Is

c3pn-~+~

KP

where p=min(2a.

2--

“1 > o .

P

From these inequalities and Eq. (16.1), it follows that n

(16.24)

222

L I N E A R EQUATIONS

From inequalities (16.24), which a r e valid for arbitrary spheres K, contained in Kp, of arbitrarily small radius p (as Lemma 4.1 of Chapter 2 asserts), it follows that Iuxn!i s bounded and ux, satisfies a H6lder condition in a sphere Kp-that is concentric with K,,, where p” < p’. In this way, we study the behavior of the derivatives uxi close to the surface of discontinuityl’,. Inessentially the same way, we can study the behavior of uXi in a neighborhood of the intersection n S. Let us formulate the result that we have obtained a s Theorem 16.2. Swpose that 8 i s the union of Q,, . . ., 8,and the boundaries I?,, separating them, where I, j , = 1, t. Suppose that the boundary S of the entire region 8 satisfies condition ( A )and that the coefficients in L. p ( x ) , and f (x) satisfy the conditions

...,

and

Then, a generalized solution u ( x ) of the problem (16.1), (16.3) in the class W;(Q) belongs to the classes Co.m(G)and C?,l ( Q R i , where k = 1, t. I f the ri, are surfaces in the class C,, ,, then u ( x ) has first derivatives that are continuous in the sense of Hzlder up to with the possible exception of points of junction of two OY more of the surfaces I?ii and S. It satisfies all conditions of the problem in the form (16.1), (16.3) and i s a classical solution of the problem (16.1), (16.3). If the boundary S belongs to the class C,,,(resp. the class C2.p), then ii ( x ) has continuous first (resp, second) derivatives in the sense of HElder up to S except possibly at points of junction of S and P l i .

...,

This theorem answers the question a s to the existence of a solution of the problem (16.1), (16.3) in its classical formulation. The generalized formulation of the diffraction problems for equations of various types (expounded in the present section for the elliptic case), the possibility of reducing these problems to the problem of finding generalized solutions “with finite energy integral” (in the present case, generalized solutions in the space W: (Q)) for a single equation with discontinuous coefficients, the consequent applicability of methods and results regarding the existence, evaluation, and investigation of these generalized solutions with finite energy integral may be found in [73] by one of the authors of the present book. Reference [73] also contains a proof of the applicability of the method of finite differences to the actual finding of solutions of

THE CASE OF T W O INDEPENDENT VARIABLES

223

these problems. Furthermore, in that article and in others by the same author (cf., for example, [74, 75]), it is shown how one can investigate the differentiability properties of generalized solutions of general elliptic equations and, in particular, diffraction problems in the spaces W:(&), where 1 2. (These investigations a r e carried out in detail in 176, 771.) However, it was noted there that the results obtained by this method regarding the classical solvability of the problem (16.1), (16.3) for arbitrary n are quite crude in the elliptic (and parabolic) case: F o r us to be sure on this basis that the problem (16.1), (16.3) has a classical solution, we need to require that the coefficients in the operator L and the surfaces,,'l have bounded derivatives of order at least [n/2] + 3. The methods that we have expounded for investigating generalized solutions of the problem (16.1), (16.3) in the spaces Cl,@ a r e considerably finer. We mentioned their applicability to diffraction problems for elliptic and parabolic equations when we first constructed these methods in the reports [5], [15], etc. We note that, up to the time of publication of the article [ 731, a thorough study of diffraction problems had been made only for equations with constant coefficients (we are referring to the case n > 2), and the solutions of these equations had been sought in the form of potentials or infinite series. In this equation, so-called weighted integral equations were obtained to determine the densities of the potentials. (In the more fortunate cases, these integral equations were reduced to Fredholm equations of the second kind.) Infinite algebraic systems were obtained to determine the coefficients in the series. A proposition similar to Theorem 16.2 was also proven in a recent article [go]. The method of proof is different from the one given here.

>

17. THE CASE OF TWO INDEPENDENT VARIABLES The results expounded in the preceding sections a r e valid for an arbitrary number of independent variables, including the case ri = 2. However, certain things a r e true of the case n = 2 that a r e not true for ti > 2. Some of these were pointed out in Section 2 of Chapter 1. Among these a r e the following: Suppose that u (x) is a solution, in a region 51, of the equation (17.1)

the coefficients in which satisfy the conditions

2 24

LINEAR EQUATIONS

If the solution u vanishes on the boundary of the region Q, then the norm 11 u )I w;(p) for it is bounded from above by a constant depending only on the constants Y and p in (17.2) and (17.3) and the norms I l f / l L , ( Q ) and I I U I I ~ , ( ~ )This ~ fact was established by Bernstein [22] for solutions of Eqs. (17.1) with ai = a = 0 in a circle. Combining the device of preliminary transformation of Eq. (17.1) given by Bernstein with the device of transformation of the contour integral resulting from integration by parts and the bound for it given by one of the authors [55, 101, we shall now prove the assertion made for an arbitrary region Q. Furthermore, we shall weaken as much a s possible the assumptions regarding a, and a. Specifically, we replace condition (17.3) by the assumptions ;iailILq(Q).

llaltL2(Q~
<

00,

4

> 2.

(17.4)

We impose the same restrictions on the bounday S of the region case with n >/ 2. Under these conditions, we have Theorem 17.1. S w p o s e that a function u ( x ) belongs to the class W:, o(Q) and satisfies Eq, (17.1) almost everywhere. Suppose also that the coefficients in this equation satisfy conditions (17.2) and (27.4). Then,

8 as we did in Section 8 for the general

(17.5)

where the coefficients c and c, are determined by the constants and p in (17.2) and (17.4) and by the boundary S , Let us rewrite Eq. (17.1) in the form a..u 'l

xi"/

=F(x),

v

(17.6)

where F ( x ) E f ( x ) - a i U . r1 - U U .

We multiply Eq. (17.6) by

u x , x , / a 2 zand

write the result in the form (17.7)

On the basis of (17.2), the left-hand member of (17.7) will obviously be no l e s s than I ( u : , ~+ , uz,,,), and the absolute value of the Ir

T H E C A S E OF TWO I N D E P E N D E N T V A R I A B L E S

225

first t e r m on the right will not exceed the sum

Therefore, it follows from (17.7) that (17.8) Consider the integral J1

=

J

P

(%,XI

ux7xx,- u:,$

%%*

(17.9)

Bernstein showed [ 221 that this integral is nonnegative for arbitrary u in C,(Sz> equal to 0 on Sif 0 is a disk. Let us show that the following more general proposition holds: l e m m a 17.1. For an arbitrary region Q and arbitrayr u in

w:,o(Q) 9

(17.10)

where y, = o (y,) is the equation for S in a local coordinate system with the y,-axis directed along the tangent to S at the point y , = y , = 0 and with the y,-axis directed along the outer (relative to Q) normal to S at the same point. In the special case in which Q is a convex region ,

and hence J, 0. Formula (17.10) is a special case of the more general proposition (8.19) proven in Section 8. Specifically, 1, can be put in the form

and, by twice integrating by parts, we can reduce this integral, just as was done in Section 8, to the form

226

LINEAR EQUATIONS

from which it is clear that 2 4 coincides with the integral

in Section 8, which was evaluated for alj=S{. Therefore, Eq. (8.19) or, what amounts to the same thing (17.10), is valid for it. By hypothesis ,

(17.12)

so that

Let u s integrate both sides of inequality (17.8) over 5;! and apply (17.13):

Analogously, we can prove that

It follows from (17.14) and (17.15) that

Inequality (17.5) is derived from this inequality just a s w a s done in the general case in Section 8. Thus, we see that inequalities (17.5), obtained for an elliptic operator of the general form (17.1) with n = 2, reduce essentially to the analogous problem for the Laplacian operator. This reduction is done by representing Eq. (17.1) in the form (17.7) and shifting from (17.7) to inequality (17.8), which contains no variable coefficients a,) ( x ) .

227

THE CASE O F T W O INDEPENDENT VARIABLES

This device, discovered by Bernstein, enables us to obtain the desired bounds for u ( x ) in the case n = 2 without differentiating the coefficients a i j ( x ) . We shall use it below to obtain a prion*bounds for max I V u ( x ) l and lull, a, Q.At the moment, we note a special case u

of inequality (17.16). Corollary 17.1. I f 9 is a convex region and

u E W”,o(Q),

then

where the constants v and I-( are the v and I-( in inequality (17.2). If the boundary values of II ( x ) coincide with the boundary values of a known function y ( x ) E W i ( Q ) ,then, by applying inequality (17.5) to the function ‘u ( x ) = u ( x ) -9 ( x ) , we obtain the following inequality regarding u (x):

11’ 11

I/? 11 W;(Q),)+

c ( ~ ~ f ~ ~ L 2+ (S?)

$(Q)

‘1

~ ~ “ ~ ~ ~ 2 ( Q ) *

(17.18)

If we make no assumptions regarding the boundary value of u and the boundary S, we can find a bound for the norm llull v;(n,,for arbitrary 8 ’ c Q instead of the norm lIuIJ w;,p), and we find it not in t e r m s of llu]lL,(g)but in t e r m s of /lull W ; ( R ) . To do this, we multiply (17.8) by E2(x), where E(x) is a smooth function of compact support with values between 0 and 1 in the region 9, so that

The integral

becomes, after twice integrating by parts, w

J* =2

J (-

e

so that

ux,uxf12E€x,

+

ux,ux,x2EExJd x ,

2 28

where

L I N E A R EQUAT IONS E

is an arbitrary positive number. Therefore,

If we combine this inequality with an analogous inequality for the derivatives u ~ , and ~ , uxs2 and choose E sufficiently small, we obtain, after some manipulations, 2

(17.20) where the constant c1 depends on v and

p

and also on max(VE1. P

W e recall that we can find a bound for the norms (ulrr,P, where a > 0, and IIUII w;(8)for arbitrary finite q in t e r m s of the norm

I1lJ /I wz”

@).

Remark: Inequality (17.5) i s valid not only for the first boundary condition u I s = 0 but also for any other “regular boundary condition,” that is, any condition of the form

($+ au) Is = 0, where ard

denotes differentiation in any direction that varies in a differentiable manner from one point on S to another and a ( s ) i s a given function that is bounded on S. The outline of the proof of (17.5) is the same in this case a s with the condition u I s = 0. The basic analytic portion of it consists in proving that the integral (17.9), which, as we saw, reduces t o the boundary integral (17.11), is still transformed under other “regular boundary conditions” to integrals over S that do not contain deriatives of u of o r d e r s higher than the first. The same is true of the considerably more general transformations of boundary integrals that arise in the study of the derivatives of higher o r d e r s (cf. [ll]). Similar r e m a r k s hold for other boundary conditions and for the bounds that we obtain below for H6lder norms. Let u s now turn to the bounds for max I uxl 1 and ) u 11, ~, corresponding to the solutions of Eqs. (17.1). ’ A bound for the HGlder constant for the derivatives uxi, where 1 = 1, 2, was given by Nirenberg [28] with the aid of an extremely simple device (which we shall describe in Section 6, Chapter 9). In a much more complicated manner, a bound for max lVul has been exhibited by introducing quasiconformal transformations and special representations of solutions of two-dimensional elliptic equations

THE CASE OF T W O INDEPENDENT V A R I A B L E S

229

(see B e r s and Nirenberg [29, 301). Bounds for I u ( , , . also appear in the book [78] by Vekua. A l l these techniques a r e strictly twodimensional. We shall derive all these bounds with the aid of our method, expounded above for arbitrary n > 2. We shall also weaken the assumptions regarding the function f ( x ) and the coefficients of the operator L to as great extent as possible. (We note that the possibility of such weakening of the assumptions w a s noted by the authors of the articles [28-391.) Without loss of generality, we can confine ourselves to the case in which u E C,(G)o A s one can easily show, the bounds obtained below a r e valid for arbitrary u ( x ) in W i @). Suppose that the coefficients a,,( x ) satisfy inequalities (17.2) and that

11%

a, fllr,(Q)<-P’

(17.21)

9>2.

Let u s begin with a bound for [ uXl(.,*, assuming that we know the number M,=maxlVuJ. Let us denote the derivative 9

rewrite inequality (17.8) in the form

1 V P , l2

sc (F2 +

d , X ,

- ux,x, G,d.

u,,

by p1 and let us (17.22)

where c is a known constant that depends only on and v in (17.2). Let Ah, denote the set of points x in an arbitrary circle K , at which pl(x) > k. If K , is not contained entirely in Q , then let u s choose A k , p contained in K , n G and let u s choose a number k exceeding max p l . We multiply (17.22) by E2 ( x ) , where E(x) is a smooth funcKpns

tion of compact support with values between 0 and 1 on K,, and let u s integrate the result over dk,,.This yields (17.23)

where

By virtue of the assumptions (17.21) and the boundedness of and )Vul, we have the following inequality for I , (we recall that 0 < E < 1 everywhere):

1u(

(17.24)

230

LINEAR

EQUATIONS

Let u s t r a n s f o r m the integral I , by twice integrating by p a r t s a s follows: (17.25) 'k, p

Let u s solve for uXs, in Eq. (17.1) and substitute t h i s expression f o r it into (17.25). Then, w e c a n f i n d a bound for 1121 by using (17.21) and H6lder's inequality:

F

k ) €I V € I [ \ z -

V2IS 2 J ( P 1 -

2

2P ' X , - x

=I I

PlX,

If

*ks p

+IPI,,I]dx

4;

PlvPllZ~2dx+ p

(17.26)

+ c1 j' (pl - k ) 2 1 ~ ~ 1 2 d+x c1 rnesl-7 A ~ , 2

@.

P

If we substitute (17.24) and (17.26) into (17.23), we obtain the inequalities

J IVp,12i2dx

,y

I(P1--h)2

2

IV~12dx$-mes1-4Ak.

p

p

I

*

(17.27)

in which the constant y is determined by the constants v and p in conditions (17.2) and (16.21) and by the quantity maxlu. VuI. An0

alogous inequalities can be proven f o r the function

--

T h e r e f o r e , the functionp, belongs to t h e c l a s s B 2 ( Q , M ,y,

p l = -ur,. 00.

-1, 1

q

and

the conclusion of Theorem 6.1 of Chapter 2 holds for it; in other words, we can find bounds for the n o r m s Ipl(,.9 , , where Q ' c S 2 , f o r s o m e a > 0 in t e r m s of quantities assumed known. We can find in an analogous manner. bounds f o r 1 f i x ,la. 9,, where Q'cQ, Thus, we have proven Theorem 17.2. Suppose that u ( x ) is a solution of (17.1) in W;(Q)n O1@) that satisfies conditions (17.2) and (17.21). Then, the norms u x i J ae, , , where Q'cQ,f o r I = 1 , 2, with some a > 0 , are

I

finite and bounded from above in t e r m s of M = max I11 I, MI = max 1 Vu I, 9 Q the distance f r o m Q' to S , and the constants v, p , and q in (17.2) and (17.21).

THE CASE OF T W O INDEPENDENT VARIABLES

u Is,

23 1

If we assume some regularity of the boundary S and the values we can find a bound for I uxi I

U.

e

for all

a.

Without loss of gen-

erality, let us assume that u Is = 0. Suppose that S E W ; , where q> 2. By introducing new coordinates y k = y, ( x ) E W;,where q > 2, let u s straighten out some portion of the boundary S, and let us consider at first only the derivative of u in a neighborhood of this portion along the direction tangential to it. Let S, be represented by the equation y, = 0 and let u s take uy,. On S,, the function p”, =uy, vanishes. F o r p”, and -pl, we derive inequalities (17.27) just a s above, also, if K , intersects S,, we only take k_>,O. From Theorem 7.1 of Chapter 2 , we derive a bound for I p , ( , close to S,. More precisely, for arbitrary K, at a positive distance from S\S, but situated close to S, (where the coordinates y , and y, a r e introduced), we have

-

0,

To find a bound for

,;

-

=osc [ p , , K , n Q ) ~
(p)

0,

(17.28)

us consider in-

equ_ality (17.27) for with k = Oand the corresponding inequality for --,with k = 0. Adding these twoand using (17.28), we see that

J 1 vp,

lE2 d x

< cp28.

KpnQ

where p=min(a,

I

-, 2, 4

and hence (17.29)

It follows from this and Eq. (17.1) as expressed in the coordinates y, and y, that (17.30)

On the basis of Lemma 4.2 of Chapter 2 , it follows from these inequalities that the derivative ; , z u y , also satisfies a H6lder condition and that

%(P)

< C3PP.

(17.31)

232

L I N E A R EQUATIONS

where c3, like c1 in (17.28), is aknown constant. Inequalities (17.28) and (17.31) provide a bound for the oscillations of the derivatives uy, and uy,, and hence for u,, and uxl, close t o the portion S,. The transformation from the coordinates x , and x2 t o the coordinates y, and y2 must be reversible close to S , . Also, the derivatives of the coordinates in one system with respect to those in the other must be continuous in the sense of H6lder. On the basis of our assumption regarding S (namely, the assumption that S 6 W & ,where q > 2), such a transformation is possible in a neighborhood of an arbitrary point on the boundary and we can obtain a bound for I us,. 1 for the entire region Q . We have proven Theorem 17.3. Suppose that u ( x ) E W;,,(P)andsatisfiesEq, (17.1), for which conditions (17.2) and (17.21) are satisfied. Suppose that S E W&.where q > 2, and that the quantities M = essential rnax 1111 and Iul.,, n,

q

9

E C,, (Qh and the quantity 9 is bounded from above by a constant depending on v, /1 , and

MI = essential max ( V u1 are finite. Then, u

an conditions (17.2) and (17.21), on

II

M and M I , and

on the boundary

S O

Let us now see about finding a bound for max I V u I. Let us prove: Theorem 17.4. Suppose that a function II ( x ) E Wi(S!) satisfies Eq. (17.1), f o r which conditions (17.2) and (17.21) are satisfied, For arbitrary P c ~ the , quantity essential rnax I V u J is finite and 9’

bounded from above by a constant depending only on v, b , and q in (17.2) and (17.21), on the norm ll~Il,,+~~,,and on the distance from

to S. I f Sc W;,where q > 2, and i f u I s = 0 , then essential max(Tu1 Y is bounded from above by a constant that depends only on v, p , and q in (17.2) and (17,211, on the norm 11 u /I L2 (n), and on the boundary S. Qf

Proof: We first prove the first assertion of the theorem. On the basis of (17.21) and the injection Theorem 2.1 in Chapter 2, it follows from inequality (17.20) that, for arbitrary positive r and Q’CQ,

/I II ”;(9‘) 4 c ( r .

w.

(17.32)

where c ( r , W ) is a constant depending on r,Q’, I I U I I ~ ; ~ ~ )the , constants c and c1 in (17.20), and the constants 1-1 and q in (17.21)* In view of this and conditions (17.21), the function

F t f- aiu,, - au

is q‘-summable over 8 , where q’ is less than but arbitrarily close

to q, and

II FIILg, ( 9 ’ ),< CI(4’. Q’h

qf < 4.

(17.33)

T H E CASE O F TWO INDEPENDENT V A R I A B L E S

233

where c, (4'. Q') is a known constant, distinct in general from c ( r , Q') in (17.32), that increases without bound as Q'+ Q and q'+q. Inequality (17.23) w a s derived in an arbitrary sphere K p c Q with arbitrary k for p , . On the basis of (17.33), we have the following bounds for I , and I , : I,

2

< c (4'. 9') mes -1

4'

A,,,,

q'

< q.

(17.34)

and

(17.35) +c'

-

'k.

p

From these inequalities and inequality (17.23), we obtain

Since we can take q' arbitrarily close to q and since q > 2 , we can assume that q' in (17.36) is greater than 2. On the basis of Theorem 5.3 in Chapter 2, we obtain from (17.36) a bound for essential max pl ( x ) from above. Proceeding analogously for - pl, S'

we obtain a bound for essential minp,from below. In the same way, Q'

we obtain interior estimates for p,=ux,. This proves the f i r s t part of the theorem. To obtain a bound for I V u I close to the boundary, we proceed a s we did above in finding a bound for the oscillations of uZL in a neighborhood of S. Specifically, we introduce new regular coordinates y1 and y2 to straighten out some portion S, of the boundary S and we rewrite (17.1) in t e r m s of the coordinates y1 and y,. Assuming that the first two derivatives of x , ( y , , y z ) , for 1 = 1, 2, with respect t o yk belong to L,, where q > 2 , we obtain Eq. (17.1) for (YIS

Y2)

=p

(XI

(Yl. Yzh x!2 (Yl9

Y2))

the coefficients in that equation possessing the same properties (17.2) and (17.21) as do the coefficients in (17.1). (It is easy to see that boundedness of the derivatives dxl ( y ) / d y , implies that the

234

LINEAR EQUATIONS

derivatives d 2 x l ( y ) / d y kd y , belong to L,, where q > 2.) Let yo = 0 be the equation for the portion S , of the boundary. Then, uy,I, = 0. Suppose that K pintersects S along the portion S,. We denote by 3 uy, > k . Obviously, inthe set of points y in KPn D at -which equality (17.23) will be valid for p 1 provided we assume that everything is expressed in the coordinates y, and y,. For k >O, the integral I, =

A

j- (PTy, -plly,~y,y,)E2 d Y

(17.37)

tt* P

admits a representation of the type (17.25) since the contour integral

which results from integrating by parts twice, is equal to 0. Therefore, an inequality of the type (17.26) is valid for I , with k >/ 0. We can find a bound for the integral I , in the same way as we did in (17.34) for the interior spheres. Therefore, inequalities (17.36) for F, with k >/ 0 will also be valid for spheres K, intersecting S , (but not S\S,). They enable u s to give a bound for p",=uy, from above for regions adjacent to S , (cf. Theorem 5.3 in Chapter 2). Analogously, by considering the function -&, we obtain a bound for uy, from below and hence a bound for maxlu,, I for an arbitrary QI

region 51, adjacent to S, (though at a positive distance from S\S,). Inequalities (17.36) are also valid for the function p,ruYl-for all values of k. To see this, note that the integral of the type I , corresponding to it is of the form

When we transform it to the form Ak. p

we get the boundary integral j-E2[P"2y,(;z-k)COS(nl S

T H E CASE OF TWO INDEPENDENT V A R I A B L E S

235

which vanishes for arbitrary k. (Because of the factors cos(n, y,) and uY,,,,, the integrand vanishes on Sl.) A l l the other estimates and the considerations that lead to a bound for max luy,l a r e the same as those above for

91

ify,.

tain a bound for

Returning to the original coordinates, we ob-

1 Vul=

vu;,

+ u;,

close to S,. 1s this way, we have

a bound for l V u l in M. If all the conditions in the second part of Theorem 17.4 are satisfied, to find a bound for max IGuI, we do not I

need to know I l ~ l l ~ : ( since, ~) on the basis of Theorem 17.1, this and the known constants in norm is bounded in t e r m s of conditions (17.2) and (17.21). This completes the proof of the second part of Theorem 17.4. From Theorems 17.1-17.4, we get the followin Theorem 17.5. Suppose that a function u (x) E W 2 ( Q )and satisfies Eq, (17.1), f o r which conditions (17.2) and (17.21) a r e satisfied. Then, f o r an arbitrary interior subregion &' of the region 9 , the function u ( x ) E C,, @), f o r a > 0, and the quantity l u l l , I, 51, i s bounded by a constant that depends only on Y, p , and q in (17.2) and (17.211, the n o m I ~ U ~ ~ ~ ; ,and ~ , , the distance f r o m G' to S. The exponent a depends on the s a me quantities, If the boundary S c W;, f o r q > 2 , and i f I l l r = 0 , then 11 EC,,m(C!) and Iu 1 ,,I, i s bounded by a constant that depends only on the constants Y , p , and q in (17.2) and (17.21), on the norm I I I I I J ~ , ( ~ ) , and on the boundary S. We note again that although the bounds for max I V u / and 1 / 1 1 , , , were derived for sufficiently smooth u, they a r e also valid for arbitrary u in W i ( 9 ) . Tn fact, we used the smoothness of u ( x ) only in intermediary calculations when transforming the integral J , in (17.9) to the form (17.10) and the integrals I, in (17.23) to the form (17.25). In the final equations (17.10) and (17.25), all the integrals are meaningful and finite for an arbitrary function u (x) in W i (0).If we approximate the function u ( x ) in the norm of W i (8)by means of functions u , , ( x ) E C 3 ( G ) and use the lemmas of Section 3 of Chapter 2 , we see that the relations (17.10) and (17.25) a r e also valid for the limit function. Suppose that the conditions of Theorem 17.5 a r e satisfied. Then, Eq. (17.1) can be regarded a s the equation

F

121,

c

( x ) u , x = F ( x )=f ( x ) - a,u ' 1

- au

(17.39)

where F (x) L, (LA), with q > 2. -If we assume also that the coefficients a i , ( x ) a r e continuous in M, then, by using Schauder's gluing method (briefly described in Section 11) and the considerations mentioned in Sections 9 and 10, we can show that u ( x ) belongs to

2 36

LINEAR EQUATIONS

W : ( Q ' ) , where Q ' c 8 , for arbitrary behavior of u on S or belongs t o IV: (9)if there is a function y ( x ) agreeing on S with the boundary values of 11 ( x ) which belongs to Wg (8). The a priori bounds obtained in the present section enable u s

to investigate the solvability of the Dirichlet problem in convex regions for elliptic equations (17.40)

with arbitrary bounded coefficients a,, (x). The results given below are also valid for a n a r b i t r a r y simply connected region that is mapped onto a convex region by any nondegenerate transformation x i = x 1(y) possessing bounded first and second derivatives. Suppose that the domain of definition is already transformed into a convex region 8 over which the variables x1and x2 range and suppose that the conditions (17.2) and (17.4) are satisfied in this region for the coefficients of Eq. (17.40). Letf(x) denote a function belonging to L2(Q). It has been shown that under these assumptions, inequality (17.17) is valid for an arbitrary function v in W;,o(Q). It is also valid for the entire family of operators

Specifically, (17.42) We showed e a r l i e r (Sections 9 and 10) that the problem L ~ u ~ v A u f=,

(17.43)

UI,=O

has a unique solution in W : ( Q ) for arbitrary f in L 2 ( Q ) . This, together with (17.42), ensures the unique solution in W i ( Q ) of the problem L$Ef,

ul,=O,

TElO,

11,

(17.44)

f o r arbitrary f in L,@) (cf. Theorem 9.1). Consequently, the operator

THE CASE OF T W O INDEPENDENT VARIABLES

237

sets up a one-to-one correspondence between the complete spaces W2’.o 6 2)and LZ( Q ) . Let us now consider the complete equation (17.40) under the condition that (17.45)

ul,=O.

assuming that

Q

is convex. We transform this problem to the form (17.46)

u+Au=F,

where An L- (t?)-’ (aiu,,

+ au), and

F =(L?)-’ f.

Let us consider Eq. (17.46) in the space Wi,o(Q). In this space, the operator A is completely continuous. This is true because, if a sequence ( v , ~ ] for , m = 1 , 2, converges weakly in W,’(Q),it follows on the basis of the injection theorems that the functions { v , ~ \ converge strongly in C(&) and that their derivatives (w,~,,] converge strongly in L,(Q) for a r b i t r a r y r < 00. Fromthis and the assumptions (17.4), it follows that the functions

...,

wrn

for

m =

1, 2,

+avin.

aiV,nxL

...,converge in L2(Q). But then, the sequence

converges in W i ( Q ) . Thus, in Eq. (17.46), the operator -4 is completely continuous and therefore Eq. (17.46) is solvable for arbitrary F if and only if the corresponding homogeneous equation v+Av=O

(17.47)

possesses only the zero solution. This fact, together with Theorems 17.3 and 17.4, yields Theorem 17.6. Let 51 denote a convex region satisfying the conditions of Theorem 9.1 and suppose that conditions (17.2) and (27.4) are satisfied. Then the problem (17.40), (17.45) has a solution in Wi,,(Q) for arbitrary f in L , ~ ( O )i f and only i f the uniqueness theorem in the space w-;, ,,(S!) holds for it. I f f ( x ) and a ( x ) belong to L,(Q), where q > 2 , a solution n ( x ) of the problem (17.40), (17.45), i f it exists, belongs to Cl,=(a), where a > 0.

238

LINEAR EQUATIONS

Remark 1: If condition (17.2) is satisfied, if the a , and a belong to L,(Q), where q > 2 , and if S belongs to W i , where q > 2 , an arbitrary solution W i , o ( Q )of the equation Lu = 0 belongs toC,,.(H) Therefore, in this case, we need to prove the uniqueness theorem for (17.40), (17.45) only in the space W i , (51) n C1,a (6 Remark 2 : A s was noted above, this theorem is valid for arbitrary simply connected regions that can be mapped by a sufficiently smooth transformation (in Wf, for q > 2) onto any convex region. 18. TWO-DIMENSIONAL SADDLE-SHAPED SURFACES Consider elliptic equations of the form 2

2

1. j = 1

ulj(x)ur..,. r

/

=o

Tl8.1)

under the condition aljEl$

>0

(18.2)

where 151 # 0. The solutions of these equations possess a number of distinctive properties. In the f i r s t place, for an arbitrary function u that satisfies Eq. (18.1) in some region &, i t s greatest and smallest values in lie on the boundary S of 8 ; that is, for x G, m i n u ( x ) 4 u ( x ) ,< max u ( x ) .

xcs

XCS

(18.3)

Here, the function u is assumed to be continuous in and twice differentiable at every point in 8. Let u s prove, for example, the right-hand inequality in (18.3). Let u s suppose that this inequality is untrue, that i s , that u attains its greatest value at some pointxo in the region 8. Then, the function v ( x ) = 11 ( x )

+

E

I x-x"IZ

will also,- for sufficiently small positive E, attain its maximum value in 0 at some point x' in the region 9. But this is impossible

since the inequality

alp,.,

1 J

i .0,

must hold at that point whereas, on the basis of (18.1), aipxix,

= ai jU,rlxJ

+

L

aij2~81=

2~

2 all > 0.

i=1

TWO-DIMENSIONAL

SADDLE-SHAPED

SU R FA C ES

239

This contradiction proves that u (x)

.< inax I I (x). XE s

The first of inequalities (18.3) is proven analogously. A second peculiarity of solutions ic of equations of the form (18.1) is the fact that, for strictly convex regions Q , the quantity max lVul is determined only by the boundary values of u (specifP

ically, it is equal to max I'p' (9. (9" @)I, S

where 'p(s)= u ( x ) l x , s c s ) and is independent of the coefficients a f j . Let u s formulate the conditions regarding the boundary values of (s) of the solution of u ( x ) in the following form: Suppose that the normals to the planes in the Euclidean space (x,, x 2 , u ) that pass through three arbitrary points P,, P2, and P,of the boundary curve

form angles a with the u - a x i s such that ltan a1 does not exceed some constant M,. Let u s assume this is also true of the planes representing the limiting positions of such planes a s any pair of the points P, or all three points P,. P2,and P3 approach coincidence. W e also assume that the regionS1 is strictly convex. This condition, known in the literature as the "three-point condition," ensures that rnaxlVul< M, S

(18.4)

for any solution u of Eq. (18.1) that coincides with 'p on S. To see this, let u s take an arbitrary point xoon the curve 1 and consider the tangent to that curve at xo. Let u s pass through this tangent two planes II+ (xn) and II- (x") the normals to which make angles a* with the u - a x i s and which satisfy the condition

I tan a+ I = M,. By virtue of our hypothesis, the curve I lies between the planes I' (x") and II- (x"). More precisely, for points ( x , u ) on the curve 1, the ordinate of u lies between the ordinates of u corresponding to points on II* (xo) with the same abscissa x . But then, this will also

be true for the entire surface u = u (x), where x f n. This is true because otherwise there would be a subregion &, of the region 52 on the boundary of which the solution II would coincide with a linear function of the form u = f ( x ) 6,~( X , - X ? )

+b2( x 2 - x!$ +'7 (x"),

240

LINEAR EQUATIONS

and would be either greater o r less thanthat function inside 9,. But this is impossible since the functionf @)satisfiesEq. (18.1). Therefore, the difference f (A,)-e (.v) is a solution of the same equation, one that vanishes on the boundary of the region Q,. By virtue of property (18.3) proven above, such a solution must vanish identically in It follows from what we have shown that the inequality I u ( x )- u

(x0)

I

IX-XOI

<

MI.

is valid for points x E P and if the function u is differentiable in G , then1V u 1 does not exceed MI at the point x". This completes the proof of inequality (18.4). Let u s show that this is also true for the entire region, that i s , that inax I Vu I 51

.< MI.

(18.5)

The surfaces u = u ( x ) , where x E s, corresponding to solutions of Eq. (18.1) are so-called saddle-shaped surfaces. They possess the following properties: (1) No plane can cut from them a cap that is projected inside Q. This means that, for an arbitrary linear function u(x)

f ( 4=blx,

+ bzx, +

63

and an arbitrary solution u ( x ) of Eq. (18.1), the closed regions of values of x corresponding to u ( x ) >f ( x ) and u ( x ) < f ( x ) either are empty o r have intersections with the boundary of 9. (2) If a curve 1 bounding a portion of the surface u = u ( x ) for x E a, where u ( x ) is a solution of (18.1), satisfies the three-point condition stated above, then an arbitrary surface u ( x )= f ( x )

b1x1

+ 62x2 +b3

with slope exceeding M I (that is, such that the absolute value of the tangent of the angle between the normal to the surface and the uaxis exceeds M,) and with at least one point (xo,-uo), where x E Q , in common with the surface 11 = u ( x ) , where x E Q , intersects the boundary S at two distinct points, partitioning it into two arcs, on one of which u ( x ) > f(x) and on the other u ( x ) < f(x). The two regions corresponding to u ( x ) > f ( x ) and u ( x ) < f ( x )are each simply connected. It is in the second of these two properties of saddle-shaped surfaces (or, what amounts to the same thing, the second property together with the first property for planes with slope exceeding M,)

TWO-DIMENSIONAL

SADDLE-SHAPED

24 1

SURFACES

that we are interested. Let us prove it. We denote by G' the set of points x in s at which u ( x ) >f(x). Tt contains the point x". If the distance from M' to S were positive, the set a:, where u ( x ) f ( x ) - E , would (because of the continuity of u ( x ) andf ( x ) ) also not have points in common with S for sufficiently small positive E. But the function u ( x ) - f ( x ) + ~ is -.a solution of Eq. (18.1) that vanishes on the boundary of the region !&. Consequently, it is equal to 0 throughout But this is impossible since the difference u (xu)-f (9) E=E is positive at the point xi'. This contradiction shows that &' has at least one point in common with S. H e r e , we might have the following possibilities: (a) The boundary of P' has two points in common with J and these two points partition S into two open a r c s , on one of which u ( x ) -f ( x ) > 0 and on the other 11 ( x ) - f ( x ) < 0. (b) The boundary 9' has one point x' in common with S , with u ( x ' ) - f ( x ' ) = O , and u ( x ) - f l x ) < 0 everywhere else on S. (c) The boundary of i2' has one point x' in common with S and u ( x ) - f ( x ) > 0 at all other points on S. These are the only possibilities since, by virtue of the threepoint condition, the plane u = f ( x )cannot have more than two points in common with 1 . Case (a) represents what we wish to prove. Let u s show that case (b) is impossible. If this situation did exist, the function ti ( x ) would be equal tof ( x ) in and would be less than f(x) in Let us rotate the plane u=f(x)about the straight line lying in that plane passing through the point (x". u"), where uu=u (xn)=f (xn), that is parallel t o the tangent to S at the point x'. If we take the angle of rotation 0 sufficiently small and perform the rotation in the direction corresponding to increase in the ordinate u of the plane at the point x ' , then, a s is easily seen, the inequality f e ( x )> u ( x ) holds for the rotated plane u =fe(x), where xES. But, as was shown at the beginning of the proof of property (2), this is incompatible with the condition that f R (xn)- ii ( x " ) = 0. Case (c) is also impossible since we can reduce it to case (b) by switching from the functions u ( x ) and f ( x ) to the functions - u ( x ) and - f ( x ) . Thus, we have shown that an arbitrary plane ti == f ( x )possessing a common point (xu, u"), where xuE bl , with the surface u = tI ( x ) and possessing a slope exceeding kl, partitions S into two open portions such that u ( x ) > f(x) in one, u ( x ) <, f(x) in the other, and u ( x ) = / ( x ) at the two points defining the partition. Here, the region !b is partitioned into two openconnected sets: M,, in which u ( x ) > f ( x ) , and Q,, in which fi ( x ) < f (x). These two sets a r e separated by the closed set Q,, throughout which u ( x )= f ( x ) . The connectedness of the s e t s 8,and Q, follows from the fact that, if ti ( x , = f (1.)on the boundary of every connected region O ' C ! ~ ' , this equation is valid for the entire region 9'.

>

+

a:.

a\@.

a'

242

LINEAR EQUATIONS

Below, we shall need the following almost obvious topological fact. Let xOdenote any point in Q,[throughout which u ( x ) > f ( x ) ] and let x‘ and x” denote two points in B, [ thoughout which u ( x ) < f ( x ) ] . Let u s draw a broken line P between the points x’ and x”without leaving the region Q,. The vector connecting the point x‘lwith the point x a s x moves from x’ to d’along the broken line P rotates through some angle W. ( A s usual, in assigning values to 9, we treat the counterclockwise direction as positive.) Neither the sign nor the magnitude of this angle depends on the broken line chosen. It is easily understood that this is a consequence of the simple connectedness of the region Q,. Let u s now prove inequality (18.5). We assume the opposite. Then, there exist two points and 2 in D such that

x

(18.6)

-

x

Without loss of generality, we may assume that the points 2 and lie on an interval (0, a) of the x,-axis and that the end points of that interval belong to the boundary S. We are interested in the r_elgtive position of the straight line lv joining the points (x”, ;)and where

(x, u),

- -

- x=

x =(XI. 0) (XI.

0)

on the one hand, and the curve 1, formed by the intersection of the surface u = u (x) with the coordinate plane x 2 = 0 on the other. The equation of the curve 1, is u = u ( x , , O ) where 0 << x1,
wl-

(1) the interval XI < x, < contains a point xy at which the ordinate of the curve I , exceeds the ordinate of the straight line N

-

and the intervals 0 < x1 < XI and XI < x1< a contain points xiand x;’ at which the ordinate of the curve I , is less than the ordinate of the straight line N , or

TWO-Dl MENSIONAL SADDLE-SHAPED

243

SURFACES

(2) conversely, the ordinate of 1, at the point x','is less than the ordinate of N, and the ordinates of 1, at the points xi and x ; a r e greater than the ordinates of N , o r (3) one of the preceding two alternatives can be achieved by a small parallel displacement of N or by a small rotation [under which (18.6) is still satisfied or, more precisely, under which the tangent of the angle of inclination of the line N to the x,-axis remains greater than M,] and a subsequent parallel displacement of N . We leave to the reader the verification of this assertion. We shall prove, however, that the alternatives (1)and (2) a r e impossible for saddle-shaped surfaces. For example, let us take case (1). Let us draw through N any plane II: u = f ( x ) G b,x, b,x, + b3. It partitions Q into three subregions 8,. Q,, and M, such that u ( x ) > f ( x ) in PI, u ( x ) < f(x) in Q r , and u ( x ) = f ( x ) in 51,. The point ( x f , O ) E P,, and the points ( x i , 0) and ( x ; , 0) belong to Q,. Let w denote the angle through which the vector connecting the point(xy, 0) with (XI,0 rotates a s its terminal point moves along an arbitrary polygona line contained in 51, to the point ( x r , 0). This angle is equal to + z or -x. A s the plane Il is continuously rotated around the straight line N , the points (x!. 0), ( x i , 0), and ( x ; , 0) in the polygonal line connecting ( x i , 0) with (xy, 0) remain in regions of the type of P, [in which u ( x ) > f(x)] and 9, [in which u ( x ) < fcx)]. Therefore, the number w (which may be either 1c o r - 1c) remains unchanged. On the other hand, it is clear that the plane II can be drawn both in such a way that o = + x and in such a way that W = -z. This contradiction proves the impossibility of the first alternative. The impossibility of the second alternative is proven in the same way. Thus, we have proven the absurdity of the original assumption (18.6). We state these propositions in the form of Theorem 18.1. Let u ( x ) denote any solution of the elliptic equain Q, tion (18.1) that is continuous in G and twice di'erentiable where P is a strictly convex region. Suppose that the double inequality (18.3) holds f o r u ( x ) and the boundary S of 8. Suppose that the curve

+

1

+

I: ( x = s E S . u = ' p ( s ) J

(in xu-space) bounding the surface u =u ( x ) satisfies the threepoint condition with constant MI. Then, u ( x ) satisfies a Lipschitz condition in with constant M,. If, in addition, u ( x ) i s differentiable i n n , then rnax I V u I P

< M,.

This proposition is to be found in the articles [22, 561. The proof that we have given is due to von Neumann [56].