Chapter 11 Linear Elliptic Partial Differential Equations

Chapter 11

Linear Elliptic Partial Differential Equations 11.1 Introduction A standard classical argument ,for tackling the boundary value problem for elliptic partial differential equations is based on reformulating the equation as an integral equation by using a Green’s function and then invoking the theory of integral equations. While this argument has achieved considerable success, it is somewhat artificial to base the theory on the integral equation rather than on the differential equation itself, and recent theoretical developments show that a direct attack on the differential equation often yields more information, and at the same time avoids the tedious technical problems associated with the construction of the integral equation. The numerical solution is one context in which the advantages may be clearly seen, for the use of an integral equation does not fit in with standard numerical procedures, and is clearly unnatural. The purpose of this chapter is the introduction of the direct approach. The starting point for the analysis is the replacement of the original boundary value problem by a certain weak analogue. For illustration consider Poisson’s equation V’f - y = 0 (11.1.1) on a bounded open region R, a solution f~ W2(SZ) n %(a) vanishing on the boundary dQ being required. Multiplication of this equation by any 4 E %:(R) and integration gives ( 1 1.1.2) 283

284

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

and since %:(Q) IS dense in LZ2(Q), (11.1.1) and (11.1.2) are equivalent. Integration by parts yields (since 4 and its derivatives vanish on 8Q)

I f

V 2 4 dx

=

b

g+ dx.

(11.1.3)

Therefore (11.1.3) and (11.1.1)are equivalent for smooth$ However, (11.1.3) makes perfectly good sense for anyfin LY2(Q), although (11.1.1)cannot be recovered directly, and indeed does not have any obvious interpretation. This heuristic argument leads to the following weak version of the original problem: for given g find a function f with f = 0 on 8Q which satisfies (11.1.3)for all 4 E %?:(R). There are two attractive features of the weak problem which are immediately apparent. First, it makes sense for a wide class of right-hand sides, g, certainly for any g E LY2(Q). Second, since derivatives off do not appear in (11.1.3), the smoothness offis, at least initially, not such a pressing problem as it would be if the original equation were to be considered. In fact the discussion in the previous paragraph is rather imprecise, the statementf = 0 on aJz being meaningless for a general f~ Y2(Q), and in order to make sense of the boundary condition some smoothness must be required off, although not as much as in the original formulation. The key to success in tackling the weak boundary value problem (known as the generalized Dirichlet problem) is the choice of space. The problem fits naturally in a certain Sobolev space of functions satisfying relatively mild smoothness conditions, the fact that this space is a Hilbert space significantly simplifying the analysis. Sobolev spaces are now a basic tool in partial differential equation theory. Thus although for simplicity only a relatively simple example is considered, the homogeneous Dirichlet problem for a linear elliptic equation, the Sobolev space theory described has applications to both linear and nonlinear equations and equations of elliptic and evolution type. The advantages of posing the problem in a Sobolev space become apparent when existence and uniqueness for a general elliptic equation is tackled, for the problem may be formulated in terms of a bounded linear operator whose properties are relatively simple to study. In certain cases, for example for Poisson’s equation on a bounded domain, it is easy to show that the operator has a bounded inverse, and it follows that the generalized Dirichlet problem has exactly one solution for every reasonable right-hand side g. However, in general the homogeneous equation may have non-trivial solutions, and then existence for arbitrary g cannot be expected. The most that can be hoped for is existence and uniqueness if the homogeneous equation has only the zero solution, in other words a theorem analogous to the Fredholm alternative. Under certain conditions, the most important of which

11.2

NOTATION

285

is the boundedness of the domain, a certain related operator has a compact inverse and the required Alternative Theorem is readily deduced. It is interesting to note that compact operator theory, originally devised to treat the integral equation arising in the Green's function approach, is still the main tool needed in the direct method, where however it is applied to a "Green's operator" whose precise form need not be calculated. After a preparatory discussion of Sobolev spaces in Section 3, the proof of this Alternative Theorem will be the main business of this chapter, and will be tackled in Sections 4 and 5. The above argument yields criteria for the existence and uniqueness of solutions in a Sobolev space for an elliptic operator of order 2m. However, these solutions need not be in g2"(Q), and cannot therefore be regarded as solutions in the classical sense. A separate investigation is needed to determine when these solutions have their classical meaning, and some results in this area are given in Section 6. Agmon (1965) and Friedman (1969) are standard texts on linear partial differential equations, and the recent books by Folland (1976), Schechter (1977), Showalter (1977) and Treves (1975) are other useful references. Much of the known theory of linear and nonlinear second-order elliptic equations is contained in Gilbarg and Trudinger (1977).

11.2

Notation

The complexity of the notation presents something of a difficulty, and for convenience of reference some of the main conventions to be used are summarized below. We shall deal with partial differential equations on subsets R of R". R will always be an open set, will be its closure, and dR = 0'0 its boundary. Several spaces of functions will be needed. Since for the most part the functions will be defined on the standard domain R, in order to simplify the rather cumbersome notation, in this chapter the symbol R is omitted from the expressions for these spaces unless the domain is different from Q, Thus Y zwill denote Yz(Q). The functions will always be complex valued unless stated to the contrary. In addition to the spaces g mand V"'(n) of m times differentiable functions mapping C2 and respectively into @, frequent use is made of the space (Definition 1.3.23) of functions in %?"' with bounded support in R. Since R is open and a support is closed, it is easy to prove that given f E %: there is an E > 0 such that for every x E dR, f = 0 on S(x,E), the open ball centre x and radius E . Thus f vanishes at all points within a distance t of the boundary--

286

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

that is in a strip at the boundary. Note that %?: is dense in -Y2for 0 < m < 00 (Theorem 2.5.6). For a general partial differential equation in n dimensions the classical notation is extremely awkward. The complexity is considerably reduced by the use of multi-indices.

11.2.1 Definition. A multi-index a is an n-tuple ( a 1 , .. . , CI,,) of non-negative integers. We write I a1 = a1 + . .. + a,; this conflicts with the notation for the Euclidean distance in R",but the meaning will always be clear from the context. Multi-indices will be denoted by a and B.

1

A point in [w" will be x = (xl, . . . ,x,) with x 1' = Cs;, and xu = xy' . . . x:". We write Dj = a/axj and Da = D;' . . .D;". With these conventions the notation for a partial differential equation may be simplified by writing

c

PaD"

c c m

=

j=Oal+

IalSrn

Pz,..

. . . +z , , = j

0;' . . . 0;".

.1,,

Although the theory may be carried through under more general conditions, here the simplifying assumption that the coefficients are smooth will be made, and the operator will usually be written in divergence form.

11.2.2 Definition. Assume that for some a, p with la1 = 181 = m, p a p # 0, and that for all CI, p, p , E Vm(n).The pap are thus complex valued variable coefficients. For 4 E Wtrn define

14 =

c

iai,IPlGm

lp4 = (-

1)"

( - l)'a'Da(papDp4),

c

Ial=jpi=rn

(11.2.1)

D"(PapDB4).

1 is called a formal partial differential operator of order 2m. 1, is known as the principal part of 1.

As for formal ordinary differential operators the idea of a formal adjoint is needed. Suppose for the moment that n = 1 and R = (- 1,l).For 4, $ E g: an integration by parts gives

J-1

J-1

Note that the integrated term vanishes because 4, $ are zero near 30, this being itself a consequence of the assumption that 4, IC, E % .; Analogously, for the general case, repeated integration by parts gives for 4,t+b E %;-,

1 1.2

C

=

..

J* 4. r

( - l)l'l

lal,lbldm

=

287

NOTATION

(4, I**),

D"(j,,Da$) dx

say, where ( , ), is the inner product of Y 2(The . suffix zero will be used in this chapter for reasons which will soon become apparent). 11.2.3 Definition. The operator 1*, where

1*4 =

1

ja/,lS/Qm

(- l)'"lD"(p,,D'~)($ E %2m),

is called the formal adjoint of 1. is said to be formally self-adjoint iff 1 = I*. In order to obtain a well-posed boundary value problem for 1, some condition of ellipticity must be imposed, and the following will be used here. 11.2.4 Definition. 1 will be said to be strongly elliptic (on Q) iff there is a 0 such that

c >

Re( - l)mIp(O = Re jai=ifl/=m

5"p,,(x)tB 3 c I t 1'"

(X E

a)

+

for all 5 E R", where 15 1' = 5; . . . + ti. The term uniformly strongly elliptic is sometimes used for this condition in the literature. Evidently if 1 is strongly elliptic, so is l*. It is also worth noting that the condition of strong ellipticity is invariant under a change of coordinates with non-vanishing Jacobian (Problem 11.1). 11.2.5 Example. For 1 = -V2

-c D', n

=

1

n

-lp(t)= ti".Thus (minus) the

+

1

0:)in R2, 1 is strongly elliptic Laplacian is strongly elliptic. If 1 = -(xlD: on {(xl,x2) : x1 3 d} for d > 0, but not for d = 0. If l = D , - 0::( - l)mlp(t) = ti, but t2 = 5; 9: and the condition does not hold. The heat conduction equation is therefore not strongly elliptic.

+

288

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

11.3 Weak Derivatives and Sobolev Spaces In the classical theory of differential equations, it is usual to regard a function f as a solution only if all derivatives off appearing in the equation exist and are continuous. This prompts the following definition. 11.3.1. Definition. Let R be bounded and have a Vm boundaryt, and let 1 be the formal operator of Definition 11.2.2. For given g E %?, a function f is said to be a classical solution of If = g iff f E V 2 mand

lf-= 9

(1 1.3.1)

in R. f is said to be a classical solution of the homogeneous Dirichlet problem iff in addition f E g m P and

'(a)

ajf/avj

=

o

( j = 0, 1, . . . ,(m - 1))

(11.3.2)

on 80, where a/dv denotes differentiation in the direction normal to the boundary. The corresponding inhomogeneous Dirichlet problem may be expressed in the above form under mild restrictions on the boundary data, see Friedman (1969, p. 38). Following the tactics outlined in the introduction, the concept of a weak solution is now introduced. In view of Definition 11.2.3 of the formal adjoint 1*, a generalization of the argument leading to (11.1.3) yields the following. 11.3.2 Definition. For given g E z2, a function f solution of If = g iff ( f , 1*4)0= (9,410 for all I$ E W ,:

and we then write

If

g.

E

Y 2is said to be a weak ( 11.3.3)

(11.3.4)

No boundary conditions have been imposed on weak solutions, and these are therefore analogues of classical solutions of 1f = g rather than of classical solutions of the Dirichlet problem. We shall return to this point later, but first let us consider (11.3.4)further. Observe that it is not legitimate to interpret the left-hand side of (11.3.4) as a sum of ordinary derivatives since the existence in the usual sense of any

t

For the purposes of this chapter an intuitive understanding of this as a "very s m o o t h boundary will suffice. A precise definition would be on the following lines. For each point P of an, there are open sets S, c R"-', S, c an with 0 E S,, P E S,, and a bijection 4:s' + S, such that 4 E V"(S,), and such that the rank of the Jacobian of 4 is (n - 1) at all points of S , .

11.3

WEAK DERIVATIVES AND SOBOLEV SPACES

289

of the terms which would appear there is not guaranteed. With the eventual aim of assigning a meaning to these terms, we first show that it is sometimes possible to give a sensible interpretation to the derivatives of a function which is not smooth in the conventional sense. 11.3.3. Definition. A function f in 9 p (Definition 2.5.2) is said to have an xth weak derivative iff there is a g E Y E such that

b

g 4 dx = ( -

j n j . ( D a 4 )dx

for all I$ E %?.: g is called the ath weak derivative off, and we write D'f'

=

g.

The following remarks are intended to clarify the idea of a weak derivative. (i) Since 4,Da4 have compact support, both integrals in the definition exist. (ii) Weak derivatives are essentially an $P2 notion, and as usual in this context functions equal almost everywhere are identified. With this understanding weak derivatives are unique. For if g1 and g2 are both weak ath derivatives off,

Jabl - 9214 dx = 0

(4E

Now for any compact set S c Q, g, - g2 E T2(S), and %:(S) is dense in Y 2 ( S ) .It follows that g1 = g2 a.e. in S, and as S is arbitrary, a.e. in 0. (iii) If a function has an ath derivative g in the ordinary sense lying in then g is the weak ath derivative off. To see this simply consider the left-hand side in the definition and integrate by parts. (iv) The weak derivatives may be thought of as averaging out the discontinuities in f. However, note that f may have an ordinary derivative almost everywhere without having a weak derivative. For example iff(x) = 1 (x > 0), f(x) = 0 (x < 0), then

9y,

J:

f4'dx =

lo14'

dx

=

-

4(0),

and since there is no g E 9;' such that 4(0) = f t &I dx for all 4 E %,; f does not have a weak derivative. (v) In one dimension there is a simple characterization of functions with weak first derivative. They are just those functions which are absolutely continuous and have first derivative in 9:L (Problem 11.6). (vi) The averaging property ofthe weak derivative has the nice consequence that an exchange of order of differentiation is always permitted. For D,D@ =

290

11

DjD& when

Q, E % ,: and iff has weak derivative DiDJ, then

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

jRDiDjf.4dx=

I,

f.DjDi(Idx=

JR

f.DiDjQ,dx =JnDjDf.Q,dx,

whence DiDjf = D .D.f . ' (vii) Iff is a solution of If 2 g we cannot immediately conclude that each term in If can be interpreted as a weak derivative. Certainly this will follow from the definition if 1 has just one term, but the argument breaks down for general 1. We shall see in Section 6 that for strongly elliptic 1 the conclusion is broadly correct but non-trivial to prove. J.

The Hilbert spaces in which the analysis will be carried out are now introduced. The norm may be regarded as measuring the average value of the weak derivatives. 11.3.4 Definition. Let m be a non-negative integer. Denote by W" (or %"(Q)

if the domain requires emphasis) the set of functionsf such that for 0 < 1 CI 1 < m all the weak derivatives D*f exist and are in Y 2 ,and equip 2"with an inner product and norm as follows:

( f ,9)" =

,

1 I

,a,Sm

I,

Oaf

'

D"g dx, r

k"is known as a Sobolev space of order m. k" is a proper subset of the set of functions with mth weak derivatives, for the Oaf are required to be in Y2-and not just in 9:'Evidently . R o= Y 2 and ( , * )o, 1 are the inner product and norm of T2;for consistency with the higher order Sobolev spaces, in the notation in this chapter the suffix zero is retained. An obvious relation is

-

- 1l 0

%P c

11.3.5 Theorem.

. . . c .%"+'

c

.*"

c . .. c

8 0

=

9,

.Fmis a Hilbert space.

Proof: It is easy to check that 2'""is pre-Hilbert. To prove completeness, let ( f j ) be a Cauchy sequence in .k". Then for la1 < m,

11.3

29 1

WEAK DERIVATIVES AND SOBOLEV SPACES

Hence (D"fj)is Cauchy, and so convergent, in Y 2with limit f ( " ) say. Thus for 4E %?; (f'", D"$),= lim(f,,

=

( - 1)l":lim(D"f,,

Therefore f " ) has weak derivatives D"f'O' %'". Further

1

=f'")

4),= ( - l)c"((f 4)o. for / a / < m, and so is in '),

lD"f, - D"fo'/' dx, ilf, f ' " I1: = f ( ' ) in 11 1 ". This proves that (6)has a limit in X" and establishes -

lal
whencef, + completeness.

R

*

0

It is standard that the set V" of smooth functions is dense in Y 2 In . other words, if m = 0 the closure of %Tin em is R"itself. The following theorem, see Friedman (1969, p. 15), states that this is also true for arbitrary m, and gives an alternative description of a function in .A" as a limit of a sequence of smooth functions.

11.3.6 Theorem. The closure of gmin I/ *

/Im

is 3'".

The existence of weak solutions locally in X 2 " and so having 2m weak derivatives may be proved under minimal assumptions on 1, see Agmon (1965, p. 49). For such a solution the left-hand side of If Z' g may be written as a sum of weak derivatives. However, c X if !2 is bounded, and it is therefore evident that even smooth functions in X'" do not vanish on the boundary as is required in the classical homogenous Dirichlet problem. To obtain a weak analogue of this problem a modification of 8"is needed.

@"(a) '"

11.3.7 Definition. Let XV,m be the closure in X" of % .: Sobolev space of order m.

X;;' is also called a

As before there is a chain of inclusions Since functions in %?: vanish near an, functions in their closure %:' expected to behave at d!2 in a manner which reflects this fact.

may be

11.3.8 Example. As for %' I , in one dimension a simple description of X ,!, may be given. For R = ( - 1, l), by remark (v) above every f~ FA (c %' ') is . the definition, there absolutely continuous and has first derivative in Y 2 By is a sequence (4j)in % :' such that lim 11 $ j - f l l , = 0. Since 4 j has support in 0,

292

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

4jW

=

JIl

4j(Qdt,

and from Schwarz’s inequality

-

Hence 4j(x) -, {E I f ’ ( t ) dt in sup norm, and since q5j + f in // // and so in z2,it follows that f(x) = f ’ ( t ) dt a.e. Thus f ( - 1) = 0, and by a similar argument also f l ) = 0. The result in the opposite direction is easy (Problem if and only if it is absolutely continuous ll.S), and we conclude thatfE with .f‘ E z2and f vanishes at 1. 11.3.9 Lemma. Suppose that R is bounded and has smooth boundary. If n %m-1(!3), then djflzlvj = 0 on do for 0 < j d m - 1. On the other hand, i f f E and the normul derivatives aboce canish on 30, then f E F?.;

f E ~??;

%‘“(a)

Proof: See Friedman (1969, p. 39).

If m = 0 or R = R“ then %?: = F m ,b,it it is obvious from the lemma that in general these spaces are not equal. Except in one dimension a simple characterization of functions in ;%! is not usually possible. However, the lemma shows that a smooth function in f ; vanishes together with its first (m - 1) normal derivatives on dR. Since this property is just what is required of a solution of the classical homogeneous Dirichlet problem, a weak analogue of this problem may reasonably be posed as follows: find a solution of If 2 g in % .;‘ This is called the generalized Dirichlet problem, and its study is the primary object of this chapter. The resolution of this problem depends on the properties of Sobolev spaces which are established below. Suppose f is defined on R. For Open R’ 3 R define an extension of j to R’, which will also be denoted by f, by requiring that f = 0 on Q’\Q. We shall say that f is extended to R’ by zero. In general an f~ Fm(R) extended by zero to 0’ will not be in %“‘(!2’). In fact if such an extension of the characteristic function of R is considered, it is obvious that all that can be said is that it is in # ‘(R’). However, in view of the above remarks more may be expected of functions in x:(R). 11.3.10 Lemma. Suppose that 0,R’ are open and R’ =I R (i)

r f f ~H:(R)

is extended by zero to R’, then f E XT(R’). In this sense m;(R) c *K(R’).

1 1.3 WEAK

(ii) I f f

E

293

DERIVATIVES AND SOBOLEV SPACES

%ft(R')and f has compact support in R, then (the restriction to R of)

f E m;(Q).

Proof. (i) If (Cpj) is a sequence in V??(Q) with Cpj + f in 11 * ( I r n , the extension by zero of each Cpj to R' is in %?:(R'), and ( @ j ) is a Cauchy sequence in P:(Q'). The result follows immediately. (ii) is readily proved using a sequence of mollifiers and is left as an exercise (Problem 11.11). 0 Although the functions considered are usually defined on a proper subset of R", the properties of the Fourier transform (Theorem 2.6.1) may be usefully exploited if the functions are first extended to R" by zero. The following states that iff. .#T(R), f tends rapidly to zero at infinity. This is of course not in general true if merely f E .Xm(R).

Proof: For ~EV?;(Q) integration by parts gives (11.3.5). Now by Lemma and therefore Oaf E Y2(R"). 11.3.10the extension of a general f is in &':(Rfl), Hence for all Cp E %?:(Fin),

(W,$1,

I\-

=

( Y f4)" , = (- W f ,

W), = (-

1)'"'(f, =

m,

lRn (-

i ) ~ ~ ! < ~ f (di. t)&i)

(11.3.5) follows since %?,"(R") is dense in Y,(R"), and hence so is its image under the isometric isomorphismf -+ f.(11.3.6)is an immediate consequence of the definition of 11 (Im and the Parseval formula. The following theorem is the key to the resolution of the generalized Dirichlet problem, as it will enable us to show that the basic operator in the theory is compact. The restriction that R is bounded cannot in general be lifted, but under certain conditions on dR, the result is also true for .em, see Agmon (1965, p. 30). 11.3.12 The Rellich Imbedding Theorem. Suppose that R is open and bounded, and let m, k be non-negative integers with rn > k. Then the imbedding of 2; in # ," is compact.

294

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Proof. We must show that any sequence (fj) in the closed unit ball of X : has a subsequence convergent in I( Ilk. From the weak sequential compactness of the closed unit ball in Hilbert space (Theorem 6.4.3), there is a subsequence, d 1. denoted also by Uj),weakly convergent in .f:to an f such that Now for 0 < k < rn the imbedding of H: in -P.,kis continuous, so every may be regarded as a continuous linear continuous linear functional on functional on % . ,: and it follows that f j f in . X i . It will be shown that if k < rn, then .+ from which the result readily follows (Problem 6.11). from the definition Extend f,, f to R" by zero. Since fj f i n .f:= Y2(0), of the Fourier transform,fj + f pointwise, and so

-

llflIm

.Pi

l fjl k

Il f l k ,

c

-

-

IelzIfj(t)l2 I zcI S m lY121f(t)12 -+

IalCm

(11.3.7)

for each 5. By Schwarz's inequality /&(<)I < c for some c dependent only on Q. Therefore, for any r < GC! each term in (11.3.7) is dominated for 1 t 1 d T by a constant, and so from the Dominated Convergence Theorem 2.4.11, n

n

In view of equation (11.3.6)this is almost what is needed. If it can be shown that the contribution from 15I > T to the integrals is small, the proof will be complete. Take any F > 0. Since rn > k there is a T > 0 such that

c

lal
lel2/

c 1eI2<

lal
E

(It1 3 4.

( 11.3.9)

By Lemmas 11.3.10and 1 1 . 3 . 1 l , p f ( t ) ~L?,(R")forla/ d rn.Thus

d

( 11.3.10)

i;,

where the first inequality is obtained by using (11.3.9) and (11.3.6).A similar argument shows that (11.3.10)holds with f replaced by f j . Now from (11.3.8) there is an no such that for j > no i r

I

11.3

295

WEAK DERIVATIVES AND SOBOLEV SPACES

Thus f o r j > no,

I

< 3E, where R" has been split into the interior and exterior of the ball s(0,T ) and j~~k the last inequality and (11.3+10)have been used. This shows that l i m ~ ~ f = l l f l l k as required. 0 The next result will be used to determine when solutions of the generalized Dirichlet problem are smooth. The bounded domain Q will be said to have the cone property iff there are positive numbers 8, h such that for every x E R there is a right circular cone with vertex x,angle 8 and height h contained in 0. 11.3.13 Lemma. Suppose that Q is open and bounded, and let m > $n be an integer. Assume that either f E W : , or that f E Wm(n) and R has the cone property. Then there is a real number c (depending only on n, m, Q) such thal (11.3.11)

ProoJ: We prove the result for YE W ,: and refer to Friedman (1969, p. 22) for the other case. Let p be any number greater than the diameter of Q. Take an arbitrary point P in R as centre of coordinates and let r be the distance from P. Extend f to R" by zero. Then

on integrating by parts ( m - 1) times, b being a constant depending only on m. Integration over the angle of the ball S(P, p ) gives

where r is the element of volume and b' depends on n, m only. By Schwarz's

The first integral is finite as m > $n, and depends only on m, n, Q, while the second is not greater than llflli.The result follows as P is arbitrary. 11.3.14 Sobolev Imbedding Theorem. Suppose that Q is open and bounded, and let k be an integer less than m - $n. Iff E %:,' or f E X" and R has the

296

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

gk(n),

cone property, then f is equal a.e. to a function in and the imbeddings of .Frand ern in Wk(Q are continuous with norms determined oniy by n, m, Q. Proof: (11.3.11) applied to D‘f for la1 d k shows that the imbedding of .?if‘: in Wk@) is continuous on the dense subspace %?: of FE.The result follows on extending the imbedding by continuity (Theorem 3.4.4). The proof in the second case is similar. 0

11.4 The Generalized Dirichlet Problem We may now proceed towards our primary goal, the solution of the weak analogue of the classical homogeneous Dirichlet problem. Recall that this is the generalized Dirichlet problem, which consists of finding solutions in ??; of I f 2 g, the boundary conditions being modelled by means of the restriction that only solutions in 2: should be allowed. In fact this formulation is rather awkward, for since the order of the Sobolev space is only m, it is not possible to interpret the terms in l f of order greater than m as weak derivatives. A more convenient formulation is suggested by considering Poisson’s equation again. is a solution of the generalized Dirichlet problem for 1 = - V2, Iff E -V’f G g, (11.4.1) which from Definition 11.3.2 is equivalent to

%?A

(f,-V24)o = (9, Oh (4 E w3. Since f E .%?A, it has first order weak partial derivatives, and one integration by parts is legitimate. With the components of V interpreted as weak derivatives we obtain r

J Now if

R

Vf.V4 d x

Blf,

r

=J

L

$1 =

g$dx.

R

V f . V$ dx,

B is a bilinear form (Problem 6.18) on %LA x +Z: and IBlfAIl

=

lb,$,

Dcf.u$dx/

c l(W>DL4)01 c il I/Dt4 n

d

1

I=

n

I=

1

~ I f I l O

llfll /I 4 I 1

1

iio

(11.4.2)

297

11.4 THE GENERALIZED DIRICHLET PROBLEM

A

for some c depending only on n. Thus B is bounded, and as .%! is the closure of %?: in 1 ) * B may be extended by continuity to a bounded form on x and from (11.4.2)

%?A,

(II,

(11.4.3) The problem of finding anf E .mksatisfying this equation is thus equivalent to (11.4.1), and (11.4.3) is the required alternative form of the generalized Dirichlet problem. It is a rather natural formulation in the setting of .?if,!, for each derivative in B l f , 41 is defined in the weak sense, whereas in the original problem (11.4.1) the terms in V’f cannot be so interpreted. It is interesting to note that the last formulation is essentially variational. To see this assume for simplicity that the functions are real valued, and let Q be the quadratic functional defined by

Q(f)=

I

/ V f I 2 d x- 2 l Q g f d x .

The formal Frechet derivative of Q at 4 is 2

6,

Vf.V@dx-2

l*

g4> and the condition that this should vanish is just (11.4.2).Thus the generalized Dirichlet problem may be regarded as the Euler equation for Q. Further. it is easy to check that iff” is a solution of (11.4.2),then Q(f 4) 3 Q ( f ) for which shows that Q takes a minimum at a solution of (11.4.2). any 4 E This approach may be pursued to prove the existence and uniqueness of a function in minimizing Q, and thus to yield a proof of existence for the generalized Dirichlet problem. This is the direct method of the calculus of variations, which has its historical roots in an argument of Riemann for the Dirichlet integral (see Problem 11.17). A somewhat different line of attack will be followed here. We return to the generalized Dirichlet problem for arbitrary I, and follow the argument leading to (11.4.3).

+

.%A,

11.4.1 Definition. With 1 as in Definition 11.2.2, for all f,4 E ?if: set B [ f >41

=

c

lal,lPlSrn

( P q ? ~ YW> ) O .

B will be called the bilinear form associated with 1. 11.4.2 Lemma. B is a bounded bilinearform on ?if: x ?if:. adjoint, B is Hermitian. L

If 1 is formally self-

298

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Proof. By assumption pap E %(a). Hence

where c, c' are constants depending only on the pEoand m, n. This shows that B is bounded. The remaining assertion follows from Definition 11.2.3. [7 11.4.3 Definition. Let 1 be the formal operator of order 2m of Definition 11.2.2. Given g E LZ2 the problem of finding f E 2: such that

BCf, 41 = (9>4)0 for all 4 E A?;, is called the generalized Dirichlet problem.

(11.4.4)

This is the final version of the weak analogue of the classical homogeneous Dirichlet problem. As a first step towards a solution, the problem is recast in abstract form in the single Hilbert space 2;.By Schwarz's inequality, ((9,4)o\d Ilgl10114110, therefore, Since Il@l/,, d i141im,I(g, 4J0i < Ilgllol1411m. Thus g* defined by g*(+) = (g, 4)0is a continuous linear functional on .iy and it follows from the Riesz Representation Theorem 6.4.1 that there is a unique k E &?: such that (9, 4)o = g*(4) = (h, 4)m.The generalized Dirichlet problem is then to find, for such an h E 2;an f E 2; satisfying

r,

9"

41 = (h,41m (4 E .*;I.

(11,4.5)

The method by which this is solved may be clarified if initially the additional assumption is made that B is Hermitian. As B is bounded, by Problem 6.18 there is a unique bounded self-adjoint operator, L say, such that

BCf, 41 = (Lf,4 ) m

(f,4 E '*;),

and (11.4.5) becomes (Lf,4)m= (k,4)m for all 4 E &;,' or equivalently Lf = k. If L-' ~2(,#';), this equation, and so the generalized Dirichlet problem, has the unique solution f = L- ' h . Since this provides a complete answer to existence and uniqueness, we look for a condition on B which will ensure that L has this property. The simplest such condition is that B is strictly positive, that is there exists a c > 0 such that = Blf,f] >, c llflli, for then by Theorem 6.6.6,O E p(L). If B is not Hermitian (and so not real-valued), this condition is not appropriate, but a suitable generalization is readily found. It is the following.

1 1.4

THE GENERALIZED DIRICHLET PROBLEM

299

11.4.4. Definition. B will be said to be coercive iff there is a real number > 0 such that for all f. P:

c

Re B [ f > f l2

(.llfllrs.

11.4.5 The Lax-Milgram Lemma. Let B be a bounded bilinear form on %: x #:. Then there is a unique L E 9(%$') SUC h thut B [ f , 41 = (LJ,#),for all f,# E %.: f f B is coercive, L- E 9( #:).

-

Proof: Since B is bounded, for fixed f,B [ f , * ] = B*( ), say, is a continuous antilinear functional on A ' : . Hence by the Riesz Representation Theorem 6.4.1, there is a unique k E X : such that BCf, Cp] = (k, Cp), for all 4 E % . ,; and 11 k/lm= I/ B* 1 . The relation k = Lf for each f~ -8; then defines an operator L which is evidently linear, which is bounded since // B* // d d for some d E R, and which is such that B[ f , 41 = (Lf, C.p,) For B coercive, there is a c > 0 such that

llf/lm

I l L f I l m / l f / / m 2 lB[f,.fIl >, R e B [ L f I 3

ll.f/l~~

Hence l/LfIlm2 cllfll,. Also R(L) = .#:, for if h E R(L)', B[h, h] = (Lh,h), = 0, whence h = 0 by coercivity. The result follows from Lemma 3.8.18. 0 11.4.6 Theorem. Let 1 be the formal operator of order 2m ojDeJinition 11.2.2, and let B be the associated bilinear form. Then if B is c0erc.it.e there is exactly one solution (in .X:) of the generalized Dirichlet problem f o r any given righthand side g E 9,.

Proofi Apply the Lax-Milgram Lemma to (11.4.5). 11.4.7 Example. Take 1 B[f,

=

and Re B [ f , f l where p o

=

+ k + p with k E R and p E g X ( n ) Then .

-V2

41 =

0

i,

[Vf.v$

+ (k + P)f$I

dx,

/If1: - llfll; + Re(@ + P)f,f)o 2 /If 1: + (k - 1 + Po) llf/l;, =

inf Re p(x). Thus if k 2 1 - p o the form is coercive, and the

XER

generalized Dirichlet problem has a unique solution by Theorem 11.4.6. In general more cannot be asserted, but in the important case when is bounded, the condition k 2 1 - p o can be considerably weakened by using Poincare's inequality (Problem 11.15). The next lemma shows how the argument goes in one case, and proves in particular existence and uniqueness for Laplace's equation.

300

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

On the other hand if p = 0 and k is a large negative number, B is not coercive and the argument breaks down. This is natural, for since the Laplacian has negative eigenvalues (corresponding to values of k for which If = 0 has non-trivial solutions), existence must fail for some negative k . The best that can be hoped for is a Fredholm alternative. This will be the subject of the next section. 11.4.8 Lemma. Let Q be bounded. Suppose thut 1 is homogeneous ojdegree m with constant coefficients : If= P,$D"+pf.

c

iz:,:/i:=m

Then the associated fonn is coercice i f 1 is strongly elliptic. Proof. It is evidently enough to consider functions in %," since this is dense in .X;. Take any 4 E %?; and extend 4 to R"by zero. Then Re B [ 4 , 4 ] = Re ,.

3c

J l(lZm

d<

(by ellipticity)

R"

(11.4.6) c, c' being strictly positive numbers independent of 4. Now it is an immediate consequence of Poincare's inequality (Problem 11.15) that there is an a > 0 such that for all 4 E %?:,

,.

n

Combining this with (11.4.6) we obtain Re B[4,

41 2 4 1 + 4- i1411:l.

0

When B is coercive, existence and uniqueness for the generalized Dirichlet problem are guaranteed by Theorem 11.4.6. It will be seen below that the solution f depends continuously on the right-hand side g in the sense that f = Gg where G : Y 2+ X ; is a bounded linear operator. Since %?; c %?: = Y 2 any , 4 E %?: can also be regarded as an element of Y 2and , often no confusion arises ifwe do this. However, it is now necessary

11.5

A FREDHOLM ALTERNATIVE FOR THE GENERALIZED DIRICHLET PROBLEM

301

to distinguish between these two possibilities, and to this end the following definition is introduced. 11.4.9 Definition. Let K : X: its adjoint be K* : 9, .#;.

+ Lf2 be

the imbedding of 8;in .Y2,and let

-+

Obviously K and I(* are bounded linear operators with norms not greater than unity. One advantage of defining K explicitly is that the properties of its adjoint may be exploited. From Definition 6.5.6, if g E P2and 4 E #:, (9,410

=

(9, m ) o

=

( K * & 4)m.

(11.4.7)

Now from the Lax-Milgram Lemma 11.4.5and equation (11.4.5),the solution f of the generalized Dirichlet problem is L-'h, where h is related to the right-hand side g by (h, +)m = (9, 4)ofor all 4 E %.?; Therefore from (11.4.7), h = K*g and f = L-'K*g. This proves thatfdepends continuously on g, for L- K* is bounded.

'

11.4.10 Definition. The operators G = L-'K*: Y 2+ 2; and KL-'K*: Y 2 Y 2will be called Green's operators.

G=

-+

The term Green's operator isused because G plays the role here taken in the classical theory by an integral operator with kernel the Green's function. An advantage of the present method is that all the properties of G required for proofs of existence and uniqueness may be obtained without a detailed examination of the Green's function. The results of this section are summarized as follows.

11.4.11 Theorem. Suppose that the form associated with 1 is coercive. Then for any right-hand side g E Lf2, the solution of the generalized Dirichlet problem isf = Gg, where G = L- K * is a bounded linear operator from z2into X:.

'

11.4.12 Corollary, Assume that in addition 1 is formally selfadjoint. Then = KL- K * is a bounded selfadjoint operator 92 + Y2.

'

Pro05 Since B is Hermitian, L is self-adjoint, whence so also is G.

11.5

A Fredholm Alternative for the Generalized Dirichlet Problem

Theorem 11.4.6 settles existence and uniqueness for the generalized Dirichlet problem when the form B associated with I is coercive. However, it is clear

302

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

from elementary examples (such as the operator -V2 + k of Example 11.4.7) that if B is not coercive, the corresponding homogeneous equation (that is with right-hand side g = 0) may have non-trivial solutions. These examples also suggest that uniqueness and existence for arbitrary g can be expected if and only if there are no such solutions, We shall now prove that this powerful analogue of the Fredholm Alternative is in fact true for all strongly elliptic operators on a bounded domain. The proof is based on the compactness of the Green's operator, and is reminiscent of the argument used in the classical integral equation approach. The method to be used is suggested by the tactics for the simpler equation

Mf

(11.5.1)

= 9,

where .8is a Hilbert space and M E dia(.#) is self-adjoint. If M is strictly ) (11.5.1) has the unique solution f = M - l g . positive then 0 ~ p ( M and However, suppose that instead of positivity only the weaker condition (LAf ) -bllfl/2 (b > 0 ) holds. Then for a > b, still M , = M + a l is strictly positive, 0 E p(M,), and M a has a bounded inverse. A rearrangement of (11.5.1)gives M af = g + af, and then f = aM,'f+

3

(3 = M,'g).

(1 1.5.2)

Existence and uniqueness for (11.5.1) may thus be settled by considering (11.5.2). If Ma-' is compact, the advantage of using (11.5.2) is apparent, for the Fredholm Alternative for compact operators may be applied. In adapting this argument to the generalized Dirichlet problem the fact that two Hilbert spaces A?: and LZ2 = A?: appear and that coercive forms rather than strictly positive operators must be dealt with slightly complicates matters, but the technique is similar. We show that by adding a suitable term to B a new form is obtained which is coercive (and so has a bounded Green's operator). That this is the case is a consequence of the fundamental Garding's inequality. For the somewhat complicated proof of this result see Friedman (1969, p. 34). 11.51 Theorem (Girding's Inequality). Suppose that R is bounded, and assume that the formal operator 1 of Definition 11.2.2 is strongly elliptic. Then there are real numbers c' > 0 and a such that for all $ E #:, Re B[$, $1 3

cli4ll:*

-

all4ll:.

11.52 Corollary. With the assumptions of the theorem, the form B, defined by is coerciue.

B,[L

41 = BCL $1 + atA $10

11.5

A FREDHOLM ALTERNATIVE FOR THE GENERALIZED DIRICHLET PROBLEM

303

Proof: By GBrding’s inequality, for all 4 E X:, Re B , [ h

41 = Re B[4>41 + a11411~ 3 cll4ll: - all4l: + a11411: =

cll4ll:.

0

The symbols La, G,, G, will henceforth denote the operators derived from B, in the same manner as L, G , G respectively are derived from B. The Rellich Imbedding Theorem is now invoked in order to prove the compactness of the Green’s operator. As remarked previously the condition that SZ is bounded cannot in general be dispensed with.

11.5.3 Theorem. If R is bounded, the Green’s operator G, : 5 f 2 + 5 f 2 corresponding to a coercive form B, is compact. Proof. By Theorem 11.3.12the imbedding K of .7?: in A?: = Y 2is compact. The compactness of G, = KL; K* follows since L; and K* are continuous. 0 11.5.4 Definition. A complex number A will be called an eigenvalue of the generalized Dirichlet problem iff there is a non-zero f E .#: such that

BCf, 41 = 4”t$1,

(4 E G)?

and f will be known as an eigenfunction.

11.5.5 Theorem (Fredholm Alternative). Suppose that C2 is open and bounded, and let the formal operator 1 of Definition 11.2.2 be strongly elliptic. Then either the generalized Dirichlet problem has exactly one solution for any g E S f 2 , or zero is an eigenvalue. Proof: By Corollary 11.5.2 and Theorem 11.5.3, for some a E R the Green’s operator G, is compact. We first show that the generalized Dirichlet problem is equivalent to the equation f = a G J + g, (11.5.3) where 0 = Gag and f = KJ: (f and fare the same function, but a distinction is made in the notation in the proof to emphasize that they are being regarded as elements of .%: and S f 2 respectively. To be absolutely precise, we should write the generalized Dirichlet problem as B [ f , 41 = (9,K4)o, since the first appearance of 4 is as an element of Z:,the second as an element of $P2. However, here as elsewhere this somewhat pedantic notation is avoided).

304

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

First, addition of a(x 4)oto each side of B [ f , 41

41 = ( a f +

= (g,

@loyields

9>4)0>

and from Theorem 11.4.1l,f = a e u f + Gag. Suppose on the other hand that f~ Sf2 satisfies (11.5.3),and set f = G,(af+ g). ThenfE -%;C and ~

Thus for all

f ~=~ , ( a fg+ ) = Z;,(af+ g)

4 E %,; B,[L

=f

41 = (Ju $I*

+ 9)>$Irn = (K*(af+ 4)", = ( a f + g, ~ 4 ) ~ = (L,G,(af

(af + g,$)o. It follows on subtracting (ax q5)o from each side of this equation that f i s a solution of the generalized Dirichlet problem. To complete the proof recall that G, is compact and apply the Fredholm Alternative Theorem 7.3.7 to (11.5.3). 0 =

The compactness of G, may be further exploited to show that if zero is an eigenvalue, the generalized Dirichlet problem has a solution if and only if g is orthogonal to all the (finite number of) corresponding eigenfunctions. We conclude with two results which emphasize the analogy between the spectral properties of ordinary and partial differential equations. 11.5.6 Theorem. Assume that the conditions of the last theorem hold. Then either the generalized Dirichlet problem for 1 - 2 has exactly one solution for any g E S f 2 , or /z is an eigenvalue. The eigenvatues have nofinite limit point and to each eigenvalue there corresponds only a finite number of linearly independent eigenfunctions.

Proof. Apply Theorem 11.5.5 to 1 - A and use Theorems 7.4.1 and 7.4.2. 0 11.5.7 Theorem. Assume that the conditions of Theorem 11.5.5 hold, and suppose in addition that 1 is formally self-adjoint. Then the eigenfunctions form a basis for LF2.

Proof. By the argument used in the proof of Theorem 11.5.5,to a solution of p,f = ac,f there corresponds an eigenvector f and eigenvalue in = a(p, - 1). Since G, is self-adjoint (Corollary 11.4.12), the result will follow from

1 1.6

305

SMOOTHNESS OF WEAK SOLUTIONS

the Hilbert-Schmidt Theorem 7.5.1 if it can be shown that zero is not an eigenvalue of G,. To prove this suppose that c,f = 0. Then 0 = (KL,'K*j?,j), = (L,'K*JI, K * f ) , = B , [ K * i K * y ] , and since B, is coercive, K* f whence j = 0 as

=

0. Therefore, for all 4 E #vY;,

0 = ( K * J 41n= (JI, K#), is dense in L?2.

=(

tl tb)@ 0

11.6 Smoothness of Weak Solutions The analysis of the previous sections provides natural criteria for the existence and uniqueness of solutions of the generalized Dirichlet problem. In applications it is sometimes relevant to ask further whether such a solution is smooth enough to be a classical solution of the Dirichlet problem. If this is to be the case, firstfshould have 2m continuous derivatives in R, and second f should be smooth enough for the boundary conditions to be meaningful-that is f should have m - 1 continuous derivatives in the These properties are called "interior regularity" and "regularity closure up to the boundary" respectively, and the position is broadly thatfhas both these properties if the right-hand side g and the boundary dR are reasonably smooth. The proof of this assertion presents quite considerable technical difficulties, and since our primary interest is in the generalized Dirichlet problem we shall quote the main result and refer the reader to one of the cited texts for a proof. Nonetheless in order to give the flavour of the type of argument that is used, we shall sketch a proof of interior regularity in one relatively simple but important case-when the principal part of I has constant coefficients. In outline the method is to show that the order of the k if Sobolev space to which f belongs can be raised step by step to 2m g E .Ek. The continuity of the derivatives then follows from the Sobolev Imbedding Theorem 11.3.14. In tackling interior regularity, rather than use the divergence form of Definition 11.2.2 it is more convenient to take 1 as follows.

n.

+

11.6.1 Definition. Assume that pa E Vm(n)for 1x1 < 2m. The form7.1 operator 1 of order 2m, its adjoint I*, and its principal part lP are defined as follows:

I*f

=

c

,a <2m

( - l ) ' xD"@,f), '

306

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

1 is said to be strongly elliptic iff there is a c > 0 such that for all real all x E a. ( - 1)" Re I,(<) = ( - 1)" Re p,? 2 C I ~ / ~ " .

t, and

1

:a:=2m

If 1, 1 , , . . . , l r are formal operators, the relation I f g l,gl

+...+ I,&

on R will mean that for all 4 E %;,' ( f t l*4)0

=

(Sl, lT4) 0 . . . +

kA.7

Y+)O.

Interior regularity is a local property.fis differentiable on the open set R if it is differentiable on a neighbourhood of each point of R. It follows intuitively that the boundary of R should not have any significance in the discussion. In particular it should not matter whether f lies in A?: or .%?", and indeed interior regularity will be established for all solutions of If g if g is smooth enough. In order to exploit the fact that only a local result is sought, for a point P in R an equation for $-is formulated where t,b = 1 on a neighbourhood S of P and t,b E %:(a). Then iff€ X k ,it follows that t,bf E Zk,, and by Lemma 11.3.10 t,bfmay be extended by zero to R", and $ f ~&k,(R"). The advantage of this approach is that it is quite easy to tackle regularity on R" since the Fourier transform may be used. The behaviour offon S itself is readily deduced since t,b = 1 on S. We start then with two results in R". 11.6.2 Lemma. Suppose that ~ E T J R " ) . Then f ~ . m ' ( R " ) if and only (1 + E92(W

l
Proof. Recall that .%?",R")

if

.%fk(R")and proceed as in the proof of Lemma

=

11.3.11. 11.6.3 Lemma. Assume that lp has constant coefficients and is strongly elliptic. Suppose that on R"

where gaE .8"(R"). Then f

E

.8"jRfl) where

j = min (2m la:<2m

Proof. It is given that for all

+ k, - .1)

4 E %:(R"),

and since %,"(R") is dense in Y2(Rn),it follows from the Plancherel formula

11.6

that

[I

307

SMOOTHNESS OF WEAK SOLUTIONS

+ lP(t)]f(<) = c (-W :a:<2m

t%,(O.

Using the ellipticity, we deduce after rearranging that for some c > 0,

By Lemma 11.6.2 each term in the curly brackets lies in -Y2(Rn), while the factors multiplying these terms are bounded for the stated values ofj. Hence each term on the right-hand side, and therefore the left-hand side, is in Z2(R"), and the result follows from Lemma 11.6.2. 0 Following the tactics outlined above, to recast the equation If g in a form to which the last lemma can be applied, a new equation for $fis formulated. This requires nothing more in principle than liberal use of Leibnitz's Theorem for weak derivatives (Problem 11.12). We spare the reader the tedious details. 11.6.4 Lemma. Let S be an open subset of 0, and assume that $E%?~(S). Let g E Xk(Q), and suppose that f E .mt(S)satisfies If g on S . Then there are functions $, E %':(S) such that on R",

c1

+ M*f 1 2L *s +

and (the extensions by zero o f ) $g

E

c

la'<2m

e k ( R " ) ,$, f

(11.6.1)

D"(*,f), E

X'(R").

The highest derivative on the right-hand side of (11.6.1)is of order (2m - 1). Therefore, from Lemma 11.6.3, $ f E &j(R") where j = min(2m k , t 1). This proves that unless already t = 2m + k, the order of the Sobolev space to which $f belongs can be raised from t to t + 1. Now by Problem 11.10, for any bounded open S' with S' c S there is a $ E W(;S) with $ = 1 on S'. Hence f E A?'+l(S). An inductive application of this argument proves that f~ 22m+k(Q') for any bounded open Q' such that c Q. Finally the Sobolev Imbedding Theorem 11.3.14 is applied to $YE c % ? ~ m + k ( Q ) to prove continuity of the derivatives. This establishes interior regularity.

+

+

a'

11.6.5 Theorem. Let Q be an open set in R".Assume thut the formal operutor 1 of order 2m in Definition 11.6.1 is strongly elliptic and that the principal part of 1 has constant toeficients. Suppose that g E Fk(Q) and that f is a solution in Y 2 of l f g g. Then f E P2m+k(R') for (in\ hoirntIetJ n p n w t R' T I I C It thtrt 51' c a. Further f EW@') for integers s < 2m + k -in, and f ~ % ' ~ ( 5 if1 ' ) g E Urn(Q).

308

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

The assumption that 1, has constant coefficients simplifies the technicalities. For a proof in the general case see Friedman [1969, Section 1.151. For regularity up to the boundary some smoothness is required of dsZ. For the somewhat complicated proof of the following and further discussion see Friedman (1969, Section 1.17).

11.6.6 Theorem. Let rZ be a bounded domain with boundary, and let the formal operator 1 of Definition 11.2.2 be strongly elliptic. Assume that f is a solution of the generalized Dirichlet problem with right-hand side g E .#'(sZ). Then f E c#zm+k(Q). Also f E P(a)for integers s < 2m k - $n.

+

Problems Throughout R will be an open subset of R".

11.1 Suppose that 1 in Definition 11.2.2 is strongly elliptic on the bounded domain R. Make the change of variable y = $(x) where +: + R" is in Wm@, R"). If has non-vanishing Jacobian determinant in 0,show that the new operator is also strongly elliptic.

+

n

11.2 Show that ( - l)k(V2)kis strongly elliptic. 11.3 The operator 1 in Definition 11.6.1 is said to be elliptic if for any real 5 and any X E ~ If. complex coefficients are allowed, ellipticity is not enough for the Dirichlet problem to be well posed. For take R to be the open unit ball in R2 and consider I f = f,, + 2i& - f,,. Prove that 1 is elliptic but not strongly elliptic. With z = x iy show thatf(x, y ) = (1 - IzI')u(z) is a solution of If = 0 for any analytic function u, but that f vanishes on dR.

+

11.4 Let R' be a compact subset of R. Prove that there is an S(x, E ) n R' = 0for every x E aR. 11.5 Take R = ( - 1 , l )and let also 4 E Y:, where

4(4 =

f:,

tE

8

> 0 such that

W r be such that indx = 1. If + E W F show that

+(t) dt -

1;

T

$(t) dt .

: ' and Jn h4' dx = 0 for all Deduce that if h E 2 constant.

f:, 4 s )

4 E W:,

ds.

then h is (equal a.e. to) a

9 p has weak first derivative ifffis abso-

11.6 With R = ( - 1, 1) prove that a n f ~ lutely continuous and f' E 2p.

309

PROBLEMS

%?'(a)

11.7 For R = (0,l) prove that the closure of in 1) * ( 1 is Afl itself (cf. Theorem 11.3.6). One possibility is to take 4(x) = cos nnx (n = 0,1,. . .) in

(.A

4 1 1 =

Sd

f(T

-

6")dx + T'(l)f(l)- 6'(0)f(O).

11.8 Complete the proof of Example 11.3.8. 11.9 Let {j,} be a family of mollifiers (Definition 2.6.6). Show that iff€ %""(R), and R, is a compact subset of Q then as E + 0, j , * f - t f i n .F"(R,). 11.10 Assume that R, c R is compact. Show that there is a real-valued g EV'," such that g(x) = 1 (x E R,) and 0 < g(x) < 1 ( x E R). (Hint: use mollifiers). 11.11 Prove that i f f € .Z""(R) has support in a compact set contained in R, then f€

rnt(R).

11.12 Let

4 E V.:

Iff has weak derivatives up to order rn, show that

4j; and prove the n dimensional version of Leibnitz's Theorem: where

SO

also does

< CI means pi < ui for 1 < i < n, and where

11.13 Suppose I)E Vt(Rn) is equal to 1 on s(0,1) and has support in s(0,2), and set I)j(x) = I)(x/j). If f~ .%""(R") show that as j + m, t,bjf-f in 11 I/,,. Deduce that .@;(Fin)

-

= mm(Rn).

11.14 Show that iff€

,#A,

SnlIfl'

dx


So1If'['

dx

where b = n-', and prove that this value of b is best possible. - k f i ) dx associated with 1 = Show that the bilinear form B [ f , 41 = lA(f'~$' - d2/dx2 - k ( k E R) is coercive iff k <: n2. Interpret this in terms of the homogeneous Dirichlet problem for - d2/dx2. 11.15 Suppose that R c R" is bounded and let d = sup- / x - yI be its diameter. X,).tR

(i) Iff€%?;, f(x,, . . . ,x.) =

J':

g ( t , x 2 , .. . , x,,) dt,

where (x, x2,. . . ,x,) lies in R but outside the support off: Deduce that IlX'axiIIo. < ( 2 n ) - ' d l f l l , where

llfllo G 2-+ d

Hence prove that forfe X ; , llfll,,

IfK

=

11 n l.l=*

ID"f12dx.

310

11

LINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Show that the bilinear form jn(Vf. V$ - kf$) dx associated with - V2 - k ( k $ R) is coercive on if k < d2/2n. Deduce a lower bound for the smallest eigenvalue of the generalized Dirichlet problem for - Vz. (ii) PoincarP‘s Inequality. Prove that there is a constant a dependent only on n, rn such that for allfE .8:,

&‘A

11.16 For k a positive integer, show that there are coercive forms associated with 1 and (1 - V’)’.

( - 1)k(V2)k

+

11.17 Suppose that I (Definition 11.2.2) is formally self-adjoint and the associated form B is coercive. Define ( X d)E= B [ f , 41 and llflli = ( X f ) , for all J 4 E Z‘;. Show that 11 and 11 . , l m are equivalent norms and deduce that & : : is a Hilbert space X Ewith the new inner product ( * , *), and norm I( . and for all f E X Edefine the “total energy” Take given g E z2,

- /IB

IIE.

Q ( f ) = IIf IIi - ( g > f ) o - ( X 910-

Let

A?be a closed subspace of 2,.Show that there is exactly one?€ A‘ such that

and deduce that the generalized Dirichlet problem has a unique solution. Historical note. This is an existence proof by the direct method of the calculus of variations. Its history starts with the Dirichlet Principle from which important results in a number of fields were deduced by Riemann and others. The original argument on which the principle was based was simply that since a certain integral related to Q has a lower bound, there must be a function for which the minimum is attained. Because of the evident fallacy in this and other difficulties with the class of admissible functions, the method fell into disrepute until rescued by Hilbert. For details of its interesting history see Courant (1950) and Monna (1975).

11.18 (Continuous dependence of the solution on the right-hand side). With the notation of Problem 11.15(i), take k < a, and set a = 2nd-’. If g E Y 2show that the solutionfof the generalized Dirichlet problem satisfies the relations l l f l l o < (a - k 1 - l I l S l l O ~ llflll < c I I Y l l 0 ~ where c 2 = ( a + l)/(a - k)’ for - 1 < k < a, c z = l/(a - k ) for k

< - 1.

11.19 Extend Theorem 11.5.5 as follows. If zero is an eigenvalue, show that the generalized Dirichlet problem has a solution for g E z2iff (g, i,hi),, = 0 for all the eigenvectors +hiof the adjoint problem B [ d , $i]= 0 (4 E X;).