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6. Strongly Degenerate Elliptic Differential Operators
6.
STRONGLY DEGENERATE OPERATORS
6.1.
Introduction
ELLIPTIC DIFFERENTIAL
Beside regular elliptic differential operators, degenerate elliptic differential operators have been extensively considered in the last years. There are many, very different, types of degeneration. On the one hand the domain 9 c R, may be unbounded, or on the other hand the coefficients of the differential operator may be singular if 1x1 + 00 or x + aQ, or they have singularities in 9. Further, it may be happen that the ellipticitiy condition (4.9.1/2) degenerates if x -+ a 9 . Clearly, there are many possibilities of variation, and one cannot expect to catch all these different types by a uniform theory. This chapter and the following one are concerned with some (comparatively comprehensive) classes. The class considered in this chapter is characterized in a very strong degeneration of the coefficients of the operators near the boundary (and at infinity). This has the consequence that the boundary does not play any role (in contrast to the regular elliptic differential operators of the last chapter). For instance, a formally self-adjoint operator of this type with the domain of definition C$'(sZ) is essentially self-adjoint in A&?).
6.2.
Definitions and Preliminaries
In this section, the classes of operators coiisidered here are defined. Further, some
related locally convex spaces are introduced. Finally, there are proved some simple properties.
6.2.1.
Definitions
I f Q c R,,, then, as usual, Cm(Q) denotes the set of all infinitely differentiable complex-valued functions defined on Q. Definition 1. Let Q be a n arbitrary domain in R,,.Further, let p(x) E Cw(Q) be a positive function such that : 1. For all multi-indices y , there exist positive numbers c, such that IDr@(X)I
cr@'+lqx) for all
2 E9.
(1)
406
6.2. Definitions and Preliminaries
2. For any positive number K , there exist numbers cK > 0 and rlc > 0 such that e(z) > K if d ( x ) 5 cK or if 1x1 ( d ( z )is the distance to the boundary). Then ~!5'@(~)(9) denotes the locally convex space
Se(r,(f2)
=
{f If
IIfIIt,a
E CoD(Q),
=
1 rI,, ( X E D )
(2)
e6(z)l o " f ( ~ ) < I
(3)
SUP X€Q
for all 1 = 0, 1, 2 , . . ., and a11 multi-indices a } . Remark 1. Comparison with Definition 3.2.3/1 shows that (3.2.3/1) is replaced by the sharper assumption (1). The functions e(z) considered in Remark 3.2.3/1 satisfy also the above assumptions. In particular, in every bounded domain there exist functions e ( x )such that e-l(x) coincides essentially with d ( x ) . Definition 2. Let f2 c R, be a n arbitrary domain, and let e(z) be a weight function in the sense of Definition 1. Further, let m be a natural number, and let ,u and v be real numbers such that v > p + 2m. One sets 1 x i = -(v(2m - I ) + p l ) , 1 = 0, 1 , . . ., 2 m . (4) 2m (a) The class %&(9; e ( x ) )consists of all differentia2 operators of the form
Here, ba(x)E C m ( 9 )are real functions, la1 = 21 with 1 = 0 , 1, . . ., m, where all their derivatives (inclusively the functions themselves) are bounded in 9. Further, it is assumed that there exists a positive number C such that for all 6 E R, and all x E 9
(ellipticity condition). Let as(x) E C " ( 9 ) , 0 5 IBI < 2wk, and Dyas(x) = o( e x l p l + 1 ~ (1x ) ) for all multi-indices y .
> 0 there exists a natural number j ( E ) such that EpxioI+ I I ~ ( x ) for z E a - , W e ) ) , (7b)
(This means that for a n y number lDYas(x)l
s
E
where 9 ( j ) has the same meaning as in Definition 3.2.3/1.) (b) The subclass '&&(9; e ( x ) )of %&(i2; &)) consists of of all operators of %&(9; e(x)) for which there ex%& a positive number 6 > 0 such that
s
DYas(x) =
(8)
o(@x161+lYl-8)
for 0 IpI < 2 m and for all multi-indices y . Remark 2. Clearly, (8) is sharper than (7). Remark 3. %Ev(f2;e ( x ) ) and % E y ( 9 ;e(z)) are comparatively comprehensive classes of degenerate elliptic differential operators. We describe two simple examples. (a) If 0 is an arbitrary bounded domain, and if e-l(z) d ( x ) in the sense of Remark
-
6.2.2. Powers of Strongly Degenerate Elliptic Differential Operators
3.2.311, then Au = f(z) ( - & I
+ e”(z) a,
u
v >. p
407
+ 2?n,
belongs h@”(Q; e(z)). If Q is a bounded Cm-domain,then one can put e(s)= d - l ( z ) near the boundary. (b) If Q = R, , then any operator of the form
AU = (1
+ Izls)*l
u
+ (1 +
q2 > ql,
1 ~ ” ) ” s ~ ~
rlr (R,; (1 + I Z ~ ~ )provided ~), that 6 > 0 is sufficiently small. a a
belongs to the class
R e m a r k 4. * The above definition goes back to H.TRIEBEL[24]. Differential operators of the above type are closely related to investigations on the structure of nuclear function spaces. We shall return to this problem in Chapter 8. In this connection the special case
A ~= L -Au
+ e”(s)
U,
v > 2,
in bounded domains was considered in H. TRIEBEL [2]. Extensions of these investigations to unbounded domains as well as generalizations and improvements can [l,21, B. LANQEMANN [l, 21, D. KNIEPERT [l], be found in E. MULLER-PFEIFFER and H. TRIEBEL [lo, 241. Similar differential operators are considered in L. A. BAQIROV [l] and L. A. BAGIROV, V. I. FEJQIN[l].
6.2.2.
Powers of Strongly Degenerate Elliptic Differential Operators
We need for the later considerations that powers of strongly degenerate operators in the sense of De€inition 6.2.112 are also operators of such a type. L e m m a . (a) If A E%E,(Q; e(z)) in the sense of Definition 6.2.1/2, then Ak E % ~ E ~ ~e(s)) ( Q for ; k = 1,2, . . .
(b) I f A ~iflE,(Q;e(x)),then Ak E&;&(SZ; e(z))for k = 1 , 2 , . . . Proof. The proof is given by induction. Assume that the lemma is true for k = 1 , . . ., j. Then we have
c c nij
Aju
=
1=0 l a l = 2 1
xp’ = y
@4{)(Z a$’(.) )
2rnj - 1 2rn
+P
1 G Y
D*U
+
c
c2nrj
1 = 0, 1 , .
UP@)
DBU,
(1)
. ., 2mj.
The coefficients have the properties mentioned in Definition 6.2.112. xjj)
+ x , ~= x$gi), 1 = 0 , 1 , . . .,2rnj,
s = 0,1,
. . ., 2m,
(2)
yields that the ‘‘ main part ” of A j + L = Aj(Au)has the desired structure (inclusively (6.2.1I S ) ) . Using and
lDe”(s)I5 cex+lyI(z), x real number, xij)
+ x, + 1y1 = xjcl) + ~ y
(3) for
o < 1y1 5 z + s,
(4)
408
6.2. Definitions and Preliminaries
then it follows that the second summand in (1)has also the desired form. This is also ( x;) ) . true if A is an element of i i ; ~ e~
6.2.3.
Properties of the Spaces S,(,)(sd)
A complete locally convex space where the topology is given by a countable set of semi-norms is said t o be a n (P)-space. T h e o r e m . (a) Se(,)(Q)from Definition 6.2.1/1 is an (F)-space.Corn@) i s a dense subset. (b) If there exists a number a > 0 such that e-"(x) E L,(Q),then we have for all p with 1 5 p 5 oc) ( i n the sense of continuous embedding) Se(.x)(Q)c L p ( Q ) . (1) (c) If (1) i s true for a suitable number p with 1 5 p c 00, then there exists a aumber a > 0 such that e-"(x) E L,(Q). Proof. Step 1 . We prove that C$(Q) is dense in Se(&2). If the domains Q(J) have the same meaning as in Definition 32.311, then there exist functions q ~ j ( xE) C ~ ( Q ~ + l ) ) such that v j ( x ) = 1 for x E QCj) end lDyvj(z)l 5 cY2Jlrl (2) for j = N , N + 1, . . . and for all multi-indices y. See Remark 3.2.312. (Outside of Q(J+l)we set p i ( x )= 0.) I f f E Se(,)(Q), then v j ( x )f ( x ) E Corn (Q) approximates the function f(s)in Sec,)(9).It is easy t o see that X,(,)(Q) is an (P)-space. Step 2. The part (b) of the theorem is clear. To prove (c), we show at first that there exists a number b > 0 such that I Q ( j + l ) - Q ( j ) l 5 bj, j = N , N + 1 , . . . (3)
Assume that there does not exist a number b having the property (3). We fix a natural number k and a sequence O < a l < a 2 < . . . < a l < ..., a l + c o if Then there exist natural numbers jz > N + 1 such that IQth+l)
1-oc).
(4)
- Q ( j q > ail, jz+l- j l 2 k . I
We choose k sufficiently large and set
Using Sobolev's mollification method described in the proof of Lemma2.5.1 and setting ( U ) ~ ~ - ~ for ~ ( Zx ) E SZ(jlt3) - SZ(jl-z), v(x) = otherwise ( x E 9 ), ( 0 where c > 0 is sufficiently small, then it follows that v E Cm(sZ)and
6.2.3. Properties of the spaces A!?~&?)
409
for x E Q ( j i + j ) - D(ji-2).Hence v E Se(,,(Q). On the other hand, one obtains by (5)
j
J)
W
Iv(s)lpdx
2C
1= 1 J)Ul+') -&I,
Iv(s)lp dx
W
2 C
-
~ j i l Q ( j ~ " j Q(jljl = 03.
Z=1
This is a contradiction to (1). This proves (3). Now it follows for a > 0 with 2a > b
R e m a r k 1. If 52 is bounded, then e-"(x) belongs t o L,(Q) for all a 2 0. One of the main aims of this chapter is the development of a n Lp-theory for operators from the Definition 6.2.112. For this purpose, (1) is a natural assumption. The theorem shows that (1) is equivalent t o
3a > 0 where ,p-"(x) E L , ( 9 ) .
(6)
R e m a r k 2. We shall see later that under the hypothesis (6) S,(,)(D) is a nuclear (F)-space isomorphic to the space s of rapidly decreasing sequences. It is easy to see that the Schwartz space S(R,,) is a special case. L e m m a . Let D c R,,be a bounded domain, and let e-l(x) d ( x ) be a smoothed distance function i n the seme of Remark 3.2.311. Then it holds set-theoretically and topologically (7) Se(3j(Q) = CO"(52) = { f I / E CF ( ~ n;)SUPPf c N
.n7
where the topology i n CF(52) i s given by the semi-norms sup lDaf(x)l,0 Z€Q
=< la1 < 03.
P r o o f . It is not very hard t o see that Se(,)(D)andZF(52) coincide set-theoretically (here the functions of ~ S ~ ( ~ ) (are 5 2 extended ) by zero outside of 52). Further, E$(Q) is an (F)-space where the topology of Se(,)(52)is finer than the topology of c$(D). Then one obtains as a consequence of the closed graph theorem (see N. DUNFORD, J. T. SCHWARTZ [l,I], 11.2, Theorem 5) that fYe(&) and @(Q) coincide also in the topological sense.
6.3.
A-Priori-Estimates
The main aim of this chapter is t o give a treatment on mapping properties for the operators of Definition 6.2.112 in the framework of an L,-theory. A priori-estimates and an L,-theory for special self-adjoint operators of such a type (Section 6.4) are essential for these investigations. On this basis and with the aid of the SobolevLebesgue-Besov spaces with weights, one can prove in the following sections theorems on isomorphic mappings.
410
6.3. A-Priori-Estimates
6.3.1.
Equivalent Norms in the Spaces W,(sd; 8’;
e’)
Lemma. Let e(z) be a weight function in the sense of Definition 3.2.311. Further, let k = 0, 1, 2, . . ., 1 < p < 00, Y 2 p kp, and {yj(x)],ZN EY(Q;e) in the sense of Definition 3.2.311. Then there exist: (a) Balk K f ) = {z I 1z - zj,ll .c d 2-j} such t h d
+
-
s,
Qjc U Klj) c Q;-luQ j + l , 1=1
j = N,N
+ 1 , . . .,
(QN-l = Qx), where at most L balls have a mn-empty intersection ( L i s a suitabk natural number), and d > 0 i s independent of j. (b) Systems {~$)(X))E’~, j = N , N + 1, . . ., s w h that
ID“p71U)(~)l ~,,2”“’, j
=
N,N
+ 1 , . . .,
1 = 1 , . . ., N ; , IyI
> 0 . (3)
i s an equivalent norm in W”,Q; e”; e’) (Definition 3.2.312, Theorem 3.2.412). Proof. The existence of balls KiJ)and functions yiJ)(x)is a consequence of the properties of the domains Qj,see Remark 3.2.312. Lemma 3.2.411 is also valid for the functions rpjJ)(x).Whence it follows in the same manner as in the proof of Theorem 3.2.411 that (4) is an equivalent norm in W#2; Q ” ; e”). Remark. The magnitude of d determining the radius of the balls can be chosen arbitrarily small. We shall use this fact in the next subsection.
6.3.2.
A-Priori-Estimates
Theorem. Let A E %EV(Q;~ ( s ) )in the sense of Definition 6.2.112, where v 2 0. Further, let x be a real number and 1 < p < 00. Then there exists a real number c1 such that, for every complex number il with Re 3, c l , there exist two positive numbers c2 and c3 with the property that for all u E Wtrn(Q;e X + P ” ; ex+p’)
P r o o f . Step 1. Theorem 3.2.412 and Theorem 3.2.4/3(a) yield that the left-hand side of (1) is true. (Here one needs the assumption v 2 0.)
6.3.2. A-Priori-Estimatee
41 1
Step 2. To prove the right-hand side of (1)we assume temporarily that the support of u E W;rn(12)is contained in one of the balls K g ) of Lemma 6.3.1. Let xj,,.be the centre of KAj).We set rn
A1u
Then me have
AU
=
c c z=o
l al = 21
D% - h,
@xal(Xj,k)ba(Xj,k)
- ;lu = A,u + AZu+ A,u.
Extending u ( x ) oufside of K,O"by zero, then it follows that
and taking into consideration xzl the Fourier transform that
21 +(v - p ) = v , then it follows with the aid of 2m
IlA14Ep(~;e~)
c
> 0 is independent of j, k, and A. Remark 2.2.414 and Definition 6.2.112 yield that
and
are multipliers for Re A 5 0, where the number B appearing in Remark 2 . 2 4 4 is independent of A, j,and k, while it depends on the constant of ellipticity C in (6.2.1/6). Then one obtains by Remark 2.2.414 that
412
6.3. A-Priori-Estimates
where the constant c > 0 depends only on C , but not on A, i, and k. Putting this result in (3), and returning to the original coordinates x in ( 2 ) ,then it follows that
Here cl, c2, c3 ,and c, are poaitive numbers depending only on the constant of ellipticity C , but not on 2, j, and k ; Re1 5 0. To estimate A2u we choose (in the sense of Remark 6.3.1) d sufficiently small. Then one obtains for x E KP) that @ q s )ba(z)-
6
@X?'(Xj,k) ba(Zj,$)l
C2.i("2'+')
Ix - X j , k J
6 E2jXZ'.
Here d = d ( E ) is a given number, E > 0. Theorem 3.2.412 and Theorem 3.2.413 yield
s Here 6c
E"
> 0 is a given number ; d = d ( d r ) > 0. If
w is a bounded domain such that
52, then we have
ll4lw;m-1(")
6
(5)
E " I I ~ l l ~ ( ~ ; e ~ + ~ ~ ; e ~ + ~ ~ ) .
E l l 4 w;%J)
+ C(E)
5 E'IJ~IIw~~(~;
Il~lILp(w)
ex+wp;e"+vp)
+
ll~ll~,(~; ex).
Here E' > 0 is a given number. (This follows from a formula similar to (4.10.1/13).) Using this estimate and the assumptions on the coefficients as(.), then it follows from Theorem 3.2.412 and Theorem 3.2.413 that
IIA~uIIE~(Q; e x ) S EIIuII&;~(Q; e x + w p ;
ex+vp)
+ C(E) l I ~ l l & ( ~ ;
IIAu - ~
L
~ l I Q ~ ; e x )
ciIIull&im(Q;
ex+pw;ex+pv)
(6)
ex).
Finally, one obtains by (4), ( 5 ) , and (6), and by a suitable choice of
+( ~ 2 14 ~
E
and E" that
~ 3 IIUll!ip(n;ex). )
(7)
where R e d 5 0. Here, the positive numbers c1 and c2 depend only on C , while c3 depends on C and on the constants of estimates for DYbJx) and D Y a s ( x ) in the sense of (6.2.117). (Clearly, these numbers depend also on the fixed domam 52 and on the fixed function e(s).) Step 3. Let u E W26n(SZ;ex+pJ'; e X + " P ) . Using Lemma 6.3.1, then it follows by (7) 00 .vj for c2(LIp - c3 2 0 and u = C C yj&)u that j=iV
k=l
413
6.3.2. A-Priori-Estimates
We have
A(vjq$'~)- 1vjq$~= vj~)($(Au -1 ~ )
+
C
c@,a(x)DYvjd!')flu*
IS1 2m lJlnl$2m-ISI
Using xlal+lp~
(9)
+ la1 < xlslfor (a12 1, then it follows that
C S , & ( X ) D"(yJj'P$) = O(@xl.+I#l@'"l) =
o(@"lBl-a),
where 6 > 0 is a n appropriate number. Similarly t o the second step, one obtains
Putting this in (€9,and choosing E the right-hand side of (1).
C
= 2and Re
2
1 sufficiently small, then one obtains
R e m a r k 1. The proof yields a sharper result than formulated in the theorem: T h r e exist a real number c1 and a positive number c 2 , depending only on the con-stant of ellipticity C i n (6.2.1/6), the constants of estimate for DYbJx) and DYas(x), and the o-behaviour of DYas(x) i n the seme of (6.2.1/7) (and In, e ( x ) ,p , and x ) such that for all complex numbers 1with R e il 5 c1 and all u E Wim(sZ;@ + p P ; e x + p v )
llAu -
~UIIL~(Q;
ex)
2 czllull +yg;
+ c2Vl
px+v)
llullLp(n;ew) -
(11)
Later on, we shall use this sharper version of (1). R e m a r k 2. The assumption v 2 0 is needed only for the proof of the left-hand side of (1). It is easy t o see that v 2 0 is a natural assumption. Namely, if v < 0 and il < 0, then A - ilE belongs t o '%Eo(Q;e ( x ) ) , but not to ?l$(In; e ( x ) ) .In this case the term -ilu belongs t o the ''main part" of A - AE, and one cannot expect i l ~ estimate l of the form (1). Let v >= 0. Since x in (1) is an arbitrary number, it is easy t o see that one can replace ilu by ile"(x)u with 0 5 v. Afterwards it follows that, in such a formulation of the theorem, the assumption v 2 0 is not necessary. We shall return t o this question later on, Theorem 6.5.1.
6.4.
&-Theory for -
-I-gU(x),v
>2
The results of this section and the a-priori-estimates of Section 6.3 are the basis for the further considerations. But the investigations of this section are also of selfcontained interest. Together with Theorem 6.6.1, they are the basis for the structure theory for the spaces Se,,,(In)in the eighth chapter.
-A
414
6.4. &Theory for
6.4.1.
Self- Adjointness
+ e'(z),
Y
>2
Lemma. Let 52 c R, be an arbitrary domain, and let Au = 0 for u E D'(Q). Then u is a harmonic function in the cbsicu2 sense. P r o of. Let p E C$ (52).Then pu can be interpreted in the usual way as a distribution belonging to E'(R,) c S'(R,). (See for instance H. TRIEBEL[17], p. 49/50, p. 103.) From the properties of distributions of E'(R,) and from (2.8.1/16), it follows for y E Corn(R,) that
l(pu)(w)l 6
Cz~~~~IW~(R,)*
c~lIWIICk(J?,)
Here k is a suitable natural number, and 1 is a natural number such that 1 > k Theorem 2.6.1 yields pu E Wiz(Rn). Further we have (in the sense of D'(R,))
n +. 2
Using Theorem 2.3.4 with s = -2 (or the usual rules for Fourier transforms in S'(R,)), then one obtains pu E Wi'+l(R,). Putting this in (l), then it follows m
.
Wi1+2(R,).Iteration and application of (2.8.1/16) yield guu E n Wi(R,) j=-m c C"(R,). Whence it follows the lemma. Remark 1. * Differentiability properties of the above type are well-known in the literature (theorems of Weyl-type, properties of hypo-ellipticity). One may generalize the lemma essentially. See, for instance G. HEUWIQ [l], IV, 3.4/3.5, and the references [3]. given there, and L. HORMANDER Theorem. Under the hypotheses of Definition 6.2.112 the operator A ,
pu
E
+ e'(s)u,
> 2, B ( A ) = C$(Q), is essentially selj-adjoint i n L,(Q).Its closure A is an operator with pure p i n t spectrum. Au = -Au
Y
Proof. Step 1. Clearly, A is a symmetric positive-definite operator. We want to show that it holds N ( ( A orE)*) = (0) for sufficiently large values of a > 0. Then, by well-known theorems, it follows the self-adjointnessof A (see for instance H. TRIEBEL [17], p. 206-207). Let A*v av = 0. Then in the sense of the theory of distributions we have -du e'@) v av = 0, v E L,(S). (2)
+
+
+
+
If o is a bounded Cm-domainsuch that 6 c 52, then -dw
= -&V
- eY(s)vEL&)
(3)
has a solution w E Wg(o)A W i ( o ) . (Dirichlet's boundary value problem for - A . We note that - A satisfies the hypotheses of Theorem 4.9.1. See also Remark 4.9.1/3. Since -d is positive-definite on W i ( o )A @(o),then it follows by Theorem 5.2.3/1, that 0 is an element of the resolvent set.) Applying now the lemma to v - w, then it Eollows v E Wg(o). Now, (3) and Theorem 5.2.2(c) yield that w belongs to Wi(w). Application of the lemma gives w E W,"(o).Using (2.8.1/16),then it follows by iteration w E C"(52).
6.4.1. Self-Adjointness
416
Step 2. For the proof of the self-adjointnessit is sufficientto show that for sufficiently large a > 0 any function w(x)with
-dw
+ e’(x) w + aw
0, w(x) E P ( QA)L2(Q), (4) vanishes identically. Assume without loss of generality that w(x) is a real function. For { ~ j } j ” E, ~!P (Definition 3.2.3/1) we set vj(x.0 = (e’(x)
=
+ a)-* C yl(x)
Then we have
j
1= N
E
Cz(Q), j = N , N
+ 1, *
n
n
Using (4) and
av 2 C - -avJ . -vjw ’l
l i = l axk
axk
1 2
=-
c n
k=l
a,p? - I - ,
axk
a$ axk
then it follows by (5)that
(z(x))2dz
Using the properties of e ( x ) and y&), partial integration yields
J-
n
fi yr)2axs
w2( l = N
J-wz n
2
k=l
Since v > 2, one can make the factor in the last integral arbitrarily small, independently of j, if one chooses a sufficiently large. Transfering this summand to the left-hand side and considering j * a,one obtains w(x) I 0.
Step 3. Now we have to prove that A is an operator with pure point spectrum. It follows from Theorem 6.3.2 and Theorem 3.2.4/1 that D ( A ) = W,”(Q;1 ; e2’). (8) By the theorem of F. RELLICR (see for instance H. TRIEBEL [17], p. 277), one has to prove that the embedding from D(A)into L2(Q)is compact. If xj(x)is the characteristic function of Q ( j ) from Definition 3.2.311, then it follows from the compactness of the embedding from W i ( S ( i )into ) L,(Q(j))(Theorem 3.2.5) that
Mj = { x j ( x )~ ( xI )IIU(~)IIw~(n;l;e’v)5 1) is a precompact set in L2(Q).Now it follows from
\ 11 - X , j ( X ) 1 2 lu(x)12 ax 5 2 4 ’
ir
[ @2”lU12a x fi
that M,; for j 2 j&) is a pre-compact &-netfor the image of the unit ball of D ( A )in L2(Q).Whence one obtains that the embedding from D(A)into L2(Q)is compact.
416
6.4. &-Theory for
+ e'(r). v > 2
-d
Remark 2. * The above proof is due to H. TRIEBEL[2].The used method, in particular estimates of the type ( 6 ) and (7), goes back to E. WIENHOLTZ [ l ] (see also I. M. GLAZMAN [2],Chapter 1 , Theorem 3.5). 6.4.2.
Eigenfunctions
Theorem. If A is the operator of Theorem6.4.1, then the eigenfunctions of A belongto SQ[=) (52) (Definition 6.2.1/1). Proof. Step 1 . We start with preliminaries. Let v E L2(Q),and let
-dv
+ @'(X)
v
=
g
E L,(52)
in the sense of the theory of distributions. We want to show that v belongs to D ( A ) . If 9 E C$(52), then (v, AV)L, = (9, d L , *
Whence it follows v E D(A*) = D ( A ) . Step 2. Let il be an eigenvalue of A, and let u(x)be an eigenfunction, Au Then, in the sense of the theory of distributions, we have
-du
=
ilu.
+ $(z)u = ilu.
(1)
Further, one obtains from (6.4.1/8)and from the first step of the proof of Theorem 6.4.1 that u E W,"(52;1 ; e2') A C@(52). (2) We want to show that @ ( x ) u ( x ) belongs to D ( A ) = % 1( ; e2') $ forIeach ; number oc 2 0. Let 26 = v - 2 > 0. The proof w i l l be given by induction. Assume that e e ( j - l ) ubelongs to D ( A ) for a natural number j . We shall show e"ju E D ( A ) .It holds
Since eE(j-l)u belongs t o D ( A ) ,it follows from (6.4.1/8)and Theorem 3.2.4/3 that
e&j+2I I < = ce8(j-1)+vIuI
E L2(Q),
(4)
Whence it follows that v = eeju and the right-hand side of (3) belong to L,(Q). Now it follows, from the first step, e8juE D ( & . Repeated application of Thec j . Hence e% E D ( A ) for all a 2 0. orem 3.2.413 yields @" E D ( A ) for y Step 3. If u is the eigenfunction of the second step, then we want to show that @DYu belongs to D ( A )for every number oc 2 0 and every multi-index y. We use again induction, firstly by Iyl = 0, 1 , 2 , . . .,and secondly for fixed y by oc = E j , j = 0,1,2,. .
.
417
6.4.3. Domains of Definition of Fractional Powers
Here, E has the same meaning as in the preceding step. Assuming that the statement is true for 0 IyI k - 1, then it follows for I/?I= k that -ADs, + @(z)Dsu = W U+ 1 c,,D"$"'fl-'h E L,(Q).
s
s
llllZ1
The first step and Theorem 3.2.4/3 yield Dsu E D ( A ) .Then one obtains in the same manner as in the second step that e"D% E D ( A ) ,OL 2 0. Step 4 . If the functions y j ( x )have the same meaning as in Definition 3.2.3/1, then it follows from (2.8.1/16) and from the above results
R e m a r k 1. One can show that, for v = 2, the Theorem 6.4.1 as well as the above [2], p. 165. This shows that theorem are generally untrue. We refer t o H. TRIEBEL v > 2 is a natural assumption. R e m a r k 2. Considering the above theorem and Theorem 6.2.3(a) it seems to be meaningful t o extend the domain of definition of the operator A from C$(Q) t o S e ( & ) ( 9I)f. one wants t o remain in the framework of L,-theory (or L,-theory), then (6.2.3/1)must be valid. But Theorem 6.2.3 shows that this is equivalent t o (6.2.3/6). Hence, for a n L,-theory (or L,-theory), (6.2.3/6) is a natural additional assumption. 6.4.3.
Domains of Definition of Fractional Powers, Isomorphic Mappings
I n the following considerations, we shall assume that there exists a number a 2 0 such that e-"(x) E L,(Q) (see Remark 6.4.2/2). T h e o r e m . Let A be the operator of Theorem 6.4.1. Further, let a 2 0 such that
e-"(x) E Ll(Q). (a) For s 2 0 it holds
*(a;
D(A8) = 1 ; e2'"). (b) For s 2 0, the differential expression -Au from @ + a @ ; I ; e 2 8 v + 2 ) onto W,88(Q;1 ; e2'"). (c) It holds
(1)
+ e'(x)u i s an isomorphic
(2) mapping
OD
D(A") =
( D l ( A j ) = AS'~(~)(Q)
j=O
(3)
(set-theoretically and topologically). The differential expression -Au + eV(x)uyields an isowwrphic mapping from Se,, (Q) onto itself. Proof. Step 1 . Let j = 0, 1, 2 , . . . It follows by Theorem 3.2.4/1, Lemma6.2.2, and Theorem 6.3.2 that IIuIID(zj) llull wijm 1; P"") for u E W,2J(Q;1 ; @jr)c ~(.li). (4)
-
27
Triebel, Interpolation
418
6.5. Lp-Theory
On the other hand, one obtains with the aid of (1) t h a t S@(,)(Q) c W,2j(S;1 ; e 2 J V ) .
(5)
But by Theorem 6.4.2, there exists a subset of A ! ~ ~ ( ~namely ) ( Q ) , the set of all finite linear combinations of eigenfunctions, which is dense in D ( A j ) . Then, ( 2 ) with s = j = 0 , 1 , 2 , . . . is a consequence of (4). Step 2. Now one obtains (2), for arbitrary values s 2 0, from Theorem 1.18.10 and from (3.4.218).(It holds HE = W2 .) Since A is positive-definite, whence it follows also the part (b) of the theorem. Step 3. Formula ( 5 ) yields (set-theoretically and topologically)
D(A")'
'Q(.Z)(')
(6)
Now, let u E D(2"). Then Theorem 3.2.413 yields e'DYu E La@)for arbitrary numbers 01 5 0 and arbitrary multi-indices y. Now one obtains in the same manner its in the fourth step of the proof of Theorem 6.4.2 the conversion t o (6). This proves (3). Clearly, -Au + @ ( x )u gives an isomorphic mapping from D(A") = SQ(r)(Q) onto itself.
6.5.
L,-Theory
In this section, anLp-theoryfor operators A belonging to %Ev(Q;e ( x ) )will bedevelop-
ed on the basis of the previous considerations. Differentiability properties and the behaviour near the boundaxy of solutions of the equation Au = f (or Au - Ae"(x)u = f ) will be described as before with the aid of isomorphic mappings between function spaces generated by A (or by A - Ae'(z)).
6.5.1.
A-Priori-Estimates (Generalization of Theorem 6.3.2)
Using the results of Section 6.4, one can generalize Theorem 6.3.2 essentially. Theorem. Let A E 2iEv(Q; e ( x ) ) in the sense of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E L,(Q). Let (T v, 1 < p < 00,
'
0, 1 , 2 , . . . Let z = - > 2. Then there exists a real number c1 m such that for all complex numbers A with Re 1 5 c1 there exist two positive numbers c2 and c3 such that for all u E W:m+2k(Q;px+pJ'; ex+p(v+ks)1
x real, and
k
=
C31JU11w;'"+k'(n;
@ x + P r ; ex+P(v+kr))
2 IIAu - Ae'(x) u l l ~ r e(x ;~q x;+ P * r ) 2 c211uJIw2'"+k'(Q; Q x + P r ; @X+P(Y+kZ)).
Proof. Step 1. We start with preliminaries. Let
Bu
=
(-A
+ @'(X))kU
- '1u, '1 5 0 .
6.6.1. A-Priori-Estimates(Generalizationof Theorem 6.3.2)
419
Theorem 6.4.3 yields that B is a n isomorphic mapping from Se(z)((sz)onto itself. Then it follows from Lemma 6.2.2, Theorem 6.3.2, and Theorem 3.2.411, that B for rj 5 c is a n isomorphic mapping from W;"(S; ex;e x + P T k ) onto LJQ; ex). Step 2. By the first step, it holds for u E C$((S)
IIAu - Ae"(z)ull W F ( Q ; e x ;
ex+pk+)
-
IIB(Au - le"(z)~ ) l l ~ e~x ) .( n ;
(3)
Considering the operators p s ( - A + p y (z))and A , then one has the same situation as in the proof of Lemma 6.2.2. Whence it follows that V-P
(-A
+ @')A = e--
P
=
Iteration yields
BA E 91;;:k'p;
P
ern ( - A
?L,":,',(Sz;
+ em)A v-P
-L
Ee
rn
e(z)).
n1+1
) I l P + ~ , . + ~ ( ( s ze(z)) ; rn
nr
e(4).
Since CT 5 v, the left-hand side of (1) is a consequence of Theorem 6.3.2. Step 3. We prove the right-hand side of (1). Since x is an arbitrary number, one may assume 0 = c 5 Y without loss of generality. If B has the above meaning, then it follows, in the same manner as in the second step, with the aid of the proof of Lemma 6.2.2, that BAu = ABu + 2 afl(z)D8u. (4) 161 < 2m+2k
Here, the last term is a perturbation of BA (or AB) in the sense of (6.2.1/5). Then, from Theorem 6.3.2, the counterpart t o (6.3.2/6),and (3) it follows that
One obtains by the second step that
[(-A
+ e')"
- @I B E q g ; : m + k ) ( Q ; e(4).
Applying Theorem 6.3.2, putting the result in ( 5 ) , and choosing E in a suitable way, then it follows that llAu - ~ u ~ ~ w Fe(x D ; e x;t p k r )
>= c I I J u J ( ~ ~ ( ~ + ~ ) ( Q ;
extwp; extp(v+kr))
- C,IIUIIL,(O; p x t v q .
(7)
(Here we used Y 2 0 and Y > p.) Transfering the last term t o the left-hand side and estimating it with the aid of Theorem 6.3.2, then one obtains the right-hand side of (1). 27*
420
6.5. Lp-Theory
R e m a r k 1. The special choice t
'
=- makes
the proof easier. But probably m the theorem remains also valid if one assumes only t > 2. R e m a r k 2. The proof is based on Theorem 6.3.2, and it uses the special operator -A + e"(x). This shows that one may carry over the important Remark 6.3.2/1: There exist a real number c1 and a positive number c2 depending only on the constant of ellipticity C from (6.2.1/6), the constants of estimates for DYb,(x) and Dyas(x),and the odehaviour of Dyas(x)in the sense of (6.2.117) (and Q, e(z), p , x , k,and a), such that for all complex numbers 1 with Re 1 5 c1 and all u E ~ ~ m + ' c ) ( Qe x;+ P p ; ex+P(v+kr) )
l!Au - A@"(%) u l l w ~ ( n ; ex;
QX+Pk)
2 C211UIIw2(m+.t)(n; e x t p r ; eu+P("+kr)).
(8)
Here, c1 is also independent of k. This improvement w i l l be very useful, later on.
6.6.2.
Isomorphism Theorems
In this subsection there are proved some of the main results of this chapter. Theorem 1. Let A E UEv(Q; e(x)) in the sense of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E Ll(Q).Let 0 < v, 1 < p < co, 1 5 q 5 03, and let x be a real number. Further, let c1 5 0 be a real number in the sense of Theorem 6.5.1 and Remark 6.5.112. (a) If Re 1 5 cl, then A - 1e"(x)gives an isomorphic mapping from tSp(x)(Q)onto S d X ) (Q). (b) If Re 1 5 cl, and if s
2 0 , then A - ?&x)
H ; m + S ( Q ; @ x + p P ; ex+pv+sp
( c )If Re 1
5
cl,
and if
s
'
P r o o f . Step 1 . Let
+ e"(X))"
BU = ( - A
s)onto
H;(Q;
@x+sp
s).
> 0, then A - le'(x) gives an isomorphic mapping from
BS+zm(Q.ex+PP ; @X+PV+sP P.P
gives an isomorphic mapping from
5) onto
B;,,(Q; @*;@ X + s P
s)
u - 1@(x)u = B0u - I@(x) U ,
,
(1)
where t > 2, rj < t m , and 1 < 0. We want to show that B gives an isomorphic mapping from Secz) (Q) onto itself. By Theorem 6.4.3, B,, where D(Bo)= Gm(SZ; 1 ;e Z r m ) ,is self-adjoint and positive-definite. Further, it holds that
D(B:)
=
W,"mk(Q;1 ; eZrkm),k = 1 , 2 , . . .
By the previously developed technique of estimates it follows that
Bku = B ~ u+ Du, IIDUIlr,,(n) 5
EIIuII~?'~Q;
1; e * r h )
+ c b ) Ilull~,~~)
5 E'IIB~UllL,(Q) + C'(&')llUll&(Q).
6.5.2. Isomorphism Theoreme
42 1
Here E > 0 (resp. E' > 0) is a given number. Now, the criterion of self-adjointnessof T. KATO(see H. TRIEBEL[17], p. 209) yields that Bk, where D(BIC)= D(B$),is a self-adjoint positive-definite operator. By Theorem 3.2.411, CF(Q) is dense in D(Bk). Hence, Bk, D(Bk) = Wirnk(Q; 1 ; elrkm),is the k-th power of the positive-definite Now, (6.4.3/3) yields that B self-adjoint operator B , with D ( B ) = Wim(Q;1 ; eZrrn). gives an isomorphic mapping from Spcz. (Q) onto itself. Step 2. Let A be the operator of the theorem, and let B be the operator of (1) V - P where t = n2
and 7 = c - p. We set for 0
a
5
1
Aau = &(A - A@(x))u + (1 - a)@(z) ( B , - Re A * @(x)) u = aAu + (1 - a) @(z)B,u - (an + (1 - a ) Re A) @(z) u .
(2)
+
Since Re (an (1 - a)Re A ) = Re A, it follows by Theorem 6.5.1 and Remark 6.5.1/1 that there exists a number c1 independent of a and k such that (6.5.1/2) holds with IXA (1 - a)e"(z) B, instead of A . Remark 6.5.1/2 yields that c, is independent of a. Assuming that for a given number 0 5 a, c 1 the operator Amogives an isomorphic mapping from
+
R 1 --
J+'2(m+jC)(Q; @ + P P ; e x + P ( v + k r ) )
then it follows that
P
onto R, = W F ( Q ; e x ;
f+Pk),
A,u = f E R, is equivalent to ~1
It holds that
+ A,f(Aa - Am,)u = A,:/
E R,
(4)
IlA;f(Am - ~ m , ~ l l I ~ 5 l +1 I01 ~~ O1ol c < 1 for
la - a01 < C 1
Here, c is independent of 01 and a,. Now one obtains that (4) has a unique solution. For these values of a, there exists A i l . The first step and Theorem 6.5.1 yield that A,' exists in the above sense. Now it follows, by iterated application of the above procedure, that A - Ap'(x) is an isomorphic mapping from R, onto R,.For x = 0 and p = 2, one obtains from (6.4.3/3) that A - A@(x) gives an isomorphic map(Q) onto itself. ping from Step 3. Part (b) and part (c) of tthe theorem are consequences of the second step and of Theorem 3.4.2. Remark 1. As a special case the theorem contains the assertion that A - &"(z) gives an isomorphic mapping from wgrnts
(Q; @ x + p p ; e x + P v + s P q
onto
W> (Q; p; p
~ + ~ p s ) .
For 0 = 0 this result was proved in H. TRIEBEL[24]. R e m a r k 2. The above theorem is comparatively general. We describe two simple conclusions.
422
6.F. Distributions of Eigenvalues, h c i a t e d Eigenvectors
c((Q)
-
1. Let Q c R , ,be a bounded domain, has the meaning of Lemma 6.2.3, and e - l ( x ) d(z) is the smoothed distance function in the sense of Remark 3.2.3/1. Then it follows from the above theorem and from Lemma 6.2.3 that Au + ( u + e"(x)u, v > 2m,
c$
is an isomorphic m p p i n g from (Q)onto itself. (Here one has t o use that A is positivedefinite in L2(Q).)The same holds for @(x) ( u + @"x) u where v - ,u > 2m. 2. If Q = R,,, then Remark 6.2.113 yields that
AU
=
(1
+ lxl2)V1(-d)l~lu + (1 +
is an isonzorphic mapping from
I~(')qzU,
2
> q17
S(R,,) onto itself.
T h e o r e m 2. Let A E 21Ev(Q; e(x))i n the seme of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E Ll(Q).Let x be a real number. Then A , considered as a mapping from W;m(Q;e x + P C ; e x + P v ) into Lp(Q:en), is a @-operator with the index O.*) P r o o f . If CT < v and if c1 has the meaning of Theorem 1, then A , considered as a mapping from W?(Q, e X + P P ; QX+P") into &(Q: e x ) , is a @-operator if and only if A ( A - c1@'(x))-l, considered as a mapping from L P @ ):'Q into itself, is a @-operator. This is a consequence of Theorem 1. Both the operators have the same index. It holds that A ( A - CleU(x))-l= E
+ cleu(x)( A - Cl@(Z))-l.
(5)
In the same manner as in the third step of the proof of Theorem 6.4.1, it follows that ZL + @(x) u is a compact mapping from W?(Qn;
@X + P P ;
ex+"")
into Lp(Q;ex).
Then, ~ " ( x( )A - cle"(x))-l is compact operator in L,(Q; ex).Now, (5) yields that A ( A - cleU(x))-l is a @-operator with the index 0. R e m a r k 3. The theorem is the counterpart to Theorem 5.2.2(b).
6.6.
Distributions of Eigenvalues, Associated Eigenvectors, and Green Functions
This section is the counterpart t o Section 5.4. The investigations on distributions of eigenvalues are of interest, later on, in the framework of the structure theory for the spaces h z )
can,.
*) The index of a @-operatoris defined as the difference between the finite codimensionof the range and the finite dimension of the null apace.
6.6.1. Distributions of Eigenvalues
423
Distributions of Eigenvaliies and Domains of Definition of Fractional Powers
6.6.1.
The used symbols have the same meaning as in 5.4.1. T h e o r e m 1. Let the differential expression A belonging to the class '%Ev(!2;e ( x ) ) (Definition 6.2.1/2) be formally self-adjoint. Let v > 0. Further, it is assumed that there exists a number a 2 0 such that e-"(x) E L,(Q).Then A ,
D ( A ) = W$"(O;$"; $'), (1) is a self-adjoint operator, bounded from below, with pure point spectrum in L,(sZ).There exist two positive numbers c1 and c2 such that inin (p,0). I f s 2 0, then D(A") = Wp"(f2; pa""; eaSv), (3) provided that A , without loss of generality, is positivedefinite. P r o o f . Step 1 . Theorem 3.2.4/1 yields that C,"(!2) is dense in D ( A ) . Hence, A is a symmetric operator. Now it follows from Theorem 6.5.211 that A is a self-adjoint operator. (For appropriate A, there holds R ( A - ilE) = L,(Q)).Further, one obtains from Theorem 6.5.211 that (3) is valid for s = I = 0, 1, 2, . . . For general values s 2 0 , (3) follows by interpolation from Theorem 1.18.10 and (3.4.2/8). (It holds Wg = H g . ) One obtains in the same manner as in the third step of the proof of Theorem 6.4.1 that the embedding from D ( A )into L2(f2)is compact. (Here, one uses the assumption v > 0.) By the theorem of F. RELLICH(see for instance H. TRIEBEL [17], p. 277), A is an operator with pure point spectrum. Step 2 . We prove the left-hand side of (2). Let K , and K , be two open balls such that I7,c K , and If, c 0.Let S be an extension operator from W,2"(K1)into fki'"(K2)c D ( A ) in the sense of Theorem 4.2.2. (The extension operator of Theorem 4.2.2 is multiplied with ~ ( xE)C$(K2)where ~ ( x=) 1 in a neighbourhood of K , . The functions are extended by zero outside of K 2 . ) Let R be the restriction resp. from L,@) onto L,(K,). Denoting embedding operator from D ( A )onto Wim(K1), operators by I , then Here ,C
=
I W ~ m ( I \ l ) - t L t ( K ,= )
RID(A)+L2(i$.
Theorem 1.16.1/1 yields sI ( I w, 2mwI)-tf2wI)) 6 cs,(IwwLm))-
The left-hand side of ( 2 ) is proved by Theorem 3.8.1 and Theorem 5.4.1/1 (and (5.4.1/5) and (5.4.1/6)). Step 3. We prove the right-hand side of (2). Let ilbe a (sufficiently large) positive
+
number, and let j A = [log, A;] 1. If Q ( Jhas ) the same meaning as in Definition 3.2.3/1, then Q ( j l ) will be covered by cubes Ql of the side-length d2-jA which are parallel to the axes. By (3.2.3/7), one can choose d > 0 such that
f2Wc
(J Q1 c Q ( J A + ~ )
(4)
424
6.6. Distributions of Eigenvalues, Associated Eigenvectors
(ais independent of in).It holds that Using (3.2.3/7), then it follows that one needs for the above covering a t most a -+-
a
LA 5 c p 2.iA'L I - c21
n
(6)
cubes. It holds that
In the sense of Remark 5.4.113the self-adjoint operators belonging t o the ECilbert spaces Wim(Q;ew;e2') (with respect t o L,(Q)), Wim(Q - U Q 1 ; e2"; e2") (with respect t o L,(Q - U QJ), and f l m ( Q 1 ear, ; eaU)(with respect t o L2(Qz)) are denoted by A,, , A , , and A z, respectively. It holds t h a t j5J-Q) = Lz(Q -
UQd
LA
CB
C @ L2(Qd
>
Here, E is a sufficiently small positive number. Hence N A , ( A )=0. Setting ,ii =min(p,O), one obtains that 2
I I ~ wzm(Q1; II
q2p; e z v )
2
2
2ii -
2
c ~ ~ ~ ~ " ~ ~ I I ~ I~ I ' 'YL1 ~I I~~ (I I Q ~ V~ ~)~ (2 Q , )
> 0, cr > 0. If B is the operator belonging t o the quadratic form IIUll'$im(p,) respect to L,(QJ),then, using (6) and Theorem 5.4.1/2, it follows that
c
(with
If Q is the unit cube and if D is the operator belonging t o the quadratic form I I u ~ ~ $ ; ~ ( Q ) , then NB(7) 5 cND(7).
(9)
This is a consequence of the transformation of coordinates mapping Q1 onto Q, and a comparison of the corresponding quadratic forms. Theorem 5.4.111 and Theorem 3.8.1 yield N&)
II
5 ~ 7 % .Then, it follows from
(8) and
(I))that
6.6.2. Associated Eigenvectors
425
Remark 1. The estimates of the third step can be improved. On the other hand, formula (2) shows clearly the influence of the different parameters, in particular of a and v. An asymptotic formula cannot be expected under these general assumptions. Theorem 2. Let Q c R, be a bounded C”-domain. Let e ( x )be a function in the sense of Definition 6.2.111, where e-l(x) d ( x ) near the boundary. Here d ( x ) denotes the distance of a p i n t x E Q to the boundary. Then it holds for the operator A from Theorem 1, where p > -2m, that n-1 2m if - 2 m < p < - - , n N
2m
n
1% log I
if p = if
-
/A>--,
7
2
9
c > 0.
(10)
2m n
Proof. The theorem is a consequence of ( l ) , Theorem 3.8.2, Theorem 5.4.1/2, Theorem 5.4.1/1, (5.4.1/5), and (5.4.1/6). Remark 2. The theorems show that N ( 1 )
2m
n
N
2% holds, provided that p > - -.
n See Theorem 5.4.2. The differential operators of Theorem 2 are closely related to a special class of Tricomi differential operators. We shall return to this question in Remark 7.8.312.
6.6.2.
Bssociated Eigenvectors
The used notations have the same meaning as in Subsection 5.4.1. Theorem. Let A E 2?Ly(Q; e(x)) in the sense of Definition 6.2.1/2. Further let v > 0 , 1 < p < C Q , and e-a(x) E L,(Q) for an appropriate number a 2 0. Then A with the domain of definition
D ( A ) = W;m(s2;e p r ; e P ” ) is a closed operator in Lp(Q).Its spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The eigenvalues and the associated eigenvectors are independent of p . The associated eigenvectors are elements of Secx.(Q), their linear hull is dense in ~Y~(~)(s2). Further, the linear hull of the associated eigenvectors is dense in all the spaces Wi(Q; ex;e’) where 0 s < co, 1 < q < CQ, - m < x + sq z c m (hence, it is also dense in Lq(Q)). Proof. Step 1. Let p = 2. We set,
s
1
-4u = 2
-
C C z=o Ial=21
b,(x) PU+ P [ ( e x 2 1 ( x )b,(x) u)] + BU
111
[ex21(s)
AU
+ Bu.
(1)
426
6.6. Distributions of Eigenvalues, Associated Eigenvectors
It follows from the proof of Lemma 6.2.2, that A belongs also to kFy(Q;e(z)) and that B is a perturbation operator, where its coefficients have the property (6.2.1/8). A is formally self-adjoint. Hence, it follows by Theorem 6.6.1/1 that A , where D ( A ) = WB,m(Q; p ;p ) , isself-adjointinL,(Q). Let k A where
D(A)=
= 0 , 1 , 2 , . . . Now,
oneobtains by Theorem 6.6.1/1 that
@ z ( k + l ) r ;@Z(k+l)V)
(2) is a self-adjoint operator with pure point spectrum in the Hilbert space H = Wikm(Q;ezkr;eWv)(after introduction of a suitable norm). To apply Theorem 5.4.113 to the operator A with the domain of definition (2), we use the decomposition (1). It holds that w;(h-+l)m(Q;
IlBuIlfI = I ( B U I I M ~ , ~ ~S~ c; ~ *~ 2~/ i~c ; , Z ~ JW@“18’-6(5) ) rll+2nr-1
1
p u 1 2
dz)T,
R
(3)
= Ej
m
V + -(2km + 2m - j ) . m
(4)
Here 6 > 0 is a suitable number. The last formula is a consequence of the technique of estimates developed in the proof of Lemma 6.2.2, in particular (6.2.Z.lZ) and (6.2214). (Here one has t o take into consideration that A belongs not only t o %Ev(Q; e(z)), but also t o $&(Q; e(z)).) Setting 6 = Zvs, then
Without loss of generality let 0 < orem 3.2.413 that
1 s < -. 2m
IIBuIJUS cllull w 22( B t l - l ) m ( n ; e r ( k + i - a ) r :
If I is the embedding operator from w;(k+l)m(Q;
$(k+l)~
;
e2(’”+1’”)
into
+
(5)
Then it follows by (3), (5), and The(6)
ez(~+i-a)v).
J,j7?4(’”tI-*)m(Q;
@Z(ktl-.)p.
ea(rtl-a)v
),
and if ilis a complex number with I m il 0, then B ( A - AE)-l considered as an operator from H into H can be represented a s B(A - ilE)-l = B I ( A - ilE)-l. (7) Here, B on the right-hand side is a bounded operator in the sense of (6), while (d - U3-l is a bounded operator acting from H into D(A) in the sense of (2). Theorem 6.6.1/1 and the proof of Theorem 5.4.111 yield that I from the right-hand side of (7) belongs t o Gr where r is a suitable number, 1 < T < 01). Then it follows from Theorem 5.4.113 that A , with the domain of definition (a), is a closed operator in H where the spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The linear hull of the associated eigenvectors is dense in H .
6.6.3. Green Functions
427
Step 2. Let again p = 2. It follows from the first step that the linear hull of the associattecl eigenvectors of the operator A , where D(A) = qm(5 ear; 2; e”), is dense in the Hilbert space L,(Q).Theorem 6.5.211 and Theorem 3.2.413 yield
D(A”) =
m
n D ( A ~=)
j -0
n 00
p i ;p
~zjm(52;
j-0
j ) =
S~(~)(L?).
(8)
(See t,he third step of the proof of Theorem 6.4.3 and the fourth step of the proof of Theorem 6.4.2.)Here, the last equality must be understood not only set-theoretically, (52) but also topologically. Hence, the associated eigenvectors are elements of SQ(=) = D ( A m ) .Now one obtains that these associated eigenvectors coincide with the associated eigenvectors in Ekm(Q; e 2 k p ; ew’). Together with (8) whence it follows that the linear hull of the associated eigenvectors of the operator A is dense in (52). S t e p 3 . L e t O S s c 0 0 , lc q c oo,and-oo < x + s q s z < c o . T h e n w e h a v e fJ&)
(52) = W p - 2 ; ex;e’) .
ByTheorem 3.2.4/1, C$(52) (and hence also SQcx)(52)) is dense in W$2; e x ; e‘). Now one obt#ainsfrom the second step that the linear hull of the associated eigenvectors of d is dense in W @ ; ex; e‘). Here, in the sense of the theorem all values 1 < p < 00 are admissible. This follows from (8),after replacing there 2 by p , and the fact that ( A - iZE)-l, where D ( A ) = WEm(Q;e p w ; el’”) for suit,able values of ?, is a compact operator in L,(Q). (See Theorem 6.3.2 and Theorem 6.9.1.) R e m a r k . The theorem was formulated in H. TRIEBEL[25] without proof. A variat,ion of the theorem was proved by H . KRETSCHMER [l]. The proof of the theorem is also based on the criterion of I. C. GOCHBERC, M. G. KREJN by H. KRETSCHMER from Theorem 5.4.1/3. 6.6.3.
Green Functions
The methods developed in Subsection 5.4.4 are valid for general operators in Hilbert spaces. In this subsection they are applied t o operators belonging t o %K,(SZ; e(z)). Let, v > 0 and let E L,(SZ) for a suitable number a 2 0. By Theorem 6.5.2/2, ,@p; e2”), is a @-operator every operator A E %K,(SZ; e(z)), where D(A) = V,””(SZ; in L,(Q). With the aid of Theorem 6.5.211, it follows that A* is the formally adjoint operator to A with the domain of definition D(A*) = D(A).In particular, A* is also a @-operator in L,(Q).Now, one can apply the considerationsin front of Theorem 5.4.4 and introduce the operator A,
Au = Au, D(A) = D ( A )n B(A*). (1) A generates an isomorphic mapping from D( A) onto R(A).I n this sense we construct the operator A-l. T h e o r e m 1. Let A E fA;,(SZ; e(z)) (Definition 6.2.1/2), v > 0, and e-“(z) E L,(Q) for a suitable number a 2 0. Further, one sets in t h seme of Theorem 6.6.1/1
7
u = -V a + n + ( v -
a&j.
428
If
6
6.6. Distributions of Eigenvalues, Associated Eigenvectors
< 2 , then k1 can be represented in the form
(A-Y)(4=
1 G ( x ,Y)f ( Y )d?/,
(3)
n
6
G(x, y ) E WT(D; Q’”; Q’”) $ L 2 ( 0 )A L2(D) K”(0;e’”; e”), (4) where 0 t < 2 - cr. Proof. The first step and the beginning of the second step of the proof of Theorem 5.4.4 may be carried over without any changes. The operator A*A is selfadjoint , D ( A * A ) = Wim(D;e4P; e4’).
In the sense of (5.4.4/6), its positive eigenvalues are denoted by 2.. The counterpart to (5.4.4/9) is
Applying Theorem 6.6.1/1 to A*A, and putting il = il; in (6.6.1/2), then one obtains j 5 c(A; + 1). Whence it follows that (5) converges if N + co. By Theorem 6.6.1/1, it holds that
D ((A*A)$)= WGm(D; Q’”; e ” ) . This proves the theorem. (See the second step of the proof of Theorem 5.4.4.) Theorem 2. Let D c R,,be a bounded C”-domain. Let e(x) be a function in the sewe of Definition 6.2.1/1 where Q-l(x) d ( x ) near the boundary. Here, d ( x ) i s the distance of a p i n t x E I2 to the boundary. If A is the operator from Theorem 1 where p > -2m, then A-l can be represented in the form (3),(a), provided that n - 1 2m n 1 0 t m < 2m - m if - 2 m < p < - and 2 m - - > -p---, 2m. p n 2 2 N
+
2m
n 2
5 zm < 2m - -
n
and 2m - - > 0 . if p z - - n 2 Proof. By the proof of Theorem 1, one has to show that (5) converges if N -P 03. Applying Theorem 6.6.112 to the operator A*A with the eigenvalues A;, then one obtains that
0
i .:’ 1 -
j s c + c
AIKlogAj
if if
-2m
2m n
2m
Under the formulated conditions for z,whence it follows the convergence of (5) if
N+
00.
Remark. Both the theorems are the counterpart to Theorem 5.4.4. For some important special cases A E ‘91Ev(12;&)), differentiability properties for Green functions are considered by B. LANQEMANN [ l , 21.
STRONGLY DEGENERATE OPERATORS
6.1.
Introduction
ELLIPTIC DIFFERENTIAL
Beside regular elliptic differential operators, degenerate elliptic differential operators have been extensively considered in the last years. There are many, very different, types of degeneration. On the one hand the domain 9 c R, may be unbounded, or on the other hand the coefficients of the differential operator may be singular if 1x1 + 00 or x + aQ, or they have singularities in 9. Further, it may be happen that the ellipticitiy condition (4.9.1/2) degenerates if x -+ a 9 . Clearly, there are many possibilities of variation, and one cannot expect to catch all these different types by a uniform theory. This chapter and the following one are concerned with some (comparatively comprehensive) classes. The class considered in this chapter is characterized in a very strong degeneration of the coefficients of the operators near the boundary (and at infinity). This has the consequence that the boundary does not play any role (in contrast to the regular elliptic differential operators of the last chapter). For instance, a formally self-adjoint operator of this type with the domain of definition C$'(sZ) is essentially self-adjoint in A&?).
6.2.
Definitions and Preliminaries
In this section, the classes of operators coiisidered here are defined. Further, some
related locally convex spaces are introduced. Finally, there are proved some simple properties.
6.2.1.
Definitions
I f Q c R,,, then, as usual, Cm(Q) denotes the set of all infinitely differentiable complex-valued functions defined on Q. Definition 1. Let Q be a n arbitrary domain in R,,.Further, let p(x) E Cw(Q) be a positive function such that : 1. For all multi-indices y , there exist positive numbers c, such that IDr@(X)I
cr@'+lqx) for all
2 E9.
(1)
406
6.2. Definitions and Preliminaries
2. For any positive number K , there exist numbers cK > 0 and rlc > 0 such that e(z) > K if d ( x ) 5 cK or if 1x1 ( d ( z )is the distance to the boundary). Then ~!5'@(~)(9) denotes the locally convex space
Se(r,(f2)
=
{f If
IIfIIt,a
E CoD(Q),
=
1 rI,, ( X E D )
(2)
e6(z)l o " f ( ~ ) < I
(3)
SUP X€Q
for all 1 = 0, 1, 2 , . . ., and a11 multi-indices a } . Remark 1. Comparison with Definition 3.2.3/1 shows that (3.2.3/1) is replaced by the sharper assumption (1). The functions e(z) considered in Remark 3.2.3/1 satisfy also the above assumptions. In particular, in every bounded domain there exist functions e ( x )such that e-l(x) coincides essentially with d ( x ) . Definition 2. Let f2 c R, be a n arbitrary domain, and let e(z) be a weight function in the sense of Definition 1. Further, let m be a natural number, and let ,u and v be real numbers such that v > p + 2m. One sets 1 x i = -(v(2m - I ) + p l ) , 1 = 0, 1 , . . ., 2 m . (4) 2m (a) The class %&(9; e ( x ) )consists of all differentia2 operators of the form
Here, ba(x)E C m ( 9 )are real functions, la1 = 21 with 1 = 0 , 1, . . ., m, where all their derivatives (inclusively the functions themselves) are bounded in 9. Further, it is assumed that there exists a positive number C such that for all 6 E R, and all x E 9
(ellipticity condition). Let as(x) E C " ( 9 ) , 0 5 IBI < 2wk, and Dyas(x) = o( e x l p l + 1 ~ (1x ) ) for all multi-indices y .
> 0 there exists a natural number j ( E ) such that EpxioI+ I I ~ ( x ) for z E a - , W e ) ) , (7b)
(This means that for a n y number lDYas(x)l
s
E
where 9 ( j ) has the same meaning as in Definition 3.2.3/1.) (b) The subclass '&&(9; e ( x ) )of %&(i2; &)) consists of of all operators of %&(9; e(x)) for which there ex%& a positive number 6 > 0 such that
s
DYas(x) =
(8)
o(@x161+lYl-8)
for 0 IpI < 2 m and for all multi-indices y . Remark 2. Clearly, (8) is sharper than (7). Remark 3. %Ev(f2;e ( x ) ) and % E y ( 9 ;e(z)) are comparatively comprehensive classes of degenerate elliptic differential operators. We describe two simple examples. (a) If 0 is an arbitrary bounded domain, and if e-l(z) d ( x ) in the sense of Remark
-
6.2.2. Powers of Strongly Degenerate Elliptic Differential Operators
3.2.311, then Au = f(z) ( - & I
+ e”(z) a,
u
v >. p
407
+ 2?n,
belongs h@”(Q; e(z)). If Q is a bounded Cm-domain,then one can put e(s)= d - l ( z ) near the boundary. (b) If Q = R, , then any operator of the form
AU = (1
+ Izls)*l
u
+ (1 +
q2 > ql,
1 ~ ” ) ” s ~ ~
rlr (R,; (1 + I Z ~ ~ )provided ~), that 6 > 0 is sufficiently small. a a
belongs to the class
R e m a r k 4. * The above definition goes back to H.TRIEBEL[24]. Differential operators of the above type are closely related to investigations on the structure of nuclear function spaces. We shall return to this problem in Chapter 8. In this connection the special case
A ~= L -Au
+ e”(s)
U,
v > 2,
in bounded domains was considered in H. TRIEBEL [2]. Extensions of these investigations to unbounded domains as well as generalizations and improvements can [l,21, B. LANQEMANN [l, 21, D. KNIEPERT [l], be found in E. MULLER-PFEIFFER and H. TRIEBEL [lo, 241. Similar differential operators are considered in L. A. BAQIROV [l] and L. A. BAGIROV, V. I. FEJQIN[l].
6.2.2.
Powers of Strongly Degenerate Elliptic Differential Operators
We need for the later considerations that powers of strongly degenerate operators in the sense of De€inition 6.2.112 are also operators of such a type. L e m m a . (a) If A E%E,(Q; e(z)) in the sense of Definition 6.2.1/2, then Ak E % ~ E ~ ~e(s)) ( Q for ; k = 1,2, . . .
(b) I f A ~iflE,(Q;e(x)),then Ak E&;&(SZ; e(z))for k = 1 , 2 , . . . Proof. The proof is given by induction. Assume that the lemma is true for k = 1 , . . ., j. Then we have
c c nij
Aju
=
1=0 l a l = 2 1
xp’ = y
@4{)(Z a$’(.) )
2rnj - 1 2rn
+P
1 G Y
D*U
+
c
c2nrj
1 = 0, 1 , .
UP@)
DBU,
(1)
. ., 2mj.
The coefficients have the properties mentioned in Definition 6.2.112. xjj)
+ x , ~= x$gi), 1 = 0 , 1 , . . .,2rnj,
s = 0,1,
. . ., 2m,
(2)
yields that the ‘‘ main part ” of A j + L = Aj(Au)has the desired structure (inclusively (6.2.1I S ) ) . Using and
lDe”(s)I5 cex+lyI(z), x real number, xij)
+ x, + 1y1 = xjcl) + ~ y
(3) for
o < 1y1 5 z + s,
(4)
408
6.2. Definitions and Preliminaries
then it follows that the second summand in (1)has also the desired form. This is also ( x;) ) . true if A is an element of i i ; ~ e~
6.2.3.
Properties of the Spaces S,(,)(sd)
A complete locally convex space where the topology is given by a countable set of semi-norms is said t o be a n (P)-space. T h e o r e m . (a) Se(,)(Q)from Definition 6.2.1/1 is an (F)-space.Corn@) i s a dense subset. (b) If there exists a number a > 0 such that e-"(x) E L,(Q),then we have for all p with 1 5 p 5 oc) ( i n the sense of continuous embedding) Se(.x)(Q)c L p ( Q ) . (1) (c) If (1) i s true for a suitable number p with 1 5 p c 00, then there exists a aumber a > 0 such that e-"(x) E L,(Q). Proof. Step 1 . We prove that C$(Q) is dense in Se(&2). If the domains Q(J) have the same meaning as in Definition 32.311, then there exist functions q ~ j ( xE) C ~ ( Q ~ + l ) ) such that v j ( x ) = 1 for x E QCj) end lDyvj(z)l 5 cY2Jlrl (2) for j = N , N + 1, . . . and for all multi-indices y. See Remark 3.2.312. (Outside of Q(J+l)we set p i ( x )= 0.) I f f E Se(,)(Q), then v j ( x )f ( x ) E Corn (Q) approximates the function f(s)in Sec,)(9).It is easy t o see that X,(,)(Q) is an (P)-space. Step 2. The part (b) of the theorem is clear. To prove (c), we show at first that there exists a number b > 0 such that I Q ( j + l ) - Q ( j ) l 5 bj, j = N , N + 1 , . . . (3)
Assume that there does not exist a number b having the property (3). We fix a natural number k and a sequence O < a l < a 2 < . . . < a l < ..., a l + c o if Then there exist natural numbers jz > N + 1 such that IQth+l)
1-oc).
(4)
- Q ( j q > ail, jz+l- j l 2 k . I
We choose k sufficiently large and set
Using Sobolev's mollification method described in the proof of Lemma2.5.1 and setting ( U ) ~ ~ - ~ for ~ ( Zx ) E SZ(jlt3) - SZ(jl-z), v(x) = otherwise ( x E 9 ), ( 0 where c > 0 is sufficiently small, then it follows that v E Cm(sZ)and
6.2.3. Properties of the spaces A!?~&?)
409
for x E Q ( j i + j ) - D(ji-2).Hence v E Se(,,(Q). On the other hand, one obtains by (5)
j
J)
W
Iv(s)lpdx
2C
1= 1 J)Ul+') -&I,
Iv(s)lp dx
W
2 C
-
~ j i l Q ( j ~ " j Q(jljl = 03.
Z=1
This is a contradiction to (1). This proves (3). Now it follows for a > 0 with 2a > b
R e m a r k 1. If 52 is bounded, then e-"(x) belongs t o L,(Q) for all a 2 0. One of the main aims of this chapter is the development of a n Lp-theory for operators from the Definition 6.2.112. For this purpose, (1) is a natural assumption. The theorem shows that (1) is equivalent t o
3a > 0 where ,p-"(x) E L , ( 9 ) .
(6)
R e m a r k 2. We shall see later that under the hypothesis (6) S,(,)(D) is a nuclear (F)-space isomorphic to the space s of rapidly decreasing sequences. It is easy to see that the Schwartz space S(R,,) is a special case. L e m m a . Let D c R,,be a bounded domain, and let e-l(x) d ( x ) be a smoothed distance function i n the seme of Remark 3.2.311. Then it holds set-theoretically and topologically (7) Se(3j(Q) = CO"(52) = { f I / E CF ( ~ n;)SUPPf c N
.n7
where the topology i n CF(52) i s given by the semi-norms sup lDaf(x)l,0 Z€Q
=< la1 < 03.
P r o o f . It is not very hard t o see that Se(,)(D)andZF(52) coincide set-theoretically (here the functions of ~ S ~ ( ~ ) (are 5 2 extended ) by zero outside of 52). Further, E$(Q) is an (F)-space where the topology of Se(,)(52)is finer than the topology of c$(D). Then one obtains as a consequence of the closed graph theorem (see N. DUNFORD, J. T. SCHWARTZ [l,I], 11.2, Theorem 5) that fYe(&) and @(Q) coincide also in the topological sense.
6.3.
A-Priori-Estimates
The main aim of this chapter is t o give a treatment on mapping properties for the operators of Definition 6.2.112 in the framework of an L,-theory. A priori-estimates and an L,-theory for special self-adjoint operators of such a type (Section 6.4) are essential for these investigations. On this basis and with the aid of the SobolevLebesgue-Besov spaces with weights, one can prove in the following sections theorems on isomorphic mappings.
410
6.3. A-Priori-Estimates
6.3.1.
Equivalent Norms in the Spaces W,(sd; 8’;
e’)
Lemma. Let e(z) be a weight function in the sense of Definition 3.2.311. Further, let k = 0, 1, 2, . . ., 1 < p < 00, Y 2 p kp, and {yj(x)],ZN EY(Q;e) in the sense of Definition 3.2.311. Then there exist: (a) Balk K f ) = {z I 1z - zj,ll .c d 2-j} such t h d
+
-
s,
Qjc U Klj) c Q;-luQ j + l , 1=1
j = N,N
+ 1 , . . .,
(QN-l = Qx), where at most L balls have a mn-empty intersection ( L i s a suitabk natural number), and d > 0 i s independent of j. (b) Systems {~$)(X))E’~, j = N , N + 1, . . ., s w h that
ID“p71U)(~)l ~,,2”“’, j
=
N,N
+ 1 , . . .,
1 = 1 , . . ., N ; , IyI
> 0 . (3)
i s an equivalent norm in W”,Q; e”; e’) (Definition 3.2.312, Theorem 3.2.412). Proof. The existence of balls KiJ)and functions yiJ)(x)is a consequence of the properties of the domains Qj,see Remark 3.2.312. Lemma 3.2.411 is also valid for the functions rpjJ)(x).Whence it follows in the same manner as in the proof of Theorem 3.2.411 that (4) is an equivalent norm in W#2; Q ” ; e”). Remark. The magnitude of d determining the radius of the balls can be chosen arbitrarily small. We shall use this fact in the next subsection.
6.3.2.
A-Priori-Estimates
Theorem. Let A E %EV(Q;~ ( s ) )in the sense of Definition 6.2.112, where v 2 0. Further, let x be a real number and 1 < p < 00. Then there exists a real number c1 such that, for every complex number il with Re 3, c l , there exist two positive numbers c2 and c3 with the property that for all u E Wtrn(Q;e X + P ” ; ex+p’)
P r o o f . Step 1. Theorem 3.2.412 and Theorem 3.2.4/3(a) yield that the left-hand side of (1) is true. (Here one needs the assumption v 2 0.)
6.3.2. A-Priori-Estimatee
41 1
Step 2. To prove the right-hand side of (1)we assume temporarily that the support of u E W;rn(12)is contained in one of the balls K g ) of Lemma 6.3.1. Let xj,,.be the centre of KAj).We set rn
A1u
Then me have
AU
=
c c z=o
l al = 21
D% - h,
@xal(Xj,k)ba(Xj,k)
- ;lu = A,u + AZu+ A,u.
Extending u ( x ) oufside of K,O"by zero, then it follows that
and taking into consideration xzl the Fourier transform that
21 +(v - p ) = v , then it follows with the aid of 2m
IlA14Ep(~;e~)
c
> 0 is independent of j, k, and A. Remark 2.2.414 and Definition 6.2.112 yield that
and
are multipliers for Re A 5 0, where the number B appearing in Remark 2 . 2 4 4 is independent of A, j,and k, while it depends on the constant of ellipticity C in (6.2.1/6). Then one obtains by Remark 2.2.414 that
412
6.3. A-Priori-Estimates
where the constant c > 0 depends only on C , but not on A, i, and k. Putting this result in (3), and returning to the original coordinates x in ( 2 ) ,then it follows that
Here cl, c2, c3 ,and c, are poaitive numbers depending only on the constant of ellipticity C , but not on 2, j, and k ; Re1 5 0. To estimate A2u we choose (in the sense of Remark 6.3.1) d sufficiently small. Then one obtains for x E KP) that @ q s )ba(z)-
6
@X?'(Xj,k) ba(Zj,$)l
C2.i("2'+')
Ix - X j , k J
6 E2jXZ'.
Here d = d ( E ) is a given number, E > 0. Theorem 3.2.412 and Theorem 3.2.413 yield
s Here 6c
E"
> 0 is a given number ; d = d ( d r ) > 0. If
w is a bounded domain such that
52, then we have
ll4lw;m-1(")
6
(5)
E " I I ~ l l ~ ( ~ ; e ~ + ~ ~ ; e ~ + ~ ~ ) .
E l l 4 w;%J)
+ C(E)
5 E'IJ~IIw~~(~;
Il~lILp(w)
ex+wp;e"+vp)
+
ll~ll~,(~; ex).
Here E' > 0 is a given number. (This follows from a formula similar to (4.10.1/13).) Using this estimate and the assumptions on the coefficients as(.), then it follows from Theorem 3.2.412 and Theorem 3.2.413 that
IIA~uIIE~(Q; e x ) S EIIuII&;~(Q; e x + w p ;
ex+vp)
+ C(E) l I ~ l l & ( ~ ;
IIAu - ~
L
~ l I Q ~ ; e x )
ciIIull&im(Q;
ex+pw;ex+pv)
(6)
ex).
Finally, one obtains by (4), ( 5 ) , and (6), and by a suitable choice of
+( ~ 2 14 ~
E
and E" that
~ 3 IIUll!ip(n;ex). )
(7)
where R e d 5 0. Here, the positive numbers c1 and c2 depend only on C , while c3 depends on C and on the constants of estimates for DYbJx) and D Y a s ( x ) in the sense of (6.2.117). (Clearly, these numbers depend also on the fixed domam 52 and on the fixed function e(s).) Step 3. Let u E W26n(SZ;ex+pJ'; e X + " P ) . Using Lemma 6.3.1, then it follows by (7) 00 .vj for c2(LIp - c3 2 0 and u = C C yj&)u that j=iV
k=l
413
6.3.2. A-Priori-Estimates
We have
A(vjq$'~)- 1vjq$~= vj~)($(Au -1 ~ )
+
C
c@,a(x)DYvjd!')flu*
IS1 2m lJlnl$2m-ISI
Using xlal+lp~
(9)
+ la1 < xlslfor (a12 1, then it follows that
C S , & ( X ) D"(yJj'P$) = O(@xl.+I#l@'"l) =
o(@"lBl-a),
where 6 > 0 is a n appropriate number. Similarly t o the second step, one obtains
Putting this in (€9,and choosing E the right-hand side of (1).
C
= 2and Re
2
1 sufficiently small, then one obtains
R e m a r k 1. The proof yields a sharper result than formulated in the theorem: T h r e exist a real number c1 and a positive number c 2 , depending only on the con-stant of ellipticity C i n (6.2.1/6), the constants of estimate for DYbJx) and DYas(x), and the o-behaviour of DYas(x) i n the seme of (6.2.1/7) (and In, e ( x ) ,p , and x ) such that for all complex numbers 1with R e il 5 c1 and all u E Wim(sZ;@ + p P ; e x + p v )
llAu -
~UIIL~(Q;
ex)
2 czllull +yg;
+ c2Vl
px+v)
llullLp(n;ew) -
(11)
Later on, we shall use this sharper version of (1). R e m a r k 2. The assumption v 2 0 is needed only for the proof of the left-hand side of (1). It is easy t o see that v 2 0 is a natural assumption. Namely, if v < 0 and il < 0, then A - ilE belongs t o '%Eo(Q;e ( x ) ) , but not to ?l$(In; e ( x ) ) .In this case the term -ilu belongs t o the ''main part" of A - AE, and one cannot expect i l ~ estimate l of the form (1). Let v >= 0. Since x in (1) is an arbitrary number, it is easy t o see that one can replace ilu by ile"(x)u with 0 5 v. Afterwards it follows that, in such a formulation of the theorem, the assumption v 2 0 is not necessary. We shall return t o this question later on, Theorem 6.5.1.
6.4.
&-Theory for -
-I-gU(x),v
>2
The results of this section and the a-priori-estimates of Section 6.3 are the basis for the further considerations. But the investigations of this section are also of selfcontained interest. Together with Theorem 6.6.1, they are the basis for the structure theory for the spaces Se,,,(In)in the eighth chapter.
-A
414
6.4. &Theory for
6.4.1.
Self- Adjointness
+ e'(z),
Y
>2
Lemma. Let 52 c R, be an arbitrary domain, and let Au = 0 for u E D'(Q). Then u is a harmonic function in the cbsicu2 sense. P r o of. Let p E C$ (52).Then pu can be interpreted in the usual way as a distribution belonging to E'(R,) c S'(R,). (See for instance H. TRIEBEL[17], p. 49/50, p. 103.) From the properties of distributions of E'(R,) and from (2.8.1/16), it follows for y E Corn(R,) that
l(pu)(w)l 6
Cz~~~~IW~(R,)*
c~lIWIICk(J?,)
Here k is a suitable natural number, and 1 is a natural number such that 1 > k Theorem 2.6.1 yields pu E Wiz(Rn). Further we have (in the sense of D'(R,))
n +. 2
Using Theorem 2.3.4 with s = -2 (or the usual rules for Fourier transforms in S'(R,)), then one obtains pu E Wi'+l(R,). Putting this in (l), then it follows m
.
Wi1+2(R,).Iteration and application of (2.8.1/16) yield guu E n Wi(R,) j=-m c C"(R,). Whence it follows the lemma. Remark 1. * Differentiability properties of the above type are well-known in the literature (theorems of Weyl-type, properties of hypo-ellipticity). One may generalize the lemma essentially. See, for instance G. HEUWIQ [l], IV, 3.4/3.5, and the references [3]. given there, and L. HORMANDER Theorem. Under the hypotheses of Definition 6.2.112 the operator A ,
pu
E
+ e'(s)u,
> 2, B ( A ) = C$(Q), is essentially selj-adjoint i n L,(Q).Its closure A is an operator with pure p i n t spectrum. Au = -Au
Y
Proof. Step 1. Clearly, A is a symmetric positive-definite operator. We want to show that it holds N ( ( A orE)*) = (0) for sufficiently large values of a > 0. Then, by well-known theorems, it follows the self-adjointnessof A (see for instance H. TRIEBEL [17], p. 206-207). Let A*v av = 0. Then in the sense of the theory of distributions we have -du e'@) v av = 0, v E L,(S). (2)
+
+
+
+
If o is a bounded Cm-domainsuch that 6 c 52, then -dw
= -&V
- eY(s)vEL&)
(3)
has a solution w E Wg(o)A W i ( o ) . (Dirichlet's boundary value problem for - A . We note that - A satisfies the hypotheses of Theorem 4.9.1. See also Remark 4.9.1/3. Since -d is positive-definite on W i ( o )A @(o),then it follows by Theorem 5.2.3/1, that 0 is an element of the resolvent set.) Applying now the lemma to v - w, then it Eollows v E Wg(o). Now, (3) and Theorem 5.2.2(c) yield that w belongs to Wi(w). Application of the lemma gives w E W,"(o).Using (2.8.1/16),then it follows by iteration w E C"(52).
6.4.1. Self-Adjointness
416
Step 2. For the proof of the self-adjointnessit is sufficientto show that for sufficiently large a > 0 any function w(x)with
-dw
+ e’(x) w + aw
0, w(x) E P ( QA)L2(Q), (4) vanishes identically. Assume without loss of generality that w(x) is a real function. For { ~ j } j ” E, ~!P (Definition 3.2.3/1) we set vj(x.0 = (e’(x)
=
+ a)-* C yl(x)
Then we have
j
1= N
E
Cz(Q), j = N , N
+ 1, *
n
n
Using (4) and
av 2 C - -avJ . -vjw ’l
l i = l axk
axk
1 2
=-
c n
k=l
a,p? - I - ,
axk
a$ axk
then it follows by (5)that
(z(x))2dz
Using the properties of e ( x ) and y&), partial integration yields
J-
n
fi yr)2axs
w2( l = N
J-wz n
2
k=l
Since v > 2, one can make the factor in the last integral arbitrarily small, independently of j, if one chooses a sufficiently large. Transfering this summand to the left-hand side and considering j * a,one obtains w(x) I 0.
Step 3. Now we have to prove that A is an operator with pure point spectrum. It follows from Theorem 6.3.2 and Theorem 3.2.4/1 that D ( A ) = W,”(Q;1 ; e2’). (8) By the theorem of F. RELLICR (see for instance H. TRIEBEL [17], p. 277), one has to prove that the embedding from D(A)into L2(Q)is compact. If xj(x)is the characteristic function of Q ( j ) from Definition 3.2.311, then it follows from the compactness of the embedding from W i ( S ( i )into ) L,(Q(j))(Theorem 3.2.5) that
Mj = { x j ( x )~ ( xI )IIU(~)IIw~(n;l;e’v)5 1) is a precompact set in L2(Q).Now it follows from
\ 11 - X , j ( X ) 1 2 lu(x)12 ax 5 2 4 ’
ir
[ @2”lU12a x fi
that M,; for j 2 j&) is a pre-compact &-netfor the image of the unit ball of D ( A )in L2(Q).Whence one obtains that the embedding from D(A)into L2(Q)is compact.
416
6.4. &-Theory for
+ e'(r). v > 2
-d
Remark 2. * The above proof is due to H. TRIEBEL[2].The used method, in particular estimates of the type ( 6 ) and (7), goes back to E. WIENHOLTZ [ l ] (see also I. M. GLAZMAN [2],Chapter 1 , Theorem 3.5). 6.4.2.
Eigenfunctions
Theorem. If A is the operator of Theorem6.4.1, then the eigenfunctions of A belongto SQ[=) (52) (Definition 6.2.1/1). Proof. Step 1 . We start with preliminaries. Let v E L2(Q),and let
-dv
+ @'(X)
v
=
g
E L,(52)
in the sense of the theory of distributions. We want to show that v belongs to D ( A ) . If 9 E C$(52), then (v, AV)L, = (9, d L , *
Whence it follows v E D(A*) = D ( A ) . Step 2. Let il be an eigenvalue of A, and let u(x)be an eigenfunction, Au Then, in the sense of the theory of distributions, we have
-du
=
ilu.
+ $(z)u = ilu.
(1)
Further, one obtains from (6.4.1/8)and from the first step of the proof of Theorem 6.4.1 that u E W,"(52;1 ; e2') A C@(52). (2) We want to show that @ ( x ) u ( x ) belongs to D ( A ) = % 1( ; e2') $ forIeach ; number oc 2 0. Let 26 = v - 2 > 0. The proof w i l l be given by induction. Assume that e e ( j - l ) ubelongs to D ( A ) for a natural number j . We shall show e"ju E D ( A ) .It holds
Since eE(j-l)u belongs t o D ( A ) ,it follows from (6.4.1/8)and Theorem 3.2.4/3 that
e&j+2I I < = ce8(j-1)+vIuI
E L2(Q),
(4)
Whence it follows that v = eeju and the right-hand side of (3) belong to L,(Q). Now it follows, from the first step, e8juE D ( & . Repeated application of Thec j . Hence e% E D ( A ) for all a 2 0. orem 3.2.413 yields @" E D ( A ) for y Step 3. If u is the eigenfunction of the second step, then we want to show that @DYu belongs to D ( A )for every number oc 2 0 and every multi-index y. We use again induction, firstly by Iyl = 0, 1 , 2 , . . .,and secondly for fixed y by oc = E j , j = 0,1,2,. .
.
417
6.4.3. Domains of Definition of Fractional Powers
Here, E has the same meaning as in the preceding step. Assuming that the statement is true for 0 IyI k - 1, then it follows for I/?I= k that -ADs, + @(z)Dsu = W U+ 1 c,,D"$"'fl-'h E L,(Q).
s
s
llllZ1
The first step and Theorem 3.2.4/3 yield Dsu E D ( A ) .Then one obtains in the same manner as in the second step that e"D% E D ( A ) ,OL 2 0. Step 4 . If the functions y j ( x )have the same meaning as in Definition 3.2.3/1, then it follows from (2.8.1/16) and from the above results
R e m a r k 1. One can show that, for v = 2, the Theorem 6.4.1 as well as the above [2], p. 165. This shows that theorem are generally untrue. We refer t o H. TRIEBEL v > 2 is a natural assumption. R e m a r k 2. Considering the above theorem and Theorem 6.2.3(a) it seems to be meaningful t o extend the domain of definition of the operator A from C$(Q) t o S e ( & ) ( 9I)f. one wants t o remain in the framework of L,-theory (or L,-theory), then (6.2.3/1)must be valid. But Theorem 6.2.3 shows that this is equivalent t o (6.2.3/6). Hence, for a n L,-theory (or L,-theory), (6.2.3/6) is a natural additional assumption. 6.4.3.
Domains of Definition of Fractional Powers, Isomorphic Mappings
I n the following considerations, we shall assume that there exists a number a 2 0 such that e-"(x) E L,(Q) (see Remark 6.4.2/2). T h e o r e m . Let A be the operator of Theorem 6.4.1. Further, let a 2 0 such that
e-"(x) E Ll(Q). (a) For s 2 0 it holds
*(a;
D(A8) = 1 ; e2'"). (b) For s 2 0, the differential expression -Au from @ + a @ ; I ; e 2 8 v + 2 ) onto W,88(Q;1 ; e2'"). (c) It holds
(1)
+ e'(x)u i s an isomorphic
(2) mapping
OD
D(A") =
( D l ( A j ) = AS'~(~)(Q)
j=O
(3)
(set-theoretically and topologically). The differential expression -Au + eV(x)uyields an isowwrphic mapping from Se,, (Q) onto itself. Proof. Step 1 . Let j = 0, 1, 2 , . . . It follows by Theorem 3.2.4/1, Lemma6.2.2, and Theorem 6.3.2 that IIuIID(zj) llull wijm 1; P"") for u E W,2J(Q;1 ; @jr)c ~(.li). (4)
-
27
Triebel, Interpolation
418
6.5. Lp-Theory
On the other hand, one obtains with the aid of (1) t h a t S@(,)(Q) c W,2j(S;1 ; e 2 J V ) .
(5)
But by Theorem 6.4.2, there exists a subset of A ! ~ ~ ( ~namely ) ( Q ) , the set of all finite linear combinations of eigenfunctions, which is dense in D ( A j ) . Then, ( 2 ) with s = j = 0 , 1 , 2 , . . . is a consequence of (4). Step 2. Now one obtains (2), for arbitrary values s 2 0, from Theorem 1.18.10 and from (3.4.218).(It holds HE = W2 .) Since A is positive-definite, whence it follows also the part (b) of the theorem. Step 3. Formula ( 5 ) yields (set-theoretically and topologically)
D(A")'
'Q(.Z)(')
(6)
Now, let u E D(2"). Then Theorem 3.2.413 yields e'DYu E La@)for arbitrary numbers 01 5 0 and arbitrary multi-indices y. Now one obtains in the same manner its in the fourth step of the proof of Theorem 6.4.2 the conversion t o (6). This proves (3). Clearly, -Au + @ ( x )u gives an isomorphic mapping from D(A") = SQ(r)(Q) onto itself.
6.5.
L,-Theory
In this section, anLp-theoryfor operators A belonging to %Ev(Q;e ( x ) )will bedevelop-
ed on the basis of the previous considerations. Differentiability properties and the behaviour near the boundaxy of solutions of the equation Au = f (or Au - Ae"(x)u = f ) will be described as before with the aid of isomorphic mappings between function spaces generated by A (or by A - Ae'(z)).
6.5.1.
A-Priori-Estimates (Generalization of Theorem 6.3.2)
Using the results of Section 6.4, one can generalize Theorem 6.3.2 essentially. Theorem. Let A E 2iEv(Q; e ( x ) ) in the sense of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E L,(Q). Let (T v, 1 < p < 00,
'
0, 1 , 2 , . . . Let z = - > 2. Then there exists a real number c1 m such that for all complex numbers A with Re 1 5 c1 there exist two positive numbers c2 and c3 such that for all u E W:m+2k(Q;px+pJ'; ex+p(v+ks)1
x real, and
k
=
C31JU11w;'"+k'(n;
@ x + P r ; ex+P(v+kr))
2 IIAu - Ae'(x) u l l ~ r e(x ;~q x;+ P * r ) 2 c211uJIw2'"+k'(Q; Q x + P r ; @X+P(Y+kZ)).
Proof. Step 1. We start with preliminaries. Let
Bu
=
(-A
+ @'(X))kU
- '1u, '1 5 0 .
6.6.1. A-Priori-Estimates(Generalizationof Theorem 6.3.2)
419
Theorem 6.4.3 yields that B is a n isomorphic mapping from Se(z)((sz)onto itself. Then it follows from Lemma 6.2.2, Theorem 6.3.2, and Theorem 3.2.411, that B for rj 5 c is a n isomorphic mapping from W;"(S; ex;e x + P T k ) onto LJQ; ex). Step 2. By the first step, it holds for u E C$((S)
IIAu - Ae"(z)ull W F ( Q ; e x ;
ex+pk+)
-
IIB(Au - le"(z)~ ) l l ~ e~x ) .( n ;
(3)
Considering the operators p s ( - A + p y (z))and A , then one has the same situation as in the proof of Lemma 6.2.2. Whence it follows that V-P
(-A
+ @')A = e--
P
=
Iteration yields
BA E 91;;:k'p;
P
ern ( - A
?L,":,',(Sz;
+ em)A v-P
-L
Ee
rn
e(z)).
n1+1
) I l P + ~ , . + ~ ( ( s ze(z)) ; rn
nr
e(4).
Since CT 5 v, the left-hand side of (1) is a consequence of Theorem 6.3.2. Step 3. We prove the right-hand side of (1). Since x is an arbitrary number, one may assume 0 = c 5 Y without loss of generality. If B has the above meaning, then it follows, in the same manner as in the second step, with the aid of the proof of Lemma 6.2.2, that BAu = ABu + 2 afl(z)D8u. (4) 161 < 2m+2k
Here, the last term is a perturbation of BA (or AB) in the sense of (6.2.1/5). Then, from Theorem 6.3.2, the counterpart t o (6.3.2/6),and (3) it follows that
One obtains by the second step that
[(-A
+ e')"
- @I B E q g ; : m + k ) ( Q ; e(4).
Applying Theorem 6.3.2, putting the result in ( 5 ) , and choosing E in a suitable way, then it follows that llAu - ~ u ~ ~ w Fe(x D ; e x;t p k r )
>= c I I J u J ( ~ ~ ( ~ + ~ ) ( Q ;
extwp; extp(v+kr))
- C,IIUIIL,(O; p x t v q .
(7)
(Here we used Y 2 0 and Y > p.) Transfering the last term t o the left-hand side and estimating it with the aid of Theorem 6.3.2, then one obtains the right-hand side of (1). 27*
420
6.5. Lp-Theory
R e m a r k 1. The special choice t
'
=- makes
the proof easier. But probably m the theorem remains also valid if one assumes only t > 2. R e m a r k 2. The proof is based on Theorem 6.3.2, and it uses the special operator -A + e"(x). This shows that one may carry over the important Remark 6.3.2/1: There exist a real number c1 and a positive number c2 depending only on the constant of ellipticity C from (6.2.1/6), the constants of estimates for DYb,(x) and Dyas(x),and the odehaviour of Dyas(x)in the sense of (6.2.117) (and Q, e(z), p , x , k,and a), such that for all complex numbers 1 with Re 1 5 c1 and all u E ~ ~ m + ' c ) ( Qe x;+ P p ; ex+P(v+kr) )
l!Au - A@"(%) u l l w ~ ( n ; ex;
QX+Pk)
2 C211UIIw2(m+.t)(n; e x t p r ; eu+P("+kr)).
(8)
Here, c1 is also independent of k. This improvement w i l l be very useful, later on.
6.6.2.
Isomorphism Theorems
In this subsection there are proved some of the main results of this chapter. Theorem 1. Let A E UEv(Q; e(x)) in the sense of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E Ll(Q).Let 0 < v, 1 < p < co, 1 5 q 5 03, and let x be a real number. Further, let c1 5 0 be a real number in the sense of Theorem 6.5.1 and Remark 6.5.112. (a) If Re 1 5 cl, then A - 1e"(x)gives an isomorphic mapping from tSp(x)(Q)onto S d X ) (Q). (b) If Re 1 5 cl, and if s
2 0 , then A - ?&x)
H ; m + S ( Q ; @ x + p P ; ex+pv+sp
( c )If Re 1
5
cl,
and if
s
'
P r o o f . Step 1 . Let
+ e"(X))"
BU = ( - A
s)onto
H;(Q;
@x+sp
s).
> 0, then A - le'(x) gives an isomorphic mapping from
BS+zm(Q.ex+PP ; @X+PV+sP P.P
gives an isomorphic mapping from
5) onto
B;,,(Q; @*;@ X + s P
s)
u - 1@(x)u = B0u - I@(x) U ,
,
(1)
where t > 2, rj < t m , and 1 < 0. We want to show that B gives an isomorphic mapping from Secz) (Q) onto itself. By Theorem 6.4.3, B,, where D(Bo)= Gm(SZ; 1 ;e Z r m ) ,is self-adjoint and positive-definite. Further, it holds that
D(B:)
=
W,"mk(Q;1 ; eZrkm),k = 1 , 2 , . . .
By the previously developed technique of estimates it follows that
Bku = B ~ u+ Du, IIDUIlr,,(n) 5
EIIuII~?'~Q;
1; e * r h )
+ c b ) Ilull~,~~)
5 E'IIB~UllL,(Q) + C'(&')llUll&(Q).
6.5.2. Isomorphism Theoreme
42 1
Here E > 0 (resp. E' > 0) is a given number. Now, the criterion of self-adjointnessof T. KATO(see H. TRIEBEL[17], p. 209) yields that Bk, where D(BIC)= D(B$),is a self-adjoint positive-definite operator. By Theorem 3.2.411, CF(Q) is dense in D(Bk). Hence, Bk, D(Bk) = Wirnk(Q; 1 ; elrkm),is the k-th power of the positive-definite Now, (6.4.3/3) yields that B self-adjoint operator B , with D ( B ) = Wim(Q;1 ; eZrrn). gives an isomorphic mapping from Spcz. (Q) onto itself. Step 2. Let A be the operator of the theorem, and let B be the operator of (1) V - P where t = n2
and 7 = c - p. We set for 0
a
5
1
Aau = &(A - A@(x))u + (1 - a)@(z) ( B , - Re A * @(x)) u = aAu + (1 - a) @(z)B,u - (an + (1 - a ) Re A) @(z) u .
(2)
+
Since Re (an (1 - a)Re A ) = Re A, it follows by Theorem 6.5.1 and Remark 6.5.1/1 that there exists a number c1 independent of a and k such that (6.5.1/2) holds with IXA (1 - a)e"(z) B, instead of A . Remark 6.5.1/2 yields that c, is independent of a. Assuming that for a given number 0 5 a, c 1 the operator Amogives an isomorphic mapping from
+
R 1 --
J+'2(m+jC)(Q; @ + P P ; e x + P ( v + k r ) )
then it follows that
P
onto R, = W F ( Q ; e x ;
f+Pk),
A,u = f E R, is equivalent to ~1
It holds that
+ A,f(Aa - Am,)u = A,:/
E R,
(4)
IlA;f(Am - ~ m , ~ l l I ~ 5 l +1 I01 ~~ O1ol c < 1 for
la - a01 < C 1
Here, c is independent of 01 and a,. Now one obtains that (4) has a unique solution. For these values of a, there exists A i l . The first step and Theorem 6.5.1 yield that A,' exists in the above sense. Now it follows, by iterated application of the above procedure, that A - Ap'(x) is an isomorphic mapping from R, onto R,.For x = 0 and p = 2, one obtains from (6.4.3/3) that A - A@(x) gives an isomorphic map(Q) onto itself. ping from Step 3. Part (b) and part (c) of tthe theorem are consequences of the second step and of Theorem 3.4.2. Remark 1. As a special case the theorem contains the assertion that A - &"(z) gives an isomorphic mapping from wgrnts
(Q; @ x + p p ; e x + P v + s P q
onto
W> (Q; p; p
~ + ~ p s ) .
For 0 = 0 this result was proved in H. TRIEBEL[24]. R e m a r k 2. The above theorem is comparatively general. We describe two simple conclusions.
422
6.F. Distributions of Eigenvalues, h c i a t e d Eigenvectors
c((Q)
-
1. Let Q c R , ,be a bounded domain, has the meaning of Lemma 6.2.3, and e - l ( x ) d(z) is the smoothed distance function in the sense of Remark 3.2.3/1. Then it follows from the above theorem and from Lemma 6.2.3 that Au + ( u + e"(x)u, v > 2m,
c$
is an isomorphic m p p i n g from (Q)onto itself. (Here one has t o use that A is positivedefinite in L2(Q).)The same holds for @(x) ( u + @"x) u where v - ,u > 2m. 2. If Q = R,,, then Remark 6.2.113 yields that
AU
=
(1
+ lxl2)V1(-d)l~lu + (1 +
is an isonzorphic mapping from
I~(')qzU,
2
> q17
S(R,,) onto itself.
T h e o r e m 2. Let A E 21Ev(Q; e(x))i n the seme of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E Ll(Q).Let x be a real number. Then A , considered as a mapping from W;m(Q;e x + P C ; e x + P v ) into Lp(Q:en), is a @-operator with the index O.*) P r o o f . If CT < v and if c1 has the meaning of Theorem 1, then A , considered as a mapping from W?(Q, e X + P P ; QX+P") into &(Q: e x ) , is a @-operator if and only if A ( A - c1@'(x))-l, considered as a mapping from L P @ ):'Q into itself, is a @-operator. This is a consequence of Theorem 1. Both the operators have the same index. It holds that A ( A - CleU(x))-l= E
+ cleu(x)( A - Cl@(Z))-l.
(5)
In the same manner as in the third step of the proof of Theorem 6.4.1, it follows that ZL + @(x) u is a compact mapping from W?(Qn;
@X + P P ;
ex+"")
into Lp(Q;ex).
Then, ~ " ( x( )A - cle"(x))-l is compact operator in L,(Q; ex).Now, (5) yields that A ( A - cleU(x))-l is a @-operator with the index 0. R e m a r k 3. The theorem is the counterpart to Theorem 5.2.2(b).
6.6.
Distributions of Eigenvalues, Associated Eigenvectors, and Green Functions
This section is the counterpart t o Section 5.4. The investigations on distributions of eigenvalues are of interest, later on, in the framework of the structure theory for the spaces h z )
can,.
*) The index of a @-operatoris defined as the difference between the finite codimensionof the range and the finite dimension of the null apace.
6.6.1. Distributions of Eigenvalues
423
Distributions of Eigenvaliies and Domains of Definition of Fractional Powers
6.6.1.
The used symbols have the same meaning as in 5.4.1. T h e o r e m 1. Let the differential expression A belonging to the class '%Ev(!2;e ( x ) ) (Definition 6.2.1/2) be formally self-adjoint. Let v > 0. Further, it is assumed that there exists a number a 2 0 such that e-"(x) E L,(Q).Then A ,
D ( A ) = W$"(O;$"; $'), (1) is a self-adjoint operator, bounded from below, with pure point spectrum in L,(sZ).There exist two positive numbers c1 and c2 such that inin (p,0). I f s 2 0, then D(A") = Wp"(f2; pa""; eaSv), (3) provided that A , without loss of generality, is positivedefinite. P r o o f . Step 1 . Theorem 3.2.4/1 yields that C,"(!2) is dense in D ( A ) . Hence, A is a symmetric operator. Now it follows from Theorem 6.5.211 that A is a self-adjoint operator. (For appropriate A, there holds R ( A - ilE) = L,(Q)).Further, one obtains from Theorem 6.5.211 that (3) is valid for s = I = 0, 1, 2, . . . For general values s 2 0 , (3) follows by interpolation from Theorem 1.18.10 and (3.4.2/8). (It holds Wg = H g . ) One obtains in the same manner as in the third step of the proof of Theorem 6.4.1 that the embedding from D ( A )into L2(f2)is compact. (Here, one uses the assumption v > 0.) By the theorem of F. RELLICH(see for instance H. TRIEBEL [17], p. 277), A is an operator with pure point spectrum. Step 2 . We prove the left-hand side of (2). Let K , and K , be two open balls such that I7,c K , and If, c 0.Let S be an extension operator from W,2"(K1)into fki'"(K2)c D ( A ) in the sense of Theorem 4.2.2. (The extension operator of Theorem 4.2.2 is multiplied with ~ ( xE)C$(K2)where ~ ( x=) 1 in a neighbourhood of K , . The functions are extended by zero outside of K 2 . ) Let R be the restriction resp. from L,@) onto L,(K,). Denoting embedding operator from D ( A )onto Wim(K1), operators by I , then Here ,C
=
I W ~ m ( I \ l ) - t L t ( K ,= )
RID(A)+L2(i$.
Theorem 1.16.1/1 yields sI ( I w, 2mwI)-tf2wI)) 6 cs,(IwwLm))-
The left-hand side of ( 2 ) is proved by Theorem 3.8.1 and Theorem 5.4.1/1 (and (5.4.1/5) and (5.4.1/6)). Step 3. We prove the right-hand side of (2). Let ilbe a (sufficiently large) positive
+
number, and let j A = [log, A;] 1. If Q ( Jhas ) the same meaning as in Definition 3.2.3/1, then Q ( j l ) will be covered by cubes Ql of the side-length d2-jA which are parallel to the axes. By (3.2.3/7), one can choose d > 0 such that
f2Wc
(J Q1 c Q ( J A + ~ )
(4)
424
6.6. Distributions of Eigenvalues, Associated Eigenvectors
(ais independent of in).It holds that Using (3.2.3/7), then it follows that one needs for the above covering a t most a -+-
a
LA 5 c p 2.iA'L I - c21
n
(6)
cubes. It holds that
In the sense of Remark 5.4.113the self-adjoint operators belonging t o the ECilbert spaces Wim(Q;ew;e2') (with respect t o L,(Q)), Wim(Q - U Q 1 ; e2"; e2") (with respect t o L,(Q - U QJ), and f l m ( Q 1 ear, ; eaU)(with respect t o L2(Qz)) are denoted by A,, , A , , and A z, respectively. It holds t h a t j5J-Q) = Lz(Q -
UQd
LA
CB
C @ L2(Qd
>
Here, E is a sufficiently small positive number. Hence N A , ( A )=0. Setting ,ii =min(p,O), one obtains that 2
I I ~ wzm(Q1; II
q2p; e z v )
2
2
2ii -
2
c ~ ~ ~ ~ " ~ ~ I I ~ I~ I ' 'YL1 ~I I~~ (I I Q ~ V~ ~)~ (2 Q , )
> 0, cr > 0. If B is the operator belonging t o the quadratic form IIUll'$im(p,) respect to L,(QJ),then, using (6) and Theorem 5.4.1/2, it follows that
c
(with
If Q is the unit cube and if D is the operator belonging t o the quadratic form I I u ~ ~ $ ; ~ ( Q ) , then NB(7) 5 cND(7).
(9)
This is a consequence of the transformation of coordinates mapping Q1 onto Q, and a comparison of the corresponding quadratic forms. Theorem 5.4.111 and Theorem 3.8.1 yield N&)
II
5 ~ 7 % .Then, it follows from
(8) and
(I))that
6.6.2. Associated Eigenvectors
425
Remark 1. The estimates of the third step can be improved. On the other hand, formula (2) shows clearly the influence of the different parameters, in particular of a and v. An asymptotic formula cannot be expected under these general assumptions. Theorem 2. Let Q c R, be a bounded C”-domain. Let e ( x )be a function in the sense of Definition 6.2.111, where e-l(x) d ( x ) near the boundary. Here d ( x ) denotes the distance of a p i n t x E Q to the boundary. Then it holds for the operator A from Theorem 1, where p > -2m, that n-1 2m if - 2 m < p < - - , n N
2m
n
1% log I
if p = if
-
/A>--,
7
2
9
c > 0.
(10)
2m n
Proof. The theorem is a consequence of ( l ) , Theorem 3.8.2, Theorem 5.4.1/2, Theorem 5.4.1/1, (5.4.1/5), and (5.4.1/6). Remark 2. The theorems show that N ( 1 )
2m
n
N
2% holds, provided that p > - -.
n See Theorem 5.4.2. The differential operators of Theorem 2 are closely related to a special class of Tricomi differential operators. We shall return to this question in Remark 7.8.312.
6.6.2.
Bssociated Eigenvectors
The used notations have the same meaning as in Subsection 5.4.1. Theorem. Let A E 2?Ly(Q; e(x)) in the sense of Definition 6.2.1/2. Further let v > 0 , 1 < p < C Q , and e-a(x) E L,(Q) for an appropriate number a 2 0. Then A with the domain of definition
D ( A ) = W;m(s2;e p r ; e P ” ) is a closed operator in Lp(Q).Its spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The eigenvalues and the associated eigenvectors are independent of p . The associated eigenvectors are elements of Secx.(Q), their linear hull is dense in ~Y~(~)(s2). Further, the linear hull of the associated eigenvectors is dense in all the spaces Wi(Q; ex;e’) where 0 s < co, 1 < q < CQ, - m < x + sq z c m (hence, it is also dense in Lq(Q)). Proof. Step 1. Let p = 2. We set,
s
1
-4u = 2
-
C C z=o Ial=21
b,(x) PU+ P [ ( e x 2 1 ( x )b,(x) u)] + BU
111
[ex21(s)
AU
+ Bu.
(1)
426
6.6. Distributions of Eigenvalues, Associated Eigenvectors
It follows from the proof of Lemma 6.2.2, that A belongs also to kFy(Q;e(z)) and that B is a perturbation operator, where its coefficients have the property (6.2.1/8). A is formally self-adjoint. Hence, it follows by Theorem 6.6.1/1 that A , where D ( A ) = WB,m(Q; p ;p ) , isself-adjointinL,(Q). Let k A where
D(A)=
= 0 , 1 , 2 , . . . Now,
oneobtains by Theorem 6.6.1/1 that
@ z ( k + l ) r ;@Z(k+l)V)
(2) is a self-adjoint operator with pure point spectrum in the Hilbert space H = Wikm(Q;ezkr;eWv)(after introduction of a suitable norm). To apply Theorem 5.4.113 to the operator A with the domain of definition (2), we use the decomposition (1). It holds that w;(h-+l)m(Q;
IlBuIlfI = I ( B U I I M ~ , ~ ~S~ c; ~ *~ 2~/ i~c ; , Z ~ JW@“18’-6(5) ) rll+2nr-1
1
p u 1 2
dz)T,
R
(3)
= Ej
m
V + -(2km + 2m - j ) . m
(4)
Here 6 > 0 is a suitable number. The last formula is a consequence of the technique of estimates developed in the proof of Lemma 6.2.2, in particular (6.2.Z.lZ) and (6.2214). (Here one has t o take into consideration that A belongs not only t o %Ev(Q; e(z)), but also t o $&(Q; e(z)).) Setting 6 = Zvs, then
Without loss of generality let 0 < orem 3.2.413 that
1 s < -. 2m
IIBuIJUS cllull w 22( B t l - l ) m ( n ; e r ( k + i - a ) r :
If I is the embedding operator from w;(k+l)m(Q;
$(k+l)~
;
e2(’”+1’”)
into
+
(5)
Then it follows by (3), (5), and The(6)
ez(~+i-a)v).
J,j7?4(’”tI-*)m(Q;
@Z(ktl-.)p.
ea(rtl-a)v
),
and if ilis a complex number with I m il 0, then B ( A - AE)-l considered as an operator from H into H can be represented a s B(A - ilE)-l = B I ( A - ilE)-l. (7) Here, B on the right-hand side is a bounded operator in the sense of (6), while (d - U3-l is a bounded operator acting from H into D(A) in the sense of (2). Theorem 6.6.1/1 and the proof of Theorem 5.4.111 yield that I from the right-hand side of (7) belongs t o Gr where r is a suitable number, 1 < T < 01). Then it follows from Theorem 5.4.113 that A , with the domain of definition (a), is a closed operator in H where the spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The linear hull of the associated eigenvectors is dense in H .
6.6.3. Green Functions
427
Step 2. Let again p = 2. It follows from the first step that the linear hull of the associattecl eigenvectors of the operator A , where D(A) = qm(5 ear; 2; e”), is dense in the Hilbert space L,(Q).Theorem 6.5.211 and Theorem 3.2.413 yield
D(A”) =
m
n D ( A ~=)
j -0
n 00
p i ;p
~zjm(52;
j-0
j ) =
S~(~)(L?).
(8)
(See t,he third step of the proof of Theorem 6.4.3 and the fourth step of the proof of Theorem 6.4.2.)Here, the last equality must be understood not only set-theoretically, (52) but also topologically. Hence, the associated eigenvectors are elements of SQ(=) = D ( A m ) .Now one obtains that these associated eigenvectors coincide with the associated eigenvectors in Ekm(Q; e 2 k p ; ew’). Together with (8) whence it follows that the linear hull of the associated eigenvectors of the operator A is dense in (52). S t e p 3 . L e t O S s c 0 0 , lc q c oo,and-oo < x + s q s z < c o . T h e n w e h a v e fJ&)
(52) = W p - 2 ; ex;e’) .
ByTheorem 3.2.4/1, C$(52) (and hence also SQcx)(52)) is dense in W$2; e x ; e‘). Now one obt#ainsfrom the second step that the linear hull of the associated eigenvectors of d is dense in W @ ; ex; e‘). Here, in the sense of the theorem all values 1 < p < 00 are admissible. This follows from (8),after replacing there 2 by p , and the fact that ( A - iZE)-l, where D ( A ) = WEm(Q;e p w ; el’”) for suit,able values of ?, is a compact operator in L,(Q). (See Theorem 6.3.2 and Theorem 6.9.1.) R e m a r k . The theorem was formulated in H. TRIEBEL[25] without proof. A variat,ion of the theorem was proved by H . KRETSCHMER [l]. The proof of the theorem is also based on the criterion of I. C. GOCHBERC, M. G. KREJN by H. KRETSCHMER from Theorem 5.4.1/3. 6.6.3.
Green Functions
The methods developed in Subsection 5.4.4 are valid for general operators in Hilbert spaces. In this subsection they are applied t o operators belonging t o %K,(SZ; e(z)). Let, v > 0 and let E L,(SZ) for a suitable number a 2 0. By Theorem 6.5.2/2, ,@p; e2”), is a @-operator every operator A E %K,(SZ; e(z)), where D(A) = V,””(SZ; in L,(Q). With the aid of Theorem 6.5.211, it follows that A* is the formally adjoint operator to A with the domain of definition D(A*) = D(A).In particular, A* is also a @-operator in L,(Q).Now, one can apply the considerationsin front of Theorem 5.4.4 and introduce the operator A,
Au = Au, D(A) = D ( A )n B(A*). (1) A generates an isomorphic mapping from D( A) onto R(A).I n this sense we construct the operator A-l. T h e o r e m 1. Let A E fA;,(SZ; e(z)) (Definition 6.2.1/2), v > 0, and e-“(z) E L,(Q) for a suitable number a 2 0. Further, one sets in t h seme of Theorem 6.6.1/1
7
u = -V a + n + ( v -
a&j.
428
If
6
6.6. Distributions of Eigenvalues, Associated Eigenvectors
< 2 , then k1 can be represented in the form
(A-Y)(4=
1 G ( x ,Y)f ( Y )d?/,
(3)
n
6
G(x, y ) E WT(D; Q’”; Q’”) $ L 2 ( 0 )A L2(D) K”(0;e’”; e”), (4) where 0 t < 2 - cr. Proof. The first step and the beginning of the second step of the proof of Theorem 5.4.4 may be carried over without any changes. The operator A*A is selfadjoint , D ( A * A ) = Wim(D;e4P; e4’).
In the sense of (5.4.4/6), its positive eigenvalues are denoted by 2.. The counterpart to (5.4.4/9) is
Applying Theorem 6.6.1/1 to A*A, and putting il = il; in (6.6.1/2), then one obtains j 5 c(A; + 1). Whence it follows that (5) converges if N + co. By Theorem 6.6.1/1, it holds that
D ((A*A)$)= WGm(D; Q’”; e ” ) . This proves the theorem. (See the second step of the proof of Theorem 5.4.4.) Theorem 2. Let D c R,,be a bounded C”-domain. Let e(x) be a function in the sewe of Definition 6.2.1/1 where Q-l(x) d ( x ) near the boundary. Here, d ( x ) i s the distance of a p i n t x E I2 to the boundary. If A is the operator from Theorem 1 where p > -2m, then A-l can be represented in the form (3),(a), provided that n - 1 2m n 1 0 t m < 2m - m if - 2 m < p < - and 2 m - - > -p---, 2m. p n 2 2 N
+
2m
n 2
5 zm < 2m - -
n
and 2m - - > 0 . if p z - - n 2 Proof. By the proof of Theorem 1, one has to show that (5) converges if N -P 03. Applying Theorem 6.6.112 to the operator A*A with the eigenvalues A;, then one obtains that
0
i .:’ 1 -
j s c + c
AIKlogAj
if if
-2m
2m n
2m
Under the formulated conditions for z,whence it follows the convergence of (5) if
N+
00.
Remark. Both the theorems are the counterpart to Theorem 5.4.4. For some important special cases A E ‘91Ev(12;&)), differentiability properties for Green functions are considered by B. LANQEMANN [ l , 21.