Direct and Inverse Imbedding Theorems: Applications to the Solution of Elliptic Equations by Variational Methods 082181592X, 9780821815922

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of Mathematical Monographs Volume42

Direct and Inverse Imbedding Theorems. Appfications to the Solution of Elliptic Equations by Variational Methods by

L. D. Kudrjavcev

American Mathematical Society Providence, Rhode Island 1974

TTPHMbIE l1 OBP ATHblE TEOPEMbl BJ10)KEHl1.H 0Pl1J10)KEHMH K PElliEHl1ID BAPMAUMOHHhIM METO~OM 3J1JIMOTl1qECKMX YPABHEHl1M JI.~.KY~PHBUEB AKa,L\eMtrn HayK CoID3a CoBeTCKMX Cou«aJIMCTMqecKMX Pecny6AMK TPY~bl MATEMATMqEcKoro MHCTl1TYTA v!MeHM B.A.CTEKJJOBA LV

l13AaTeAbCTBO AKa.QeMMM HayK CCCP MocKBa 1959 Translated from the Russian by S. Smith AMS (MOS) subject classifications (1970). Primary 35A05, 35Al5, 35J20, 35J25, 35J70, 46E35; Secondary 35B05, 35B30, 35B45, 35J67, 46A30, 35Cl5.

Library of Congress Cataloging in Publication Data

Kudriavtsev, Lev Dmitrievich. Direct and inverse imbedding theorems. Translations of mathematical monographs, no. 42. Translation of Priarnye i obratnye teoremy vlozheniia, originally published as v. 55 of Trudy of Mathematicheskii in-t of Akademiia nauk SSSR. Bibliography: p. 1. Differential equations, Elliptic--Numerical solutions. 2. Embedding theorems. I. Title. II. Series. QA377.K813 515'.353 73-22139 ISBN 0-8218-1592-X

Copyright © J.2Zi.by the American Mathematical Society

TABLE OF CONTENTS INTRODUCTION ·········································--··--··················--····--····--·······················--···················

1

FIRST CHAPTER. FuNCTION EXTENSIONS .....................................................................

8

The averaging operator K ... ...... ... ............................................................ The invariance of the H-classes under K ............................................. The derivatives of Kf .............................................................................. The boundary values of Kf and its derivatives ................................. Function extensions from an (n - !)-dimensional hyperplane onto the whole, space .......................................................................................... Function extensions from a domain ...................................................... Averagings relativ.e to m-dimensional hyperplanes •·························· Function extensions from smooth manifolds ..... ~,;............................... Function extensions to a domain with a piecewise smooth boundary ...................................................................................................... The variation of boundaries principle ...................................................

8 11 20 29

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

SECOND CHAPTER. WEIGHTED IMBEDDING THEOREMS..........................................

11. Weighted imbedding theorems for function classes on flat manifolds ............................................................................................................ 12. Weighted imbedding theorems for function classes on arbitrary manifolds ................................................................................................... 13. Completeness of weighted spaces ......................................................... 14. Limit theorems .......................................................................................... 15. Compact function classes ... ............ ...... .... ........ ... ......... ............ .......... ..... 16. Weighted imbedding theorems for domains with a piecewise smooth boundary ... ........................ ... ......... ............................... .. ......... ......

39 49 53 58 77 87 92 92 126 138 140 I47 151

1i-URD CHAPTER. VARIATIONAL METHODS FOR THE SOLUTION OF ELLIPTIC EQUATIONS......................................................................................................

17. General plan of the variational.method................................................ iii

156 156

iv

CONTENTS

18. General plan for the solution of degenerate equations by the variational method .......................................................................................... 19. The solution of elliptic equations degenerate on the whole boundary of a domain (0,;,,;;; a< 1) ............................................................... 20. The solution of elliptic equations degenerate on part of the boundary of a domain (0,;,,;;; a< 1) ............................................................... 21. The solution of elliptic equations degenerate on the boundary of a domain with degeneracy exponent a;;,, 1 .......................................... BIBLIOGRAPHY ..............................................................................................................................

175 181 189 199 203

INTRODUCTION The solution of elliptic partial differential equations by the direct variational method has its origins in the works of Gauss, Thomas and Riemann on the investigation of Laplace's equation when considered as the Euler equation for the Dirichlet integral. But the basic principle underlying this investigation, which is known as Dirichlet's principle, was not rigorously proved until much later (1900-1901) in the works of Hilbert, which stimulated a whole series of investigations in this direction (by Courant, Levy, Fubini, Lebesgue, Zaremba and others). Important contributions to the development of these methods were made by Soviet mathematicians N. N. Bogoljubov, N. M. Krylov, M.A. Lavrent'ev, L.A. Ljusternik, S. L. Sobolev, V. I. Kondrasov and S. M. Nikol'skii'. A closely adjoining paper in a sense is that of M. V. Keldys and M. A. Lavrent' ev [1] on the stability of the Dirichlet problem relative to a variation of the boundary of the domain. One should especially note the monograph of Sobolev [2] concerning, in particular, the investigation of Dirichlet's principle inn-dimensional space. It was shown in this work that every function having weak derivatives and a finite Dirichlet integral takes completely defmed boundary values in the sense of convergence in the mean; the statement of the problem was thereby provided with a completely fmal form. In the same place are given necessary conditions which a boundary function must satisfy in order that the first boundary problem for a polyharmonic equation might be solved by a variational method in a bounded domain. Of significance in the development of this theory were the papers of Nikol'sktl' [3-6], in which, in the case of domains with a smooth boundary, sufficient conditions coinciding "to within an arbitrary e > O" with the necessary conditions established earlier by Sobolev were indicated in terms of boundary values for the applicability of the variational method to the above mentioned problem. A development of the variational method for more general elliptic equations, both linear and nonlinear, is given in the works of Kondrasov[7-10]. We also note the works of Aronszajn [46], Gagliardo [47], Slobodeckh [48] and Babic and Slobodeckh [11], in which necessary and sufficient conditions, which are imposed on a boundary function, are obtained for the possibility of applying the variational method to the solution of the first boundary problem 1

2

INTRODUCTION

for the harmonic and polyharmonic equations in the case of a finite domain with a sufficientlv smooth boundary; much more general problems are also considered. The apparatus of the so-called direct imbedding theorems, which was created by Sobolev and his student Kondrasov, plays a large role in the application of the variational method to the problems of mathematical physics. For this reason the main part of the present monograph is devoted to the subsequent development of questions connected with direct and inverse imbedding theorems. The investigations of inverse imbedding theorems or., what is the same thing, theorems on the extension of functions and systems of functions conducted by us are a continuation of the corresponding investigations of Nikol' skil [4]. On the basis of the results obtained on imbedding theorems we apply the variational method to the investigation of boundary problems for second order partial differential equations of elliptic type, both nondegenerate and degenerate, on the whole boundary of the domain or on part of it. Here a new basis for variational principles is given which extends them to arbitrary finite domains without any restrictions on the structure of the domains and their boundaries. Existence and uniqueness theorems are proved for both the variational problem and the corresponding differential problem in the case of a selfadjoint elliptic equation of second order. The connection between the smoothness of the coefficients of an equation and the smoothness of a solution is studied. In addition, proofs of the existence and uniqueness of the solution of boundary problems with sufficiently general boundary conditions are also given in the presence of a power degeneracy with exponent a > 0. In discussing these questions we have mainly followed a method whose foundations go back to Hilbert [12] and which has been developed and refined in the monograph of Sobolev [2] and in the works ofNikol'skn [4-6] and Kondrasov [7-10]. The proof of the existence of a solution of the boundary problems for degenerate elliptic equations requires a study of the question of weighted function extensions from the boundary onto the whole domain. This is done in Chapter I of the present work. Here an essential role is played by both the methods of Nikol' skit (for example, the method of approximating differential functions by entire functions of exponential type, the methods of function extensions and others) and the methods of Sobolev (for example, the method of function averagings), which naturally receive a further development and sharpening. In this chapter we construct the theory of best extensions (in the sense of the growth of the derivatives under an approach to the boundary of the domain) of

INTRODUCTION

3

functions from the boundary of the domain onto the whole domain. The best extension problem is solved here to within an arbitrary e > 0 (see Theorems [5.4] and [8.4] as well as Theorem [12.3]). In Chapter IJ we consider spaces of functions having weak derivatives that are summable in some power with a weight; we prove theorems on the imbedding of these spaces into ordinary function spaces and investigate completeness, compactness, etc. Chapter III is devoted to the presentation of general variational principles concerning the first boundary problem for selfadjoint elliptic equations of second order, and to the proof of existence and uniqueness theorems for the boundary problems of elliptic differential equations that are degenerate on the boundary of the domain. A large part of the results of the present monograph has been published in notes of the author [ 13, 14 ]; moreover, we make use of the results of the author in [15, 16]. We now consider in some detail the contents of the individual sections. In § 1 we introduce an averaging operator K with a variable averaging radius relative to the hyperplane xn = 0 and prove its boundedness in Lin>. In §2 we prove in a sense the invariance under K of the H-classes of Nikol' skit [3]. In § 3 we establish the existence of and give formulas for the derivatives of the functions Kf, f E L~n)_ In §4 we investigate the behavior of a function Kf and its derivatives on the hyperplane xn = 0. In particular, we prove one of the basic properties of the operator K and similar operators, viz. that the boundary values of an averagable function f of a certain class coincide with the boundary values of the averaged function. We also study in this section the connection between the boundary values of the derivatives of functions and their averagings. Here also we present Nikol'skit's results on the rate of convergence in the mean of the functions of an H-class to their boundary values. In §5 we give an estimate of the growth of the derivatives of a function Kf, after which we prove the main theorem of this part of the book, which asserts that every function given on the hyperplane xn = 0 and belonging to a certain H-class can be extended onto the whole space as an infinitely continuously differentiable function whose normal derivatives on the hyperplane xn = 0 take preassigned values and for which it is possible to indicate a power order of growth for all of its derivatives under an approach to the hyperplane xn = 0. In this connection the minimal possible growth exponent is determined to within an arbitrary e > 0, i.e. the subsequent results of § I 2 imply that the order of growth obtained for the derivatives is best possible to within an arbitrary e > 0 (see Theorem [12.31). Moreover, the

4

INTRODUCTION

constructed extension preserves to within constants all of the properties of the extensions obtained by Nikol' skn [4] for the same class of boundary functions. The method of proof of this theorem consists in extending a selected auxiliary system of functions by Nikol' ski1 's method in a certain way and then applying the averaging operator K, the above mentioned properties of which ensure the required properties of the resulting extension of a boundary function. §6 contains (i) a generalization of a theorem of Nikol' skif on function extensions from a subdomain of a finite domain onto the whole space to the case of an infinite domain, and (ii) a study of function extensions by the method of "symmetry," the idea for which goes back to Whitney and Hestenes. At the beginning of §7 we give a general method for averaging functions with a variable averaging radius relative to a given m-dimensional hyperplane in n-dimensional space, 1 E;;; m < n. §8 contains a brief sketch of the part of the theory of manifolds that is applicable in the sequel. Also in this section we study the question of the connection between function extensions from flat manifolds and from arbitrary manifolds; in particular, we generalize the results of §5 to the case of arbitrary (not necessarily closed or finite) manifolds. At the end of the section we make some remarks on function extensions from domains bounded by sufficiently smooth manifolds onto the whole space with preservation of the H-class to within constants. In §9 we consider a domain G in the halfspace xn > 0 that is bounded by a finite number of pieces of (n - !}dimensional manifolds and the hyperplane xn = 0. We give a method of extending functions given on the boundary of G onto the whole domain under which the derivatives of the extension are square summable over G with weight x~, o: > 0. § 10 contains a direct application of the results. This application is based on the fact that the constructed function extensions substantially improve the properties of the functions being extended. Here we indicate the possibility of applying the above function extensions to the solution of boundary problems for partial differential equations by the method of variation of boundaries; in particular, when in conjunction with a variation of the appropriate integral we consider a variation of the boundary of the domain. In addition to one of the results of Keldys and I.avrent' ev [ 1] we indicate here the ''rate" of convergence to a solution. This method is illustrated by the well-studied Dirichlet problem for the Laplace equation. It is shown in particular that such a dual variational method can be used to solve the Dirichlet problem for the unit disk in the case of the classical example of Hadamard, when the ordinary variational method does not apply.

INTRODUCTION

5

In § 11 we prove a number of integral inequalities with weight for use in the sequel; in particular, we use them to obtain a theorem on the imbedding into H-classes of spaces of functions whose derivatives of a certain order are summable in some power with weight. Here also we continue the study of the method of "symmetric" extensions (see §6). §12 is devoted to a generalization of the results of the preceding section to the case of arbitrary manifolds. In addition, we study the connection between the values of a function in a domain and its boundary values for the case when the derivatives of the function are summable in some power with a power weight relative to the distance from a point to the boundary of the domain; we give a sufficient criterion for extending a function from a domain bounded by a smooth manifold onto the whole space with preservation of its H-class to within constants; and finally, we prove an inequality of Poincare type with weight that connects a norm of a function and its weak derivatives in a domain with a norm of the function on the boundary of the domain. We note that at the present time necessary and sufficient conditions for a function defined on the boundary of a finite domain to be equal to the boundary values of a function whose derivatives are summable with a degenerate weight have been obtained with the use of other methods and in other terms by P. I. Lizorkin [50] (and somewhat earlier for a special case of A. A. Vasarin [49]). In § 13 we prove the completeness of the spaces with weight considered by us. In § 14 we investigate by the method of diagonal sums the connection between the convergence of a sequence of functions in a domain and the convergence of their boundary values. In § 15 we give compactness criteria for both ordinary function spaces and spaces with weight. In particular, the sufficient compactness criteria presented here complement the corresponding results of Kondrasov and Sobolev [2] and Nikol'skff [3]. The complementation is due to (i) an assertion on the simultaneous convergence of a selected subsequence on all cross sections of any compact sets by m-dimensional hyperplanes, the convergence being uniform relative to the choice of hyperplane, (ii) an assumption on the summability of the derivatives not simply raised to some power but on the corresponding summability of them with a certain degenerate weight, and (iii) the consideration of normal derivatives. The results of § § 14 and 15 are based on results published in the author's note [ 15]. In § 16 some of the results of § § 11-15 are carried over to the domain with piecewise smooth boundary considered in §9.

6

INTRODUCTION

In § 17, for an arbitrary bounded domain of n-dimensional space, we construct the variational method of solving the first boundary problem for a selfadjoint elliptic equation of second order. The absence of any smoothness conditions for the boundary of the domain is achieved by understanding the boundary values not in the sense of convergence in the mean, as is usually done in this case, but in the sense of convergence "almost everywhere. " In this connection we essentially make use of the results of the author's article [16], which are formulated here. In the same section we present investigations of the smoothness of the resulting solutions, in which the results of Kondrasov play a major role. It should be noted that in this section, despite a maximal in a sense generality of presentation of the variational principles, under the classical statement of the problem, i.e. when the admissible classes are nonempty, we do not here make use of any of the imbedding theorems of Sobolev, Kondrasov, Nikol' skn and others, but give (as in the article [16], to which we refer) completely elementary proofs of the theorems [ 17 .1 ] and [ 17 .2] cited by us. An understanding of § 17 does not require a knowledge of either the methods or the results presented in § §1-16. Since we were interested in making the presentation of these questions as simple as possible, we did not seek to prove the theorems obtained in this section under minimal requirements on the coefficients. In certain cases, as is well known, under additional restrictions on the class of domains being considered the conditions imposed by us on the coefficients of the equations can be weakened. The last four sections are devoted to a study of the equation

which is the Euler equation for the functional

It is assumed in this connection that a tends to zero like the distance from a point of the domain to its boundary under an approach to the boundary of the domain.

In the case when a= xn the equation L 0 a(u) A

X11 uU

iJU + a dx = 0, n

=0

a>

0.

takes the form

INTRODUCTION

7

As is known, in the case n = 2, for example, equations of the form y 2 32 u/3x 2 + o2 u/oy 2 = 0 reduce to this same equation when a> O as well as when a< - 2. A large number of investigations have been devoted to the study of similar equations both from the point of view of equations of mixed type and from the point of view of equations of elliptic type in the upper halfplane. Of these we mention the works of Tricomi [18), Lavrent'ev and Bicadze [19), Bicadze [20), Keldys [21), Vekua [22), Visik [23), Mihlin [24), Huber [25), Germain and Bader [26) and Vasarin [49). Further references can be found in the bibliographies of these papers. The most complete investigation of boundary problems for general degenerate elliptic equations is given in a paper of Visik [23). In § 18 we give a general plan for solving the equation L c,(u) = 0 by the a variational method. We show that it leads to a representation of the desired solution in the form u = 4' + 'It, where ,t, is an averaging of u and 'It is an integral of Newtonian potential type with mass a linear function of u. The connection between the smoothness of the function a and the smoothness of the resulting solution is studied on the basis of the results of Sobolev and Kondrasov [2, 10) on integrals of potential type and certain additional investigations. In § 19 we consider the case of a degeneracy of elliptic equations on the whole boundary of a finite domain and prove an existence and uniqueness theorem for the solution of the boundary problem. We also examine the case of a halfspace studied earlier by Vekua [22) from another point of view and under other assumptions. In §20 we prove an existence and uniqueness theorem for the solution of a boundary problem when the degeneracy of an elliptic equation occurs on part of the boundary of the domain. The cases of a finite domain and a strip are investigated. It is assumed in both § § 19 and 20 that the degeneracy exponent a E [O, 1); the case a ~ 1 is studied in § 21. In conclusion we note that for an understanding of the basic contents of the present monograph one need only be familiar with the first sections of the monograph of Sobolev [2], two papers of Nikol'skiI [3, 4) and the author's articles [15, 16]. I wish to take this opportunity to express my sincere thanks to V. I. Kondrasov and P. I. Lizorkin for reading the manuscript of the present monograph and making a number of valuable remarks, all of which were taken into account in arriving at the final wording of the text.

FIRST CHAPTER

FUNCTION EXTENSIONS § l. The averaging operator K

Let Lin> denote the space of functions that are p power summable on all of one n-dirnensional space En, n = l, 2, • • • , and let 11/llin) denote the norm of a function /E Lin> in the sense of Lin>. We will always assume that 1 ..;; p..;; 00 , it being understood that by 11/llt> is meant, as usual, ess supxEEnlf(x)I. Consider the functions K(t)

K(2(~ - x)/u) = exp( - u 2 /(- u 2

=e- I/(I -t 2 >

for itl ..;; 1 and K(u, x,

4(x - ~)2)) tor I~ - x 1..;; lu 1/2, We note the following obvious formulas: -

oo

O= < u

[1.1]. For any function

1

II CoROLLARY. PROOF.

Kf II~">~ 2-,; II f 11 en>

{1.12)

If fE Ltn>, then KfE Ltn)_

Suppose p

=

00 •

Then from (1.9) we have

which implies {1.12) for p = oo. Suppose now 1 ~ p < 00 • In the following proof and repeatedly in the sequel we will apply the so-called generalized Minkowski inequality

n1 +n2 here (u, v) is a point o f some set E lying in the direct sum E u,v of 2 • We have 1 and Euclidean spaces

E:

l~l -

E:

[j_ ... jJ,:., J'... -l~/

(I,)/

(x,+ ';". ... , x. +•,;•)

,;di, ... dt.[dx, ... dx.]¾

.,; ;.J . j'.f~

K

(t,>[t . J} (

x,+ ·';'', ... , -'•

+ ••;• )]'

Xdx, ... dx.]f di, ... di•. In the inner integral we now make the change of variable

u, = x,

X 11

I; .

+ 2 -,

i

= 1, 2, ... , n.

(1.13)

I. FUNCTION EXTENSIONS

12

This is a nondegenerate linear mapping of En onto itself with determinant

so that (1.14)

and therefore

II Kt

!It)< 2!c X

I

j •·•J:fr,

iJ(x1 , iJ(

••• ,

~ n~.!.. llfll~n) 2

Pc

[l~- ·•_I, If

)

dlln

l

I

ll11)I"

/Jdl1 ••. dtn

n

5... Jlf K(t,)dt1 -I

(ll1, • · ·,

Idu1 ... ]--'--

x,,)

ll1, • • . ' Un

K(lt)

I

... dt11=211 11fll~n)_

-J,=I

We have made use here of equality (1.5). [1.2]. K is a continuous linear operator in L1n)_

The linearity of K is obvious, while its continuity follows from its boundedness proved above. § 2. The invariance of the H-classes under K

Let r > 0 . In the sequel r and a will always denote the two numbers such that r = r + a, r is a nonnegative integer and O < a ,;;;; 1. Ai~>([, h), as usual, is the difference of order k with step h of a function f l with respect to a variable xi. By

J/;(1,;)'",rn)(Mp · • • , Mn)

having weak derivatives

arif/ax?

' \

we denote the set of all functions f E L1n)

E Lin) (see [2, 31) such that

r

IL\\:>(:}, 1i) '

(n)

~ M,I h (i'

(2.1)

/J

the Mi being positive constants, i = 1, • • • , n. Here, if in the case ai < 1 the second difference is replaced by the first, i.e. (2.1) is replaced by the requirement

§ 2. INVARIANCE OF H-CLASSES UNDER K

13

(2.2)

we obtain a definition equivalent to the one given above. In place of

IIJc~·t> (M, ••• 'M)

we will simply write IIJ~n )(M). All of these definitions are due to Nikol' ski1 [3] . In the present section we show that the H-classes are in a sense left invariant under the integral transform K. To this end we need some properties of entire functions. As is well known, for every entire function of exponential type

Kv 1 ••• v/z 1 , and for any

• • • , €

zn) of degrees vi respectively in the variables zi (i = 1, • • • , n)

>0

there exists a constant A

>0

such that for all complex

(2.3) Let us prove the following lemma. [2.1.1]. In order for the entire function CO

g (z1 ,

••. ,

Zn) =

00

2i ... h

k,-o

kn

(2.4)

ak,,,,knz~•

~o

to be an entire function of exponential type of degrees vi respectively in the variables > 0 there exist a constant B > 0 such that for all complex z 1 , • • • , zn

z; (i = 1, • • • , n) it is necessary and sufficient that for any e

n oo

~ k,-o

~ < II gs- f II+ ll f-gs-1 II~)-< - - i s - ' s = 1, 2, ....

(2.16)

(2.17)

The constant d does not depend on f or the M;. We note that the expansion (2.15) with the corresponding estimates was also obtained by Nikol'ski1 ([3], page 28). By virtue of the linearity and continuity in Lin) of the averaging operator K we obtain from (2.13) the convergent in Lin) series

(2.18)

According to Theorem [2.1) the function Kq 8 is an entire function of exponential type of degree v~s) = i9!r, in x 1, i = l, • • • , n - 1, and of degree v~s) = 2n+s/r in xn, where r= mini=l,••·,nr,, since n

s

s

s

s

! ~2r;+2'n), where Af(µ) are fixed. The order of convergence to zero in equality (4.1) was found by Nikol'skn. This result, which is formulated more precisely below, was announced by Nikol' skn at a seminar on the theory of functions at the Mathematical Institute of the Academy of Sciences of the USSR in 1952 but was never published. Therefore Nikol'skn's theorem-actually a somewhat modified form of it that is more suitable for our purposesjs given here together with a proof.(6 ) [4.1)

(N1KoL 1SKII'S THEOREM).

~1=fj

Suppose f E If-;(~•;",rn)(M), 1 ...;; m

< n,

n 1 (1--P1 .:.:..i ~ -), rz. m+l

with O < f3; < 1, j = m + I, ... , n, and suppose the quantities xm + 1 , . . . , xn and hm + 1 , . . . , hn have been arbitrarily fixed. Then there exist constants c 1 > 0 and c 2 > 0 not depending on M, f or xm + 1 , • • • , xn, hm + 1 , • • • , hn such that

(6)This is done with the amiable consent of S. M. Nikol'skii.

32

I. FUNCTION EXTENSIONS

ex,

a,

-oo

-ex,

={ - f (Xi,

J ...

S I f (Xi,

iJx,._ n-l

r-

,0)

I

dx1 ...

dX11-J

tn_ 1 ), and hence lim.x ➔ 0/2 n

[4.3]. Suppose f E

.:1

,

-oo

-- 'i} (xi,

(uniformly in t 1 ,

1

0 for

Xn

-o

= 0, which proves (4.12).

and there exists the weak derivative

1" 1 r E H p(n),

1

> .!.p , ' + + ' "1

'

"n-1

• • •

:a:: I\ •

Then l

iJ

Kf (zt, ... , z,._1, 0)

:i.,._,

.,. 1. .,. u'Xi • •. u'Zn-1

l

==

iJ

f (zi,

.,. -11

.u'Xi

•.• , .,.

• • •

x,._1, 0)

(4.13)

1,._,

.,z,._,

In particular, when >.. = 0

Kf(xi, ... ,

Xn-1,

0)

= f (Xi,

... ,

Xn-1,

(4.14)

0).

These formulas are a direct consequence of [4.2], (3.14) and (1.15). The situation is somewhat more complicated with derivatives of the type o"'Kf/ax!. [4.4]. Suppose f E Lin) and all of its weak derivatives up to order µ inclusively exist, each derivative of order µ belonging to the class where r >

n is fixed the function r.p belongs to the class n(.et-l/P)tM ) where p(n-l) \:

1 '

M1-. Since O < a - 1/p

(cMife). Hence

and the use of this estimate will change the constants his> and b~s) of inequality (5.10). The theorem is proved. REMARK. For derivatives not involving differentiation with respect to xn we have the estimate (5.12)

a?)

in which the constants and a~s) do not depend on f, M or e. This follows from the fact that (see [3.51)

X

f~i'• .. ~~~11 (x1 +

Xn2t1 ' •• • Xn+ ~n ~n) dti ...

din,

where P 1 + · · · + Pn-l =rand si;;;. P;,i= I, 2, ... , n - I, and hence only the kernels corresponding to the variables t., ... , tn _ 1 can be a kernel with the singular property ( *). Inequality (5.12) follows from (5.6) and (5.10).

§5. EXTENSIONS FROM A HYPERPLANE

47

The following theorem corresponds to Theorem [5.1] in the case p = [5.2]. Suppose f E

ncn/M), with r = r + a

Then for any e

00 •

>0

and

s>O

where the constants ais) and a~s) do not depend on f or M. When inequality continues to hold for e = 0.

a
,s;,.M. 4°. a'-F(x1, ...

= tJ,,_(x1, ...

,Xn-1' 0)/ox~

,Xn-1),

>..= 0, 1, ... 'p~

5°. For sufficiently small e > 0 and s = 1, 2, · · ·

Af(s)

i}s+r F ll

II ax, ... oxnSn S1

P(

'"n ,. p(s-a)+•l

'~ - - , - ' El+

p

where M(s) ,e;;; cis)

In the case p

=

p

p

>-=O

A~o

~ M>. + c~s) ~ 11 liiA l!~n-1)_

it is only necessary to change the inequality in condition 5°

00

to

500.

II I

Xn

os+r 1s-a+• - -F- IICnl iJXsl, • . • iJxsn "

< oo

(ex = ~. -r

= -p ).

00

We remark that the existence of a function F satisfying conditions 2° -4° was proved by S. M. Nikol'skii ([4], page 87). Thus the theorem is new only in regard to conditions 1°, 5° and 500 • Its proof follows directly from [5.l]-(5.3]; it suffices to take F = Kf, where f is the function constructed in [5 .3] . We note that by way of addition to the above theorem at the present time Ja. S. Bugrov [ 17] has proved the possibility of an analogous extension from an (n - 1)dimensional hyperplane ( and even an m-dimensional hyperplane for 1 ,;;,. m ,;;,. n - 1) 0 >= LP. And in the onto the whole space in the case p = 0 if it is assumed that case p = 2 and n = 2 he showed in the same paper that in extending a function from the unit circle under assumptions corresponding to our condition 5° it is possible to put e = 0. The question of this possibility in the general case remains open.

n:,

§ 6. Function extensions from a domain

Let G be a domain in En. For a function f given on G we introduce the notation (6.1) and, in the case of a weight function a (see (5.1)),

51

§ 6. EXTENSIONS FROM A DOMAIN

(6.2)

llf 111n)
(G).

00

G

As before, when G = En the index G will be dropped. We recall that the fl-classes were defined by Nikol' not only for functions given on the whole space En but also for functions given on bounded domains [ 4]. Let us present this definition. It will not be assumed that the domain G is finite. For every domain G C En and '17 > 0 we put G,,, = {x: x E G, p(x,En\G)> '17}, it being assumed in the case G = En that G,,, = En for any '17 > 0. A function f is said to be a function of class _(M, G), r = r + a, if it

skn

n + '11k (y,!'!~ + ... + y;lk)) ,1•1

==

'11•

(x!W1 +... + x:1•1) ,1•1

58

I. FUNCTION EXTENSIONS

and hence 11c~k)I ,,;;;; 2m,,.,''0' i = 1 ' • • • c;,

'

m, k

= I'

2' • • •

.

Thus all of the points p(1k)

and Q\k) lie in the bounded parallelepiped ixil..;; 2mr, 0 , i = 1, • • • , m, lxjl,,;;;; 1, j = m + 1, • • • , n. When r, = 0, as we have already noted, the mapping (7 .9) converts into the identity mapping. Therefore, when k ➔ 00 , the distance between Pt> and Qik) tends to zero. It can be assumed without loss of generality that the sequences of points Jik) and Qt), k = 1, 2, • • • , converge to a point P0 . By virtue of the implicit function theorem and the boundedness of the second derivatives of the mapping (7 .9) the point PO has a fixed neighborhood on which (7 .9) is a one-to-one mapping for all sufficiently small r, > 0 and any ti, lt;I..;; 1, and which therefore cannot contain the points Jik) and Q{k)_ The resulting contradiction proves our assertion. We now put x,+'lr•

X,a+'lr' n

x,-'lr•

xn-'1'' /-1

J ... s IT K (11,z, X1, e/) f (E1,, ·,, e,.)

1

K2f(x)= -

c11"r211

~l - , ,

dE,..

From (7.5) and (7.10) we then have 1

11~,zfll!"' ~ 2,; 11111i">.

(7.12)

m

Formula (7.7) converts into (7.13) This implies the following more general formula (corresponding to (3.8)), which is obtained by a direct calculation: 2'i

X

's+1 's+i

,2

(

2(~1-xj)

,2

)p. II ( o'k + 2~, s

/

k-1

k

X;

k

'f.1-XJ)K(/) r•

s

__ iJ_[(2(~1-x1))P.TI(6'k+2r.1. X e1 x')KU>] iJx I r I t' ' k i k rs s+t

=j_ iJeJ

-

s

2

k-L s+t

[(2(~J-Xj))ILII (o'.k+2~, X; ,a \ k-1

I

k

k

'f.1-x1)· K(/)] rl

s

~- a'v (2(1.J-Xj))P.+lrr ( aI111 +- t'l1,. A X )t,-x, K(/) LJ lyis+t rs lk - , , ~

v-1

k-1

r

k+•

ik,

i= 1,2, ... ,n: s=0,1, ... ; p.=0,1, ....

(7.14)

§7. AVERAGINGS RELATIVE TO HYPERPLANES

59

A special characteristic of the multidimensional case (n - m > 1) is the appearance here of the second term, which was not in (3.8). As in §3, it can be shown that the function K 2 f is infinitely differentiable on en\Em when fE Lin>. In this connection we have the formula

__3.8 K 2 f(x) c)x11 •••

=

II

A

j=l

j

OX

2 ((j-Xj))Pj }211 ~ U),p.,rn°,i[( C1J r J 0 such that when

~'{' (u 1 - uf)2

+ ~:!i+ 1 uJ < Tl

we have

Iod (x

1, •.. ,

(11 1 , •••

x,,)I > O.

,IU,.)

Applying the implicit function theorem, we finally obtain [8.1]. Every point P of an m-dimensional manifold K(m) of order of smooth.•ess k ~ 2J. has a neighborhood (relative to the manifold K) in which for any

local coordinate system (u 1 ,

• • • ,

> 0 such that UJ a1W. , , = 12 , , .. , n ,

um) there exists an e

x, = x, (Ui, ••• , Um )+

n

"1 .:::.J /=m+l

62

I. FUNCTION EXTENSIONS

is a ont-to-one k - I times continuously differentiable mapping with a Jacobian that -1oes not vanish on the set

We now take an e1 such that O < e 1

. A canonical mapping is a one-to-one k - I times continuously and boundedly differentiable mapping, while the absolute value of its Jacobian is bounded from below by a positive constant:

I

o(x1, ••. ,x,,)I

>A> O.

Q (lit, • , • ,1111}

If a canonical mapping takes a point u = (u 1 , • • • , un) into a point x = (x 1 , • • • , xn), we can regard the numbers u 1 , • • • , un as curvilinear coordinates of x in V, although we will also call them local coordinates, the coordinates of P

§ 8. EXTENSIONS FROM SMOOTH MANIFOLDS

63

being denoted by (up1 • • • • , up) n · Let PO be an arbitrary point of an m-dimensional manifold K(m). There then exists a neighborhood U of PO such that for every point PE Un K(m) a system of n - m unit vectors Iii; =(a.'/), • • • , ex~>) that are orthogonal to each other and to the manifold K(m) can be chosen in such a way that the functions o.fi>(u 1 , • • • , um) (i = 1, • • • , n; j = m + 1, • • • , n) are k - l times continuously differentiable functions of local coordinates. If this choice can be effected at each point P of K in such a way that the normals N;(P), j = m + l , • • • , n, are referenced in the same way, the manifold K(m > will be called an orientable manifold and the corresponding choice of normals will be said to be consistent. In the sequel we will consider only orientable manifolds, without specifically saying so. We note only that every (n - 1)-dimensional closed bounded manifold in En is orientable, and, if it is the boundary of a considered domain, a consistent choice of normals permits one to take inward or outward normals. Let E be a subset of an m-dimensional manifold K(m) of smoothness k;;;. 2 on which a consistent choice of normals has been made, and suppose E is covered by a finite number of canonical neighborhoods vM of height e > 0. This can always be done, for example, if E is compact and Ef; K. In fact, ifwe choose a canonical neighborhood V(P) of each point PEE, we can distinguish from the resulting covering of a compact set a finite subcovering v = V(p(v>), where vM is the image under the corresponding canonical mapping of the set

~

vL%v1= [cu}v)):

(ui•>_ uf) 1 < E~, lu~•) I
I< e;j =m + 1, • • • , n}, v = l, • • • , vO, gives the desired covering of E. Let p (P) denote the radius vector of a point PE En. The locus of the endpoints of the vectors

~7!>

p

= p(P) +

±

v!';(P),

lv,1 < "l(E), while the loci of the endpoints of the vectors

(S.8)

I. FUNCTION EXTENSIONS

64

p(P)

+

n

~

v; N; (P)

+ TjN 1;

jv; I < Tj

(8.9)

1-111+1

i-,.J

(i=m+l, ... ,j-1, i (P) 1'

+

i+l, ... ,n),

n

"1

~ i=rn+l t1-/

V; N i(P)

-

PEE,

TJNJ ; Iv, I < Tj

(8.10)

(i=m+l, ... ,j-1, j+l, . .. ,n), PEE,

p(P)+

±

t•.-N;(P); [tt;I

i-111+1

will be said to be of zero measure if for any canonical neighborhood V the set Eu= {(uf): PEE n V} has an m-dimensional measure equal to zero. In the sense of this definition equivalent functions f and g given on K(m > coincide on K(m > almost everywhere. We will not distinguish between equivalent functions in the sequel. We will say that a function f defined on K(m) is of class n) if there exists a system e = {V} of canonical neighborhoods forming a locally finite covering of K(m) such that for any neighborhood VE e (see §6, (8.6) and (8.14))

lM,

(

)

0

f11EH/(ml (M, Vu). This definition is a natural one since by virtue of the corresponding results of Nikol' skii ([ 4], page 59) the H-classes of functions remain invariant in a sense under

66

I. FUNCTION EXTENSIONS

sufficiently smooth transformations of the variables, which is guaranteed in our case by requirement 3° in the definition of a manifold. We will not dwell here on the question of the dependence of the H-class of a function on the choice of the system ®, which will always be assumed to be fixed. We only note that in regard to a class Kim)),

n = K(m) is a closed bounded manifold; the stated assertion was proved for this case under assumptions not essentially different from our own by Nikol'skii ([4], §4). Suppose ® = {V} is a system of canonical neighborhoods forming a locally finite covering of a manifold K(m). We denote by A (v), v = I, 2, • • • , the members of a family of sets satisfying the following conditions: 1°. Each A (v) is entirely contained in every canonical neighborhood VE ® with which it has a nonempty intersection. 2°. The intersection of two different A (v) is of measure zero. 30_ K(m)=UvA(v)_

r

A choice of such A (v) can always be effected for any system ®. If A (v) f V, we denote by uv the canonical mapping corresponding to V, and by At) the preimage of A(v) under uv. Suppose, finally, that a function f is given on K(m)_ We put

(8.15) It is also possible to define in a natural way an invariant norm of f that does not depend on a choice of local representations of the manifold K(m), i.e. on the choice of a system ®; namely, one can put

(8.16) where dK(m > is a volume element of K(m). In the case when K\m) is a submanifold of K(m) such that R_(m) is compact and K\m) C K(m >, the passage from a system ® to another system c~uses a finite variation in the norm

llfll~m>.

K("'l(-11 vN,n+l ..• vN 11

>.

>.

.::i m+l ••• vUn .::i n vu 111 +l

(8.19)

I. FUNCTION EXTENSIONS

68 and

(8.20) (if, of course, the expression on the right exists in the sense of (8.18)). In the case when the order of smoothness k of K(m) is such that k;;.. A.+ 1 and fE )(M), where r - A. - (n - m)/p > 0, the normal derivatives defined in the above way transform under a passage from one local coordinate system to another (in particular, under another consistent choice of normals) according to the usual classical formulas. This was obtained in the same paper of Nikol'skii ([ 4], pages 101-104) under assumptions not essentially (for carrying out the proof) different from our own. Let G be a domain in Em C En, l ~ m < n, p > 0, p =p + {3, and for each l = 0, 1, • • • , p suppose given all possible systems of nonnegative integers ;\,~\ 1 , • • • , ;\.~')

I-l;(n

such that

~7:m + 1 A}') = l.

~) a function i/J A(I) m+J,

(n - m)/p =

•••

Further, suppose given in G for each system ;\_~)+ 1 , (I) E

I-1,:C;'?(M

,An

(I)

••• , (l)'

G), where r = p

••• ,

+

All

Am+l

r+ a, and suppose given a system of nonnegative functions cps(r), s=l,2, ... ,s0 which is the image of Vu under a canonical mapping x =x(u) the functions 1/1 0

(l) (l)(x(u)) Xm+l '"A n

Vu (see (8.6)), we will say that the system {1/1

(l) (l)} Am+1 ·••1'.n

is

are 4()9

4()9

extendable from

extendable from

K

means in a sense a corresponding local extendability of the considered functions. The purpose of the present section is a proof of the fact:that local 4()9 extendability of a system {1/1

(l) (l)} Am+l •• 'An

implies its extendability in the large for any

4()9 ,

s = 1, 2, • • • .

As an example, we note

[8.2i Let G be a domain in the (n - I)-dimensional hyperplane En-l = {(xi): Xn = O}. Then the system of functions 1/111.(x 1 , • • • ,xn_ 1 ) EH'-fc;~{)(M11., G), p=p+(j,X=O,l,••• , 11p,is lxnlp(s-ae)+e extendablefrom G,where r=p+

1/p=r+a,e>O,s= 1,2, •••. In fact, it follows from Theorem [6.1] that for any T/ > 0 the,!unctions 1/111.~can be extended from the domain G 71 onto all of En-l as functions 1/111. such that 1/111. =

~

,.,{

X)

~

1/11,. on G71 and 1/11,. E11fc;_ 1 /M1,.), where

'\"'h, ~ c 1riM1.

+

C2'1

ll'i',II~' · 11 , II¥, l\):'- 11 ~ c'1 ll'i'•ll~"-o • ,,

Applying Theorem [ 5 .4] to the system of functions {1/111.}, A= 0, 1, • • • , p, we obtain the function F required in the definition of the 4()9 extendability of a system of functions when l{)s 0, p = p+ ~. r = p + (n - m)/p = r + a, r+ 2 ~ k and '{)it) is a sequence of nonnegative measurable functions defined on the interval [O, t0 ], t 0 > 0, s = 1, • • • , k - r - 1. Further, for each l = 0, 1, • • • , p suppose given all possible systems of nonnegative integers X~)+ 1 , • • • , X!,1) such that r,f=m + 1 = l, and

xy>

for each admissible system X~)+ 1 , ,h

C

).(I ►..:

T ~)

n

+1 · · ·

the system of functions {-./I

• •• ,

X!,1)

suppose given a function

H(,o-l)(M p(m)

K(m)),

,.fl),

J.(l)

111+1 ... "

(t)

(t)}

being

'{)s

extendable. Finally, suppose G =

i\m+ 1 • · •i\n

En\ (K(m >\ K(m )) (clearly, G is an open set), @ is a domain in En such that @ is compact and ~ C G, GK =@ n K(m) and r(P) is the distance from a point P to K(m) "with respect to the normals" to K(m)_ Then there exists a function F= F(x 1 , • • • , xn) defined on En and satisfying the fallowing requirements. 1°. F is continuously differentiable on En\ "f(_(m) up to order k - 1 inclusively.

2°. FEil;/nlM, @), where

//M ~ c1 E M,:,!.>~ 1 ... A:,n + c2 E /I~ ,i,~~ 1 ... •:,n II j,"'>, the constants c 1 and c 2 not depending on M

i\ ( l)

••• i\ ( l)

or -./I

n

m+l

i\(l) •••i\(l) m+l n

but

generally depending on the choice of the domain @. 3°. IIFll(n) o;:;:;M. ~

p

40_

There exists an T/ = 7/(@) > 0 such that

f

I

K~IJ (©k) S1

Ir

iF+sF(P) ips[r(P)] ---..:.......:...dv 1,

+... +Sn= r + s,

< oo,

(}xr . .UX ;; 11

s = 1,2, ... , k -

r-

1, ll


) there exists a neighborhood of it which intersects only a finite number of balls Qi ( and hence a finite number of balls Q' 1 and Q;). 2°. The system {Q;) forms a locally finite covering of K(m >. Q., Qp'

=Q', i

3°. Qi C lf(i) E@i* for all i= l, 2, • • • . Let us establish the possibility of such a choice of points P, and radii r1• Since K(m) is an absolute Fu set, it has the representation K(m) = U';' A.,, where the A., are compact sets,A., CA.,+ 1 , s = 1, 2, • • • . For each point PEA 1 we choose a bounded neighborhood Wp ofit so that the intersection Wp n (K(m)\K(m)) is empty. From the system {W p} we select a finite covering and denote the intersection of K(m >

PEA,

with the union of its members by B 1 • Clearly, the intersection B1 n (Klm J\ Klm ') is empty. Suppose there have been defined open (in K) sets B., • • • ,Bil such that 0

B 1 C B 2 C • • • C Bil C K, All C Bil, 9 11 is compact and the intersection lill 0

n

(K\ K(m >) is empty, v = l, • • • , v0 • Consider the compact set li 11 U All + 1 . For 0

it, as for A 1 , there exists an open (in K(m >) set lill + 1 :J 0

Bllo+ 1 is compact while the intersection

Bil 0

U A 11

0

9110 + 1 n (K\K(m)) is empty.

Thus B 1 C B 2 C • • • C B 11 C • • • C K(m) and Uu.ill = K(m)_

0

+ 1 such that

72

I. FUNCTION EXTENSIONS

We put C1 =B 1 and Cv=Bv\Bv-i,v=2,3,···. Then the Cv are compact sets (v = l, 2, • • • ) and Uv Cv = K(m). From the above-mentioned system of all possible balls we select the subsystem consisting of those balls for which PE Cv and r < l/v. This subsystem forms a covering of the compact set Cv. We select from it a finite subcovering and denote it bye"= {Q; }. Corresponding finite sets of the balls Qp and Q~ are denoted by and e~, while the balls themselves of the system e~ and e: will be called balls of rank v. We next arbitrarily number the centers P of all of the balls contained in at least one of the systems v = l, 2, • • • . Let us show that the resulting set of points Pi together with the corresponding balls from the systems e 11 ,e~ and e: is the desired one. Suppose PEG, so that Pp= p(P, R_-(m)\K(m)) > 0. Then the ball Q(P, Pp/2) with center at P and radius p p/2 intersects only a finite number of the Cv. Otherwise

Q;

ev

ev,

ev,

Q(P, Pp/2) would contain a limit point for an infinite sequence of compact sets Cv that

clearly can only belong to the set K_(m >\ K(m), which is impossible. Therefore suppose the ball Q(P, pp/2) does not intersect any C11 for v ~ v 1 • We choose v 2 ~ v1 such that l/v 2 v2 , i.e. it intersects only a finite number of the selected balls Qi. From the fact that the balls of the system e ~ cover the compact set Cv, v = 1, 2, • • • , and the fact that K(m) = Uv Cv it follows that the set of balls Qi, i = 1, 2, • • • , forms a covering of K(m), while its local finiteness follows from the already proved condition 1°. Thus condition 2° is also satisfied, and as to condition 3°, it follows from the original choice of the balls Q. The existence of the needed system {Qi} is proved. For every U(i) Ee* there exists a VE e such that [J(i) c V. Let y(i) denote one such V and let y(~) denote its preimage under the canonical mapping P = x(u),PE y(i), u Ev. Thus with each point Pi there is associated a pair of canonical neighborhoods u(i) and y(i), either one of which may also be associated with other points P; (i = I, 2, · • • ). According to the definition of the l{)s extendability of the system of functions {if; (l) (I)} there exist functions 7;(u) defined respectively on the y(~) that ;\m+l •.

•;\n

together with all of their derivatives are continuous everywhere on the y(~) except possibly the sets V ~i) (see (8.6)). In addition,

-"'c. ( /t'1 ''f ;, V1IIi>) , M i-::::,, ,.,, C1(il""' M fi '- H(n f'(/1) ~

§ 8. EXTENSIONS FROM SMOOTH MANIFOLDS

73

(8.22)

(8.23)

(8.24)

s = 1,2, ... , It

-r -

I.

We put MP)= f(x(u )), where P = x(u). Then by virtue of a theorem of Nikol' skii on the behavior of functions belonging to fl-classes under a change of the independent variables ([ 4 ], page 59) we get (8.25) and (8.26) The indicated theorem of Nikol' skii can be applied as a result of the fact that the canonical mapping P= u(x) is k - I times continuously differentiable, r + 2,;;;;, k, and the absolute value of the Jacobian is bounded from below by a positive constant. It also follows from this according to (8.24) that /i(P) '' ~ ••• S\ K(m), there does not exist a ball Q.l 3 P• In fact 'Q. C y(i) E ~ l ~, while by assumption the intersection V n (K.(m >\ K(m >) is empty for every canonical neighborhood VE~. Therefore '-P;(P)= l,whichimplies ;(P)=O for all i= 1,2, • • • , i.e. F(P) = 0. If PEG= En\ (K(m >\ K(m >), there exists a neighborhood O(P) of P that

§8. EXTENSIONS FROM SMOOTH MANIFOLDS

75

intersects only a finite number of the balls Q;, There therefore exists an i0 such that ,p;(P) = 1 and hence i(P) =- 0 for i ;;;i, i0 . It follows that F(P) = L:o F;(P);(P). If, in addition, PE G\ K_(m), the neighborhood O(P) can be chosen so that it does not inter-

sect x_(m), and then F = L:oF;; is a k - 1 times continuously differentiable function on O(P) since each summand Fii is such a function. Thus (8.30) is meaningful and, what is more, we have shown that the function F satisfies condition 1° of Theorem [8.3]. We put o' = UiQ;. Then o' CO since c y(i) c O and O'\K(m) c O\K(m)_

Q;

If PE En\ Q;, then ,p;(P) = 1, and hence ;(P) = 0 for all i = 1, 2, • · · ,.which implies F(P) = 0 for PE O'. Requirement 6° is proved. Now let @ be a domain such that @ is compact and ® CG, and let @K = @nK(m)_

From the fact that ~K is also compact it follows that it intersects only a finite number of the balls Qi. Suppose the intersection Qi n @K is empty for i > i0 . Then W= y(i) forms a neighborhood of the compact set @K. There therefore exists an

u:o

>0

such that the cylindroid K~n >(@K) C W. Hence for s = 1, 2, • • · , k - r - 1, using (8.29) and (8.30), we get

1/

where c > O is a constant. It follows by virtue of (8.27) that condition 5° is satisfied. The compact set @ intersects only a finite number of the balls Q;- Otherwise the intersection ~ n (K.(m >\ K(m )) would not be empty. Suppose the intersection ~ n Q1 is empty for i > i0 • Then

F

= ~ F/P; i-1

on @, and, since F; satisfies conditions (8.25), (8.26) and (8.28) on Qi while ; is such that F.. is a smoothing of F.I to a function that identically vanishes outside Q 1'., I I the ; having continuous derivatives up to order k - 1 inclusively that are uniformly bounded in absolute value on En, it follows that the function F satisfies conditions 2° and 3° on G. The proof of this fact as well as the proof of condition 4°, which by

76

I. FUNCTION EXTENSIONS

definition has a local character, is an insignificant modification of the proof of the above mentioned theorem of Nikol' skif ([ 4], pages 111-115) for the case of bounded and closed manifolds K(m). We will therefore not reproduce it here. We note a special case of Theorem [8.3 ], which we will essentially make use of in the sequel. [8.4]. Suppose 1 o;;;;p 0, p= p + {3, r = p + lip= r + a and r + 2 .;;;; k. Further, for each X = 0, 1, • • • , p suppose given a function I/Ix E H(p-X) (M K(n - l )) Finally suppose G =E"\ (K(n - 1 )\ K(n - l )) ,u is a domain in p(n-1) X• • , ,"!I E" such that l~ is compact and ~ C G, @K = @n K(n - 1 > and r(P) is the distance from a point P to K(n - l) "with respect to the normals" to K(n- l >. Then there exists a function F(x 1 , • • • , xn) defined on E" and satisfying the following requirements. I 0 • F is continuously differentiable on En\ Kln - l) up to order k - l inclusively. 2°. FEH~.-o

A=O K(n-1)

the constants c 1 and c2 not depending on M x or 1/1 x·

3°. IIFlli"> o;;;;M. @

4°. aXF/aN"ll {cn-l)

5°. There exists an 11

= I/Ix, X = 0,

>0

1, • • • ,

p.

such that for any

€

>0

6°. There exists a neighborhood 0' of K(n-l > such that

0' '-.. K 0. It will be assumed that the domain + G C En and that its boundary K is representable in the form '" µ, K= "'UK"u U L,.v UMµ, K

0-

l

,

I

(9.1)

µ-I

where each set K" is a finite piece of an (n - 1)-dimensional manifold of order of smoothness k ;;i..2, each Lx is a domain in the hyperplane En-I= [xn = 0] and each Mµ. is simply an (n - 1)-dimensional bounded and closed manifold of order of +

smoothness k ;;i, 2 lying in the halfspace En. Thus the domain G is generally multipily connected. We will say that a piece K" does not intersect the hyperplane En- 1 at a zero angle if its outward normal NP at every point P E K" n En - 1 forms a positive angle o.NP with the coordinate axis Oxn. When this condition is satisfied,

o < ao:.;, a.Np< TC, PER. n £(11-1>, where OO is a constant. If this is true for all " = 1, • • • , "o, we will simply say that the part (9.2)

of the boundary does not intersect the hyperplane En - 1 at a zero angle. For every function F which together with its first order weak derivatives is defined on G we put

J. .. S ~ ::,.r II

Dm (F) =

x~

0

Thus D 0 (F)

(

dx1 ••• dx11 •

(9.3)

i-l

= D(F)

is the Dirichlet integral. Let us prove a lemma. + [9.1.1). Suppose that the boundary K of a finite domain G C En satisfies

+

the above conditions and that a part K of it does not intersect the hyperplane

En - t = [xn = 0) at a zero angle. Then there exist pairwise disjoint neighborhoods

79

§9. EXTENSIONS TO A SMOOTH DOMAIN

u, v, u, v, u

uJt 1'-.KcV~I(>,

v

and

of the sets Kl(, L-,.,. and M,.,,,

K; utA>,K cv~>, K;

u'.r,> '-.KcV~ 1 '-.K

(1 =

Al(u

c[,o,

GBo, ! 1r)]'

vr> =ALU C [,0 , ( - 0;

, 0;)] ·

v.. is not 1

empty a function

">..K ••• " . 1 J-,._

defined on En, continuously differentiable on the

LJ 1.I(.1 n L">.., equal to zero outside the set yL(">..) u 0/(K 1 ) u • • • _.{ ) (K · ) where v, v ~ 1 , • • • , V K 1">.. are pairwise disjoint neighborhoods of

domain En\ U

(K • ) V K 1">.. ,

the sets

L">..,

K" , • • • , K" 1

">..

respectively that are separated by a positive distance

from the remaining parts of the boundary, and such that Da,(">.." ••• ". ) l

addition, there must exist neighborhoods

u C v

of the sets K". and J

L ">..

(K•) UK 1 C

J-,.,_

(K·).


l

K

Theorem [6.1], for example, the functions A

PK

FiA

A

"

and ~ ~

•

and @L , and hence by virtue of the choice of @K K

Further, PK

L).•

... ,Ko, A = 1, 2, ... ,t-0 ,

The conditions of the theorem imply that It therefore follows from (9 .13) that and

F;

(913)

PL).+2(@)

FL1-(:H'2(n)

Fi,

from some domains@~ A

"

and analogously for @ ~ ) onto all of En A

with preservation of the H-class in a sense, and then applying an imbedding theorem of Nikol'skii ([3], Theorem 12, page 26),(1 1 ) we get that

which implies the existence of a constant A

- A, I p·K" I ~

>0

such that

,, ') jfL).j,(M'), where }..' is any number satisfying the condition O < }..' < }.. while M' ,s;;; aM, the constant a depending only on K and the choice of }..' but not on f ([36], Chapter V, §3). Without going into a detailed analysis of the constant a > 0 we note only that in the case when the surface K has an order of smoothness k ;;;i,, 2 there exists an e > 0 such that a single constant a > 0 can be chosen for all of the boundaries K 11 , 0 < '11 < e, of the 71-neighborhoods G11 of G. It will always be assumed in what follows in this section that the surface K satisfies all of the above conditions. We prove the following theorem. [10.1]. Suppose given on the boundary K of a finite domain G C E 3 a function [En, 0 < p < I, and suppose F is an extension ofit onto E 3 (see [8.41) satisfying the following conditions. 1°. F=f on K. 2°. F is continuously differentiable on E\ K. 3°. FE JlC_:)(M).

4°. 11 1 -p+e lloF/oxjll .. < c, i = 1, 2, 3, the constant C not depending on 71. Further, suppose 1111 > O, lim 11 ➔ 00 '11 11 = 0 and u 11 is a harmonic function defined on G., and taking the value [ 11 = F k on K . ''II

1111

1111

Then the sequence u 11 converges uniformly on G to a harmonic function u such that u lk = f, with

iu - u,J where {j < p - e, e

>0

< c'-r;~,

and the constant c' does not depend on 71.

PR.ooF. By virtue of the fact that F is continuously differentiable on En\ K. the function [ 11 = F k., is of class n(M11 , K11 ) (and even in the strong sense, ' 1 11

i.e. in the sense of first differences; see. §6), where

II

§ 10. VARIATION OF BOUNDARIES

M.

< b ,-t ±II;;~ 1 K71v

91

(10.1)

00 •

in which the constant b > 0 can be chosen so that it does not depend on v. Then for a harmonic function uv we will have according to Gjunter's Theorem

(10.2) where A is chosen so that I - p + e < A < I while M' 0 and a;a,,p-1. Then there exist constants c 1 and c 2 not depending on f or e such that 1 , r'

+ 1 + t)P r~t~~'

where a 2 = a 0 not depending on e such that in both cases

(11.3) We now estimate the integral / 2 • Applying the generalized Minkowski inequality to the following expression in brackets, we get

f

12 =

a

u

a

-a

-a

o

\S. · · ~ S· .. S[ \ ,a.-p+• l'

(11.4)

.

•,

X (}

a.--/J+•

r

•

1

p

dx,.) di

],,

. d,t 1

•••

l'p

dXn-11 ·

96

II. WEIGHTED IMBEDDING THEOREMS

We again consider two cases. Suppose first a;., p and

Then for O < xn

0

+ 1- p + c) cq

not depending on e (e ,r;;; 1)

If xl df.

0

Substituting this estimate into (11.4) and replacing the letter t by the letter x for the sake of simplicity, we get

{J. ~. s

Is0

ti

(f, h) 11 ~>

~ (2c3

11

I'(n) ~~

r•f x,. II t> + II r•f x, 11 ~>)] h l1-a, I'(n)

r), where p n

l'(H)

+an

the constants A 1 and A 2 not depending on f.

PRooF. We take the function F constructed in the proof of Theorem [6.2] (see (6.7)) and estimate the norms of its derivatives of order r + 1:

(11.39)

If o:

= 1, we have

which implies by virtue of (11.39) that

II p(r+t) xr1 •.• x 1

7n

n

Jin IJ1

< oo,

r, s .s;;; k - 2 and for some

K(m) of order of smoothness k ;;;i. 2,

r

f

~>O

II fls+l) ll(n) < oo x;i ... x~n p(r,a)

rt>

fin

for all s1 + . • . + Sn = s + 1, where >,r C G\K(m)' a = sp + 0:, s is a nonnegative integer and O .s;;; a < p. Then for any e E (0, p - a)

the constants At and B, not depending on

t

Finally,

rx+e

- T 1(M... r 1~n>) f E H 11p(11) (e can be set equal to zero when s = 0 and o:-:/= p - l). In the case p = 00 one should put s = [a] and, in all of the exponents of the distance r, take p = 1.

rt>

PRooF. By virtue of the definition of the cylindroid (see (8.8)) there exist a finite number of canonical neighborhoods (see (8.5)) U(v) and V(v) (v = 1, • • • , v0 ) of heights ~ and a respectively, 0 < ~ < a, corresponding to :mints P E

f

and such that

128

II. WEIGHTED IMBEDDING THEOREMS

•• UM, D(v) c V(•)., v = 1, 2, ... , "o· 1ti) c U Y-1

We denote by x(v)(u) the canonical mapping (see (8.3)) of the domain

v

onto

By virtue of the conditions of the theorem it is at least s + 1 times boundedly differentiable. Then for each v = 1, • · · , v0 the function f(?c(v)(u)) satisfies the y(v).

conditions of Theorem [11.7] (Theorem [11.9] when p = 00) on v if one assumes that the set v~v) corresponds to a domain r~n) in the notation of § 11. Therefore all of the assertions of our theorem are valid to within constants for the case u = r~n), a change in the constants of Theorem [11.7] (Theorem fll.91) being possible only at the exp~nse of the properties of the canonical mappings x(v)(u) and hence not depending on the function f. It follows by virtue of the finiteness of the number of canonical neighborhoods u :r;;.M11 •

t

(v)

This readily follows from theorem [11.14] and the theorem of Nikol'skii on the behavior of the H-classes under differentiable mappings ([4], page 59). Further, for each point P E G we choose a ball Qp of radius rp with center at P is such a way that it is either entirely contained in G or in one of the neighborhoods uM. Let Q~ be the ball of radius rp/3 with center at P. From the covering {Q;} of the compact set G we select a finite subcovering {Q;J, i = I, · • • , i0 , and denote by f(i) a function coinciding with f or one of its e~tensions F 11 on Qp. and equal to zero outside this ball. We take, as usual, an infinitely dif1 ferentiable function 1/J(t) on the real line such that 1/l(t) = 0 for t :r;;. 1 and 1/J(t) = 1 for t ~ 2. We put 1/1,(P) = 1/J(p(P, P1)/rp.), q,(i) = 1/1 1 1/1 2 • • • 1/1,_ 1 (1 - Vlt) and I

F (P)

=

~ f !i> (P)

q,-,.-, (P),

PEE".

i=l

The function F effects the desired extension of f from G onto En. This is proved in the same way as was indicated in the analogous cases in the proofs of Theorems [6.1], [8.3] and others. REMARK. It is not difficult to also verify that the function F satisfies the condition

II. WEIGHTED IMBEDDING THEOREMS

130

Thus when a= 1 Theorem [12.2] contains an assertion on the extendability of the Sobolev classes

wf+ 1 > ([2], page 45).

[12.3] (IMBEDDING AND EXTENSION THEOREM). Suppose 1 :5' p < 00 , r is a submanifold of an (n - 1)-dimensional manifold K of smoothness k ~ 2, :r is compact, r C K, 1/ > 0, e > 0 and there is given on the domain = (K) a function f together with its weak derivatives up to order ;\ inclusively, 1 :5' ;\ :5' k - 1, such that for some (3 ~ 0

KtJ K~nJ

I/'t., X

I

••• X

ll(tl) r,, < 00

An p tr,

for all

ti

K(II)

+11

~

~

Suppose, further, (3 = (3p + a, (3 is a nonnegative integer, 0 :5' a < p and (3 :5' ;\ - I. Then for any 11' E (0, 11) there exists a function F defined on all of En and having the following properties. 1°. F = f on r(n), = K.(r) +17 +11 • ~ 2°. FE H(r) (M' En) ' IIFll(n) :5'M• r = ;\ -(3 - 'a+ e'P,1p , (12.1) p(n) p \'

(12.2)

(r) , IL

= 0, 1, • , • , '" -

1,

r - µ - -p1 > 0. Here Mt satisfies an inequality of type (12.2) but generally\with other constants also not depending on f. If a -fa p - I and 1, it is possible to take e = 0. A sufficient condition for condition (12 .3) is the condition j,tn-1) H

=

Ken-I) H

7.1' t' -

provided

'ii'= ;\ -

1 _ix+1+e>O p •

(12.4)

131

§12. FUNCTION CLASSES ON ARBITRARY MANIFOLDS

CoROLLARY. Suppose G is a finite domain in En bounded by an (n - 1)dimensional manifold K of smoothness k ;;.,, 2 (or by a finite number of such manifolds) and suppose there is given in G (i) a continuous nonnegative function a= o(P), there existing constants r, > 0, a1 > 0 and a2 > 0 for which

and (ii) a function f together with its weak derivatives up to order A. inclusively, 1..;; A..;; k - 1, such that for some (3 > 0

Suppose, further, (3 = (3p + a, 0 ..;; a A.-tf-p- 1 (a+e) and

I

< p, (3~ is a nonnegative integer, (3 ..;; A. -

I, r =

_ix+ pt + e > 0.

Then f takes a value flK on the boundary in the sense of convergence in the mean such that

If a-::/= p - 1 and (3 ExAMPLE.

= A. -

1, it is possible to take e = 0.

Suppose A= 1, 0..;; a< 1 and p = 2. Then flK exists and

satisfies the condition flK E

HJf2-t )' where

p

=1 -

Ol/2 - 1/2

> 0.

PROOF. Let, {v}, v = l, • • • , v0 , be a finite covering of the compact set by canonical neighborhoods of height r, such that yM CK~>= K(K), and let x(v)(u) be the corresponding canonical mappings. Then the functions f(x(v)(u))

r

are defined on

+ v, and

(6)Here r(P) is the distance along the normal from P to the manifold K.

132

II. WEIGHTED IMBEDDING THEOREMS

Hence according to Theorems [ 11.1] and [ 11.2] (n)

for all A'1

+ · · • + "n ' ' = A-

"' 1-'

and

K1

+ · · • + Kn

-< Mv

(12.5)

~

= A - t'"' - 1 ' where

All of the norms of the first sum are finite by virtue of Theorem [ 11.6] and the Remark to it, while all of the norms of the second sum are finite by,!ssumption. Moreover, all of the derivatives of f of orders p = 0, 1, • • • , A - {3 - l are p + power summable on v.

In fact, from (12.5), by successively applying the Remark to Theorem [11.6] and Theorem [ 11.2] (for a = 0) we get

·I-(•) ,..,,

Thus~ function f(x(v)(u)) satisfies all of the conditions of Theorem [11.14] for r= A - {3 - l and can therefore be extended onto all of the domain v lJS a function F(v) satisfying (i) the condition f,

where and

M;

c:,

(v)

-

f

//(r)

•f' VM)

/I (11) (!Y v,

11 . ,

r

= -( +

ct+

E

1 - -p-

=" - ~~

ct+·£

-p-,

(12.8)

has the form of (12.6) under an appropriate choice of the constants c~ and (ii) by virtue of (12.7) and formulas (6.5) the condition

§12. FUNCTION CLASSES ON ARBITRARY MANIFOLDS

I

II

F (Y)

vt;>

(n)

p

~

~

b' l - t t •

..:::.i ..:::.i t-1 t,+ ... +tn=l-/l-1-t

II ii~-1-t f (X(•) (u)) t, iJ t,,

133 (n-1)

i)

u, ... "n

P

{12.9)

Let F(P)

= FM(x 0 and a partioning of the PROOF.

K

domain G11 into subdomains GM according to definition [12.7]. then, putting K(n) (K) =K(n) and K(n-l)(K) =K(n-l) we will have +11

+11

+11

+11

'

§ 12. FUNCTION CLASSES ON ARBITRARY MANIFOLDS

137

From Theorems [12.4] and [12.6] we get

I f l ~I) ¾ A [ l\f /I ~~-I)+ ~/1 f xt)/ ~nir, ct)] I( (n)

K(n-1)

+'I

(n)

K+'J

+11

n

~A/If 1/t-l) + A(c + 1) hll fx,jj~nlr,ct) i=l

I(

/((n)

+11

We next take an arbitrary v and suppose for the sake of definiteness that i11 = n (see [2.7]). We denote by Gf) the orthogonal projection of GM onto the n

hyperplane xn

=0

and for each point P

= (x 1 ,

• • • ,

xn- l) E Gf) we denote by n

'l'(_x 1 , • • • , xn _ 1) and V1(X 1 , • • • , xn- l) the ends of the interval of intersection of the straight line parallel to the Ox n axis and passing through P with GM respectively belonging and not belonging to r 0 such that O ~ VI - ..p ~ d. We now have

If l ~n> = {~ • • •~If (xi, ... , Xn) IP dxi ... dxn}¼ oM

=

oM

fi •••~If (x., • • • , Xn-1,

ip (x.,

. .. ,

Xn-1))

Q(\I)

Xn

+ ~ f xn (Xi, . . , , Xn-1, l) dt IP dX1 ... dxn}¼ 'I' (x,, .. . • Xn-1) • r0

not depending on f such that

The proof is analogous to the proof of Theorem [ 12.8]. § 13. Completeness of weighted spaces

Let r be a finite domain in the hyperplane En- l possibly coincides with En - 1 , and let

= {(xJ; ;xn = 0} that

Further, let ai'I. ;;;i, 0, A= 0, 1, • • · , r, and suppose there is defined on G a function f together with all of its weak derivatives up to order r inclusively such that

l

I\ (n)

iJ'f

'i)x'• ... ax),.n '

for all A1 + · · · + An

•

n

= A, A = 0, 1, • · ·

introduced and studied by Sobolev [2] (see [13.2] below).

140

II. WEIGHTED IMBEDDING THEOREMS

wt>(xn; a 1 , • • • , a,.; G) fk E wcxn; '¾, ••. 'ar)

[13.l]. The normal linear space

is complete.

For if a sequence of functions forms a Cauchy sequence in the sense of the norm (13.1), it can be established in exactly the same way as in the case of the ordinary space Lin> that for every considered system of numbers X1 , • • • , An (X 1 + · • • + An = X = 0, 1, • • • , r) there exists a function C,0;,., 1 , ... ,;,.,n

such that

(13.2) (13.3)

and hence for any subdomain D such that 1.IITI

A-" !I i)

i}hl' lk ).,

Xt • • •

a.\_An 11

-

DC

G

. (n) 1 = Cf>1.,, •.• -~n

I

D

0

.

n

It follows according to a result of S. M. Nikol'skii [4], Lemma 1.2, pages 43-44) that

(13.4) Relations (13.2) and (13.4) imply that c,o0 ,.. 0 E w~>(xn: a 0 , • • • , ar; G), while relation (13.3) implies that limk_. fk = c,o0 ... 0 in the sense of the norm (13.1). 00

The space w(r; a 0 , • • • , a,.; G) in the case when the domain r lies in an m-dimensional hyperplane Em, 1 ~ m < n - 1, and G = rt> (see §11) ~an be considered in exactly the same way. REMARK 1° • In the case p = 2 the space w(x p n•· aO• • • • verted into a Hilbert space by introducing the scalar product

(13.5)

141

§13. COMPLETENESS OF WEIGHTED SPACES

[13.2]. Suppose G is a bounded domain, r ;;t, 1, ar =a= (r - 1) + {3, 0 h. r

+ ~ I Qksks -

Qks-1k,-1

s->.+ I

1

Qk,.k>.

II ~II)+

tdx1,,, dxm)P

i

S=•+ l

I Qksks -

"xm+t ••·.r,i

(bs-1ks-1

l ::"' ·

0-"111+1 ... Xn

Finally, suppose an e > 0 is given. We choose ;\ so large that the sum of the last two terms in the right side of the preceding inequality becomes less than e/2 independently of xm + 1 , • • • , xn; this can be done on the basis of (14.8). After this, by virtue of the uniform continuity of the functions Qk k on G we can choose a 6 > 0 A. A.

such that the first term will also be less than e/2 when

'l:'{' Ix; i< 6. This means that

0

F =f The method of diagonal sums

Qk k

s s

employed in the proof of this theorem can

also be used to estimate the class of the limit theorem. [14.2]. Suppqse given in En a sequence of [unctions F8

eJ/; 1 ,··•,rn)(M),

IIF8 ll~n) ~M. s = 0, 1, 2, .. • , converging in Lin) to a function F. Then FE • .lr1 ,··•,rn)(aM)

HP

, where the constant

a> 0 does not depend on M.

~

145

14. LIMIT THEOREMS

PRooF. Going over, if necessary, to a subsequence, we can assume with loss of generality that

II F in

X;

F s II µ(n)

< -M2s ,

S

= 0, I, ....

W_e again consider expansions of the F 11 in entire functions qvs of degree 2v/ri (i = I, • • · , n): F11 = ··2::; qvs• so that conditions (14.1) are satisfied, and let, as

above, Q11 = ~tqvs• s = 0, I, · • • . Analogously to the way inequality (14.3) (for k 11 = s) was obtained, we have

II F s -Q s // 0

is a constant not depending on the function F 11•

REMARK.

Ifit is assumed in addition to the conditions of Theorem [14.2] that 1

II

1

/ = 1 - -p ~->0, 'i 111+1

0

0

the sequence F 8 will converge to the function F.

146

II. WEIGHTED IMBEDDING THEOREMS

For

1: [Qk-Qk-1],

Fs=iq,s=Qs+ it,F=Qs+ •=O

v-s+I

k-s+I

Therefore, using the same inequalities as in the proof of Theorem [14 .1 ] , we get

IF

0

0

~

5

-F

I

(111)

p

=

I

~qvs- ~ •~"•+I k=s+l ao

O

..

i II q,s II ~m) + ~ I Qk - Q,,_ 1I

~n)

k-s+ 1

v-s+t

~,.IIQk -

l

+ i

00

2 n2

IZ

p

0

("

Qk-Qk-1

)

1

~~

2112 P

II

(m)

p

n

k

n;I "II q,k I :n)

k-s+ 1

1,

11t+1

l

Qk -1

I

~i)

k-s+l

~ 2ncM ~ 2~~ + 2n ~ v-s+l

2k (I- x)

[II Qk -

F II t>

+II Qk-1 - F II ~n>]

k-s+l

Q.ED. From this result we can obtain among other things a somewhat weakened form of Theorem [14.1]. To this end it suffices to extend all of the functions Fk converging in Lt>(G) in "the same way" from a subdomain r onto all of E 11 with preservation of 11 >(E 11 ). class so that the convergence of the sequence of functions F8 is preserved in This can be done by means of the construction employed in the proof of Theorem [6.1].

L1

[14.3]. Suppose K is an (n - !)-dimensional manifold of smoothness k~ 2,

Ki"J =KtJ(K), 0,s;;; a< p - I and there is given on Ki"J a sequence of functions F converging in W~ >(r; a; Ki~~: lims-+ F =F, such that the sequence F, Ix converges

8

1

in

Lin-

I

00

8

>(K): lims-+..,Fs Ix = f. Then F Iv= f on the base

V (see (8.6)) of any

canonical neighborhood V C K(K). PR.ooF. We first note that by virtue of the convergence of the sequences F, and

F9 Ix they are bounded in norm in the corresponding function spaces. Hence there exists a constant M 1

>0

such that

147

§ 14. LIMIT THEOREMS

s=l,2 ... , Suppose now U is a canonical neighborhood of height ~. V C UC K~"!(K), x(u) is the corresponding canonical mapping o~ Uu onto U and Uut =K 0 for which

Therefore the functions F9 satisfy the conditions of Theorem [I 1.14] for consequently be extended onto Uu as functions F!(x(u)) such that •

(r)

F 5(x(u))EHp (M, U 11), r=

r= 0

t 1-,,>,,, CZ

and can

(14.11)

But by Theorem [11.11]

It UFs l ~n-1) < II Fa II+ B
~ ell F+ l in> ~cM2. Uu 9

(14.13)

Uu

As is easily seen, owing to the completeness of the considered spaces, all that has been said about the functions F9 is also valid for F; in particular, it can also be extended onto Uu as a function F*. Moreover, if we employ the indicated construction of an

148

II. WEIGHTED IMBEDDING THEOREMS

extension, we will have (14.14)

(14.15)

in Lin-l)(lJu). Finally, we extend the functions F! from the domain Vu, which is the preimage of the given canonical neighborhood V under the mapping x(u ), onto all of the space according to Theorem [6.1]. It then follows from conditions (14.11)(14.15) that the resulting sequence of functions satisfies all of the conditions of Theorem [14.l]for V=G. Therefore F*lv =f,andifwereturntothemanifold K bymeans u

of the canonical mapping x(u), we will have F le,= f. Q.E.D. CoMMENT. By making use of the Remarks to Theorem [14.2], Theorem [14.3] can be sharpened somewhat by dropping the requirement of convergence of the sequence F8 ~ ; namely, it can be proved that the convergence of the F, Iv is a consequence of the convergence of the sequence F8 in ~ 1 >(r; a; K.t11>).

§ IS. Compact function classes

We now formulate the main compactness theorem. As in § 14, let ct>(xm + 1 , • • • , xn) denote the intersection of a set ct> C En with the hyperplane obtained by fixing the values of xm + 1' • • • , xn. [15.1]. Suppose 1 -n, ... Xn

for which wT > 0 and XA > 0 respectively converges in L1'1')(«1>(x~L, • .. , x~0 >), this convergence being uniform relative to

x~t

1, • • • ,

x~ >. Further, the limits of the 0

normal derivatives are the co"esponding normal derivatives of the limit function and, for any bounded domain G C Em,

where 13; =r;x;\.,j = 1, • • • m, the constants MT and MA being linear combinations of the constants M, M1 , • • • ,Mn. This theorem was proved by the author in [15] and is a generalization, as has already been mentioned in the Introduction, of the corresponding theorem of Nikol'sk.ii. It can easily be carried over to the case of arbitrary manifolds by the usual method of canonical neighborhoods and canonical mappings (see, for example, the same paper [15 ]). Here we merely give some ·applications of it to spaces of functions whose derivatives are summable in the pth power with power weight. Suppose (x;) E En, r is a finite or infinite domain in the hyperplane Em =

{(x;): xm + 1

= · • · =x., = O}, possibly coinciding with

O"'l), G =((xi): (xi, ... , x,,,) tr, !xii< a, i === m + l, ... , "I, G"I = ((x,}; (x1,.,,, Xm) E r,i, IXjl < a- "fj, j = m + l, • • • 1 nj,

a (x~O))= on ((Xi): Xj=X~(/)1~ ) = n J(x,}: Xj = x~01 1. + + xzn• 1. = m + 1, ... , 11 and r = V x2 m+l

e

• • •

[15.2]. Suppose 1 ~ p < 00 , 0 ~a< p - 1 and is a system of functions f that together with their weak derivatives of first order are defined on G and bounded in w(r;

O, a; G), i.e. there exists a constant M > 0 for which

150

11. WEIGHTED IMBEDDING THEOREMS

~ M f E6 II f // w~1 ) (r; o. a; DJ~ ' •

(IS.I)

Then there exists a sequence fk E @; (k = 1, 2, • • • ) such that for any compact set ii> C G the following conditions hold: I. The sequence fk converges in Lin)(il>). 2. For any f'vced 0 ) the sequence fklx-=x{O) converges in Lin-I)(il>(xJ0 >))

xJ

I

I

and, moreover, uniformly relative to the choice of

xJ0 >, j = m + 1, • • • , n.

PRooF. Suppose first the value of j = m + l, • • • , n is fixed. From condition (15.1) it follows, in particular, that LI5.2) There therefore obviously exists an x)°) E [O, a] such that (15.3)

For -suppose the contrary; for all x 1 E [O, a] I/

f //},n-J) > M

cJ{:Kj)

Then

·

U

which contradicts (15.2). From (15.3), using [11.11],(8 ) we get

It is analogously proved that

l[f [lt-1 ' ~

1 ~

1

0(-(). and 2°. the sequence [kl

Xn

=x(O)

n

converges in

LP(n- 1 >((~~0 >))

AE~

such

for any x~0 > and,

moreover, unzformly relative to the choice of x~0 >. The proof of this theorem is essentially analogous to the proof of the preceding theorem if one takes into account that each function can be extended onto all of the space according to Theorem [ I 1.14] . These theorems can be carried over to the case p =00 by the usual method.

II. WEIGHTED IMBEDDING THEOREMS

152

§ 16. Weighted imbedding theorems for domains

with a piecewise smooth boundary In the present section we confine ourselves to the consideration of a finite domain + G C En= {(xi): xn > O}, satisfying the conditions of §9, and we will use, as a rule. all T

of the notation introduced there without additional reterences. Let h > 0, E(~) = + + {(xJ O < xn < h}, G(h) = G n E(~) and Gh = G\ G(h)' We will assume that there +

exists an h > 0 such that Gh is regular relative to its boundary (see Definition [12.7]), which contains all of the manifolds Mµ (see (9.1)). Let G(h) denote the projection of G(h) onto the hyperplane En-t = {(xJ xn = O}. We will assume in addition that the intersection with G(h) of every straight line parallel to the Oxn axis and passing through a point (x., • • • , xn- 1 ) E

G(h)

is an interval. This re-

striction is in no way connected with t~e essence of the question being studied; but it permits one to simplify quite substantially the technical aspect of some of the proofs without changing them in principle. If K is the boundary of G, a point P E K\ (K + n En - 1 ) will be called a proper point (it is not a corner point), and a neighborhood of it in K will be denoted by UK(P). [ 16.1]. Suppose given on G a function f together with its weak derivatives of first order such that

Then for any proper point P EK there exists a neighborhood UK(P) of it such that the boundary value flu K (P) exists in the sense of a mean value and satisfies the condition

This immediately follows from Theorem [ 12.3] for X = 1. If P (K), but the proof is quite tedious. Besides, it follows at once if REMARK.

Lt-•

§ 16. DOMAINS WITH PIECEWISE SMOOTH BOUNDARY

153

one assumes that the part K + of the boundary intersects the hyperplane En-t at an angle greater than 1r/2. In this case an 11 > 0 can be chosen so large that the closures of the manifold pieces K~; 1 >(K,J and K~; 1 >(LA) (see (8.9) and (9.1)) are entirely contained in the interior of G. Then by virtue of the same Theorem [12 .3] for a = 0 or, correspondingly, a theorem of Nikol' skii ([ 4], page 104) the function f is p power summable. After this it suffices to estimate the norm of f on the manifolds KK. and LA, K = 1, • • • , 1< 0 , A= 1, • • • , Ao, with the use of Theorem [12.6]. If K + does not intersect En - 1 at a zero angle, one can extend f in a neighborhood of K + n En-t from G(h) (for sufficiently small h) to a domain whose boundary intersects the hyperplane En-t at an angle greater than 1r/2 and thereby reduce this case to the preceding one. We will not explicitly reproduce this proof, since it suffices for our purposes that the function f k exist on the boundary K of G and locally belong to the corresponding H-class. In those cases when the norm of a function on the boundary plays a role its finiteness will always be included in the assumptions of the theorem. [16.2]. Suppose K is the boundary of the domain G, 1 ,;;;;, p < 00 , 0,;;;;, a< p - l and a function f together with its weak derivatives of first order is defined on G, the norm [:~~n.acl, i = l, • • • , n, being finite. Then there exist constants

ll!x,

c1

>0

and c 2

>0

not depending on f such that n

< C1 //' f IJ = ip, and hence u0 E

St (l) I{). Let us prove that K(u 0 ) = k. From inequalities (17.6) and (17.9) we have

162

Ill. VARIATIONAL METHODS FOR ELLIPTIC EOHATIONS

IK (um)- K (11 I~ I A (um)-- A(u 0) I+ 21 (f, llm -

Uo) I

0)

I

1

I

I

~ jA7 (ttm)-A 7 (u 0 ) j [A7 (tt,n)

+ A7

1

(u 0 )]

1

1

·+2 IJ fJli,IJum -u JJi.~ A :?•(am -Uo)[AT(llm) + A7 (Uo)l 0

+ 211 f /Ii, Illm -

_!.. Uo llw•t•> [ 11 2 (u,,,)

Uo llwi1> ~ T I Um -

..!..

-1- A 2

(uo)

211,11 ] +~ •

This implies that

k

= lim K (um) = K (u 0). 111-00

It is not difficult to see that the function effecting a minimum of the functional K(u) in the class tt~> is unique. For suppose u E sr~> and K(u) = k. Applying (17.13), we get

0 . We have thus proved the following theorem. [ 17 .1 ] . Suppose 'Y is a subset of the boundary of a domain G con faining one of the ith ha/jboundaries of G, 1/J is a function defined on 'Y, the class Sf (i) = J!..i>(-y) of admissible functions is not emptv and ,p ,p ll

-= inf K (u). U(:~(i)

'I'

Then k is a finite number, i.e. the functional K(u) is bounded from below on and there exists a unique function u0 E

sr

In addition, any

sequence um Est~> for which limm ➔ 00 K(um) = k (such sequences are said to be minimizing) satisfies the condition lim um = u 0 in W~l)_(3) Suppose now G * is a domain such that G* C G, and suppose given on G a (3)We note that variational problems in the case when the boundary conditions are given not on all of the boundary of the domain but only on part of it have previously been considered for domains with a sufficiently smooth boundary (see, for example, (101).

§17. GENERAL PLAN OF THE VARIATIONAL METHOD

163

continuously differentiable function that is equal to zero on G\G*. Such functions will be called variations and denoted by the symbols ou, ou 0 , etc. The function u 0 defined in Theorem [ 17 .1] satisfies the variational equation (17.14) For consider the function A(t) >.. (t)

=

A (uo

= K(u 0 +

+ to:io) -

tou 0 ). We have

2 (f,

Uo

+ touo)

= t2 A (ou0) + 2 (A (u 0 , Oif0 ) -(f, ou0 )]t + A (u0 ) - 2(f, u0 ), and since u 0

+ tou 0

ER~) for any real _t, the function i\(t) achieves a minimum

at t = 0. This together with Fermat's theorem implies (17.14). Thus we have completely studied the variational problem; namely, we have proved that in every nonempty class .ft{i) it has a unique solution which minimizes

"'

the integral in question and satisfies the variational equation (17 .14). We proceed now to a study of the corresponding differential problem. To this end we impose additional conditions on the coefficients A 1;, B1 and C. We will assume that the functions A;; and B; together with all of their first order derivatives are bounded in absolute value on G. In this case the Euler equation for K(u) can be written in the form n

n

~ ~a:.(A,;::.)+qu +f=O,

i-l=jl

J

(17.15)

'

where (17 .16) For the sake of brevity we use the notation (17.17) Further, we will assume that a boundary function ({) is given on all of the (accessible in the coordinate directions) boundary of G, and we will let simply St"' denote the set of all functions u E W ~1 ) such that u Ir = ({). We note that all of the above assertions concerning a solution of the variational problem in the classes Jr~) obviously remain true for the class .ff'"' (assuming that it is nonempty). We now put

164

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATIONS

Suppose

vEW'~2>(G71) or

U E

for any

1l

>0

and u Ir= 0,

(17.18)

W~ 1 )(G), VE W~1 )(G) and (17.19)

Then the following formula of Green's formula type holds: A(u, v)

=-

(u, L(v)).

{17.20)

To prove this formula under the boundary conditions (17.18) we first note that if a function v E W~2 >(G,.) for any '17 > 0, then it can for fixed i and k (i, k = 1, • • • , n) be modified on a set of measure zero so that for almost all x E pr; G the function ov/ox,.. will be absolutely continuous in X; on each segment contained in G and lying on a straight line perpendicular to the hyperplane x 1 = 0 and passing through one of the indicated points x E pri G. In order to see this it suffices to partition G into a countable number of cubes Q, with edges parallel to the coordinate axes, such that Q C G and hence v E W~2 )(Q). One should then modify the function v on a set of measure zero in each cube so that the function ov/oxk is absolutely continuous in X; on almost all segments perpendicular to the hyperplane X; = 0 and contained in the cube Q in question. As a result, one obtains the required modification of v on a set of measure zero in G. We proceed with the proof of (17.20). According to (17.3) we have

(17 .21) Consider a term with the first sum in the right side with fixed indices ; and k. We make the above indicated modification of v on a set of measure zero and we modify the function u so as to make it absolutely continuous in X; on almost all

165

§17. GENERAL PLAN OF THE VARIATIONAL METHOD

segments perpendicular to the hyperplane x; = 0 and contained in G. It can be proved (see the author's paper [16]) that the domain G can be represented as the union of not more than a countable number of measurable sets Em for which there exist constants hm > 0 such that the intersection of each Em with a straight line perpendicular to the hyperplane X; = 0 is either the empty ~t or an interval of length greater than hm the ends of which lie on the boundary of G. Thus G = U mEm. We fix one of the sets Em. Let (a(x), b(x)) be the interval of intersection of Em with the straight line perpendicular to the hyperplane X; = 0 and passing through a point x E pr; Em (here pr; Em denotes the orthogonal projection of Em onto the hyperplane X; = 0). For any h and / such that h + I < hm/2 let E'!,,h,l) · denote the set of all segments of the form [a(x) + h, b(x) - /] for all possible x E pr; Em. It is obvious that for any sequences hv and Iµ., v, µ. = 1, 2, • • • , such that lim,._ hv = 00

limµ._ 1µ. 00

=0

we have

We now consider the term A;k(ou/ox;) (ov/oxk) of A(u, v) on E~·l) and, using Fubini's theorem, integrate it by parts with respect to X; over [a(x) + h, b(x) -1], where x ranges over a set of total measure in pr; Ff-::/> = prkEm:

Let us estimate the first integral of the right side.(5 ) Suppose IA;k I ,s;;; A on G, where A is a constant. We have, for example,

(4)The sign

,. over a symbol means that this symbol should be omitted.

(S)The method presented below for estimating this integral in the case of domains with a piecewise smooth boundary was first encountered by the author for the Dirichlet integral in the works of Niko!' skir.

166

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATIONS

Let

Jz(h)=S

S[(:u)

••• pr 1 cm

/2

(h)

=

xk

s. ..s

2

]

x 1 -a(x)+h

dx1••·dx, ... dXn,

[u2 )x, -a(x)+h dX1•• .dx,... dXn•

pr I Em

The function I 1 (h) is summable over [O, hm] since

=

s. ..s

s (:~r

a(x)+hm

dx1 ... dx, ... dxn
p,

since p* - p

= p 2/(n/2

- p)

> 0.

170

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATIONS

PRooF. The assertion wh ELp•(G11 ) is proved in [2] (Theorem 2, pages 43-45), the condition of uniform convergence to zero of the norm II qr h llip,to 11 > following from the estimates appearing there. The same theorem implies the existence of the integral on the right side of equality (17.27) and even the membership of the function defined by it in a space Lp(G,,). Let us show that this integral actually is the derivative awh/axi. Suppose w = w(P) is a continuously differentiable function on G that is identically equal to zero outside a closed domain lying in G11 • Then

S···J[ a!, (~G) )-2) + 1¥(~) )-~ ::] dxi • •:• dxn = 0. 6>

(17.29)

r 0. Applying Green's theorem, we get

=Therefore

J•••f(

w ~:;

0 11

J·, .... Jc::~';;) o/(i)wt1s-O

for

,-o.

+ '1'11 :~) dvp = f ... Jf (Q) {f ... J[w a~, ( ~ (~) )_ o

2)

r · · · > 77 1 > 0.

> n/2, we get that o2 -.J!h/'dxf

ELP/G11 1c) for some 111c

175

>

But then o2u0/oxf E LP/G11 1c)' which implies by the same

lemma again that o2'11h/oxf E L_(G11 1c) and hence that o2u0/oxf E L.,.(G11 1c)

(i = 1, • • • , n), the number 111c > 0 being arbitrary by virtue of the arbitrariness of 111 > 0. It can easily be seen by the same method that in the case q = O there are naturally even fewer restrictions that need be imposed on f to obtain a classical solution. If the coefficients q and f are analytic it can be shown in a similar way that u0 has continuous derivatives of arbitrarily high order and hence, as is well known (see [39, 40]), is analytic.

By using the more exact results of Calderon and Zygmund [45] on integrals of potential type, one can lower the order of the derivatives of the coefficients of the equation required in the conditions of Theorem [17 .2]. We note that the methods developed in the present section can be carried over in a natural way to the case of elliptic equations of higher order, for example, to a polyharmonic equation. In conclusion we make some remarks concerning the nonemptiness of the class stop. This problem has already been studied to a sufficient extent in the literature, and necessary and sufficient conditions in the presence of a known smoothness of the boundary have been obtained for it. A necessary condition on the boundary function consists in the following (see Sobolev [2]). If a point P of the boundary r of the domain G is proper( 1 2 ) and if u E W~l)' there exists a neighborhood O of P in r such that ul0 EHJ'tj_ 1 >(O).

A sufficient condition for the nonemptiness of Rop consists in the following (see Nikol'skii [4]). If the boundary r of the domain G consists of a finite number of manifolds of smoothness k ;;;i,, 2 and I{) E ~>1>(r), e > 0, then the class R op is nonempty.

wt:

We note that in a number of cases it is possible, by using another terminology, to formulate the indicated necessary conditions in such a way that they are at the same time also sufficient (see, for example, [46, 47, 11, 48]).

(12)A point P of the boundary

r

of a domain G is said to be proper if there exists a

nl'ighborhood of this point in I" which is an (n - 1)-dimensional manifold of smoothness 7c ,;;;, 2.

176

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATIONS

§ 18. General plan for the solution of degenerate equations

by the variational method We introduce some notation which will be employed from now on in the present chapter. G is a domain of the space En, K is its boundary and a= a(x 1 , • • • , xn) is a thrice continuously differentiable positive function on G. For any pair of functions u 1 and u 2 defined on G together with their weak derivatives of first order we put (18.1) and, in particular, (18.2)

D,. (u,u) .... Da. (u). For a twice weakly differentiable function u on G we set

La. (u)

n '\.., iJ (

=

~ ax,

a ...

iJu )

ax, ·

(18.3)

i-1

Clearly, L,.(u) =0

(18.4)

is the Euler equation for the functional Da(u). The remainder of the present chapter is devoted to a study of the equations of form (18.3). The number a will be called the degeneracy exponent. As noted in the Introduction, when n = 2, for example, some equations of the type f3 iJ2z

y iJxz

+

iJ2z iJy2

= 0,

reduce to an equation of form (18.3) when a= y. We study necessary and sufficient conditions for the solution of the degenerate equations (18.4) by the variational method. The sufficient conditions are connected with questions of nonemptiness of the corresponding classes of admissible functions. Therefore the proofs given below are based on the methods of extending functions developed in the preceding sections for the case of sufficiently smooth boundaries. In connection with this we will always assume in the sequel a known smoothness of the boundary of the domain, which permits us to speak of the boundary values of a function not only in the sense of convergence almost everywhere, as was done in § 17, but also in the sense of convergence in the mean. It should be noted, however, that

§ 18. SOLUTION OF DEGENERATE EQUATIONS

177

questions analogous to the questions considered in § 17 can be discussed in the case of degenerate equations for a wider class of domains than is done below if one confines oneself to convergence almost everywhere to the boundary values. This will not be done in order to provide a uniform presentation of all of the questions which will be touched upon. For a function u defined on a domain G and a function ,p defined on the boundary f' we will write u Ir = ,p if for any point P E f' there exists a neighborhood Ur(P) of it such that

= 0

and c"

>0

such

c' 11 u lloa ~ II U llw 0 and such that Lc,_(u 0 ) = 0 almost evetywhere on G. Following the plan of the proof of Sobolev for the case of Laplace's equation ([2], page 94), we will prove a preliminary lemma. Let

and K(r) = i/J(r/h)/chn where i/1 is defined in § 17 (see [17 .2]) and c = snf"fi~i/l(t)dt. Then J • • • fr 0}. Its boundary is the hyperplane xn = 0, which we denote by En- l. Suppose a = xn, i.e. consider the equation 0t OU O Au+--= , xn dxn

which is the Euler equation for the integral

(19.4)

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATiONS

188

(19.5) +

[19.7]. If u E D()l.(En), then the boundary value u IEn-t exists and for any

finite domain

r

C

En -

t

This immediately follows from Theorem [11.7] and the Remark to [11.6]. The first peculiarity of the case being considered here consists in the fact that we cannot assert in general that the finiteness of the integral D()l.(u) implies the + membership of the boundary function in L~n-l)_ Therefore the set D()l.(En) in this case cannot be converted into a complete normed linear space. ~ + + Consider, however, the set D()l.(En) of all functions u EDOl.(En) such that

~ + [19.8]. The following definition of a norm for the functions u E D()l.(En): (19.6)

~

+

converts the set D()l.(En) into a complete normed linear space. As in the case of [19.2], the axioms of a norm and the linearity of the space

~ + D()l.(En)

+ layer E:

are verified without difficulty. To prove completeness we note that for any = {(xi): 0 < xn < a}

1-211~11 ax l

ll(n-1) + _1_ II ll l'2(n) :::::::; ,/"711 r a u ✓J ~a +n

Ea

Hn-1

a:

(n)

+ n ~(,,,. E:

(19.7) ex)

(see [11.2], (11.9) and (11.35); we have somewhat sharpened the constants here). It follows from this inequality that a Cauchy sequence of functions in the sense of the + norm (19.6) will be a Cauchy sequence in w(xn; 0, Cl.; Ean) (see § 13) and hence

189

§ 19. EQUATIONS DEGENERATE ON THE WHOLE BOUNDARY

+

[19.9]. Suppose given on the hyperplane En-I a function ,p E H'f) (1 - a)/2. Then the class of D~, A= 1, 2, ..• , A0 ,

··1J(Pjl.)(n·f) f cpM"' t , :! iv P. ' PP.> 2' Then for any a

>0

"= 1, 2, ... ,1< 0, I-'-

=

l, 2, • • • '1-'-o•

the class of D a,'() admissible junctions is nonempty.

This is a rephrasing of Theorem [9.1].

ll'P)li

[20.3]. Suppose the class of Da ,y,~ admissible junctions is nonempty and + 1>< oo . Then there exists a unique junction u0 EDa,iE n) effecting a

11 -

/(

-I-

minimum of Da(u) in the class Da,iEn): Da(u 0 )

= d.

Proof. Suppose uk is a minimizing sequence, i.e. limk__,. Da(uk) we know (see (18.1)), 00

\\(n) II cl ("Hm-uk) iJxi

2(¥

0

and since lluk+m

n

,

•>

= d.

Then, as

~ O for k-+oo,m=0,1,2, ... ,i=l,2, ... ,n, (20.3) ~

-ukw:-1>=0, according to [16.2]

/(

(20.4) By virtue of (20.3) and (20.4) the sequence of functions uk is a Cauchy sequence in the complete space W~l)(xn; 0, a; G) (see (16.4)) and hence has a limit u 0 = limk__,. uk jn this space. Finally, Theorem [14.3] implies that u0 IK = If); but then from [18.2] we have D(u 0 ) = d. The uniqueness is proved in the same way as in Theorem [19.5]. 00

[20.4]. Suppose the class of Da,'() admissible functions is nonempty. Then there exists a unique junction u0 EDa,iG) which is an analytic junction of its arguments and such that

192

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATIONS

PR.ooF. The proof of Theorem [19.6] is completely preserved for the present case with the exception of Lemma [19.6.1], which is changed to account for the presence of corner points and the absence of a, degeneracy on the whole boundary but will be completely valid if we prove the boundedness of the integral

for h ➔ 0, where v ED,/G), v~ = 0, G3 h = {P: PEG, p(P, K) Suppose 1/

>0

and >.,

(n)

K•

> 3h}.

(n) -

z'l=an >.-1 u K+'l(fa),

wr'l=Gn u K+ 11 (KK), K-1

"'"'=an

I',

(n)

u K+•

.. -1

(M ) ,. , (20.5)

Then for each point P belonging to one of the sets in the unions of (20.5) there is uniquely defined the distance r(P) along a normal to a corresponding part of the boundary K. In the sequel, if a point P turns out to belong to several sets in the unions of (20.5), so that several distances r(P) can be considered for it, we will always assume that they all satisfy the appropriate conditions considered below. We choose an 1/o > 0 so that when h < 11 0 /3 and PE G\G11 there exists a constant a> 0 for which r(P) E;; ap(P, K) (see (8.13)). Then

G "-..__ 0 311 C

cv;·aah

U ..'t'aahUMaull•

We further require that 110 be so small that two different cylindroids in the unions of (20.5) can intersect only if the closures of their bases intersect. We will always assume that h < ri 0 /3. Then

1{•

•11~

h ~- . ·. ~xii v~ dX1- • • dxn

}2-~ h1{•~ • • • ~Xn v

2

}...!.2

dX1- .. dx,,,

~ :ia/1

fi'-.0311

+h1 { \• ..

'11

2

•

11

~Xn

}...!.+ h1 {~-• .. ~x! • }.!... v 2 dx 1 • •• dxn 2 •

v 2dx1 . .. dXn ~

..'t'aah o# :iuh The boundedness of the last integral in the right side of this inequality for h ➔ 0 follows from a lemma of Sobolev ([2], page 94) as well as from [12.5] for a= 0, p = 2 in view of the fact that its integrand is not degenerate on the corresponding parts Mµ. of the boundary K (xn > 0 > > 0). The boundedness of the second integral for h ➔ 0 follows directly from Proposition [11.10].

xi

193

§20. EQUATIONS DEGENERATE ON PART OF THE BOUNDARY

It remains to estimate an integral of the form K

= ], 2, ... , K.0,

where K~n) = ~:Jah(K") n G. Suppose P0 EK" n En-l and suppose the angle formed by the inward normal Np to K" at PO with the ith coordinate axis is 0

less than a right angle. Then the manifold K" admits an explicit representation of the form x,=x,(x 1 , · · · ,x,_ 1 ,x;+i,··· ,xn) inaneighborhoodof P 0 . Therefore by virtue of the smoothness of the manifold K there exists a neighborhood Up

0

of

P0 and a number bp0 such that the intersection of a straight line passing through a point of Up

0

and parallel to the Ox; axis with K~n) is entirely contained in an

interval of length not greater than hbp Let

KK(r,)

= K" n

{(x,): 0

0

< xn < 77}

= K~n) n {(x,): 0 < xn < 77}. 77 > 0, a b > 0 and a finite

and K~"j

From what has been said it follows that there exist an number of pieces K""' v = 1, • • • , v0 , of the manifold K" satisfying the following conditions: K ( ) C U"° K . if K(n) = K(n) (K ) then K(n) C U"° K(n) K fl v=l ""' "" +3ah KIJ ' "" v=l "" • and for each piece K"" there exists a number i = i(v) such that the intersection with

K~:>

of a straight line passing through a point

axis is entirely contained in an interval

PEK~:>

and parallel to the Ox,

z,.

of length not exceeding bh. To simplify the presentation we will assume essentially (without loss of generality, as is easily seen) that the indicated intersection is an interval IP' If i(v) < n, one end of such an interval trivially belongs to K. If i(v) = n, we take those intervals Ip which do not have an end lying in K and appropriately extend them to K. These newly obtained intervals are again denoted by Ip, It can be assumed without loss of generality that the lengths of the newly obtained Ip also do not exceed bh. Let fl(") = U (n>1P CK~~). Then PEK""

The boundedness of the considered integral on the set

K~n>~i:,>

for h ➔ 0

does not require special considerations, since here xn > 11 > 0, so that we do not have a degeneracy and the necessary estimate follows from the cited lemma of Sobolev

194

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATIONS

([2], page 94). Thus the whole question reduces to an investigation of the integrals

J, {

S. .. Sx:

v 2 dx 1 •

••

dxn)

!.

,1.(v)

We denote by rM the projection of

.1,(v)

onto the hyperplane xi= 0, i

= i(v), by

l(xl' • • • ,xi-l• xi+l• • • • ,xn) the interval of intersection of .1,(v) with the straight line passing through the point (x 1 , • • • , xi-t • 0, xi+ 1 , • • • , xn) E rM and parallel to the Ox; axis and, finally, by

the lower and upper ends respectively of this interval. Clearly, w 1 ..;;; w 2 ..;;; w 1 + bh. Since one of the ends of the interval l(x 1' • • • , X;_ 1, xi+ 1 , • • • , xn) belongs to K, the relation v = 0 holds on at least one of them. We will assume for the sake of definiteness that V(Xi, ••• , Xi-1, Wi, Xt-1-1, ••• , Xn)

Suppose first i(v) < n, for example i(v)

= 1.

= 0.

Then

(20.6)

§20. EQUA1'1ONS DEGENERATE ON PART OF THE BOUNDARY

195

Suppose now i(v) = n, G" = A. Using (20.11), we have

av II("> + h ll dx1

:i(xn, Cl)~

ha ~ 2

+he .

.1 (1) 1

Substituting (20.15) and (20.16) into (20.14), we get (for e < 1)

(20.16)

200

III. VARIATIONAL METHODS FOR ELLIPTIC EQUATIONS

where c > 0 is a constant. It therefore follows from (20.13) that

Combining this with (20.12) gives us

which implies, on account of the arbitrariness of

But then inequality (19:7) implies that v

=0

E

> 0,

almost everywhere on

+ E:.

Q. E. D.

§21. The solution of elliptic equations degenerate on the boundary of a domain with degeneracy exponent o: ;;;,, 1

+ We again consider the domain G situated in the upper halfspace E" = {(xJ xn > 0} and which was studied in the preceding section. Also, we note that + all of the results obtained below are valid, as in § 20, for the infinite layer = {(x;): 0