COMPACTNESS OF ORLICZ-SOBOLEV SPACE IMBEDDINGS FOR UNBOUNDED DOMAINS by Ian Graham C a h i l l B.Sc, University of V i c t o r i a , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1975 In p r e s e n t i n g t h i s thesis in p a r t i a l f u l f i l m e n t o f the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree that the L i b r a r y s h a l l make i t freely available for r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s thesis f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department by h i s r e p r e s e n t a t i v e s . of this thesis for It financial i s u n d e r s t o o d that c o p y i n g or Department publication g a i n s h a l l not be a l l o w e d w i t h o u t my written permission. of The U n i v e r s i t y of B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5 or ABSTRACT In t h i s t h e s i s we a r e concerned w i t h the compactness of imbeddings, f o r unbounded and T r u d i n g e r domains, o f O r l i c z - S o b o l e v spaces. Donaldson [6] have g i v e n a g e n e r a l i z a t i o n o f the Sobolev imbedding theorem and the R e l l i c h - K o n d r a c h o v compactness theorem f o r Sobolev spaces t o the case o f O r l i c z - S o b o l e v spaces on bounded domains. Clark, Adams, and F o u r n i e r [ 1 ] , [ 2 ] , [ 3 ] , and [5] have g i v e n e x t e n s i o n s of the Sobolev compactness r e s u l t s t o c e r t a i n unbounded domains. t h i s t h e s i s we c o n s i d e r s i m i l a r e x t e n s i o n s f o r O r l i c z - S o b o l e v spaces. In iii TABLE OF CONTENTS Chapter Page INTRODUCTION 1. 1 1. Notations 1 2. O r l i c z and Orlicz-Sobolev Spaces 4 3. Imbedding Theory 11 IMBEDDINGS OF w \ , (G) A 1. Necessary Conditions for the Compactness of 2. W E (G) -»• L_ (G) m A where 19 A-< B 19 A S u f f i c i e n t Condition for Compactness of lA^CG) - L (G) 28 Applications to Certain Bounded Domains 34~ A 3. 2. IMBEDDINGS OF W™ L (G) ( 1. Necessary Conditions f o r Compactness of ^V 2. 37 A G ) * A L ( G ) 3 7 S u f f i c i e n t Analytic Conditions for Compactness of W ™ L ( G ) -»• L (G) 4 6 A 3. S u f f i c i e n t Geometric Conditions for Compactness of W ™ L ( G ) + L (G) A REFERENCES A 50 66 Acknowledgeme nt s I would like to give thanks to Dr., J . Fournier for his guidance and encouragement during the writing of this thesis. I would also like to thank Dr. R. Adams for reading the thesis and Maryse E l l i s for typing i t . 1 INTRODUCTION 1. Notations T h i s s e c t i o n p r o v i d e s a l i s t o f the n o t a t i o n s used. we f o l l o w Adams [ 4 ] . later. Usually U n f a m i l i a r concepts mentioned w i l l be d e f i n e d The r e a d e r may use t h i s s e c t i o n as a r e f e r e n c e and b e g i n r e a d i n g s e c t i o n 2. (a) Multi-indices We w i l l denote by a the m u l t i - i n d e x ( a ^ , I ot I = E . , 3=1 ja. . I f D. = — i 3x. 3 J a ) and l e t f o r 1 < j < n, l e t — — a i n D = D„ *•• D , which i s a d i f f e r e n t i a l o p e r a t o r o f 1 n a order (b) a |a|. Spaces and Norms (i) R w i l l be r e a l E u c l i d e a n space w i t h the norm I f x = ( x , " * , x ) e R , then Ixl = ( £ x ) ^ . 1 n - i 3 3=1 2 n 1 1 (ii) L.(G) 2 w i l l be the O r l i c z space d e f i n e d u s i n g the N- f u n c t i o n A and the open s e t G c R norm || • || . . n ,|| u | | 1 . 1 A j G n and w i t h t h e I f ueL (G) t h e n = inf{k>0| (Luxemburg norm). fA ( ^ ) d x G < 1} (iii) E A (G) w i l l b the closure i n L^(G) e of the set of bounded functions with compact (iv) support. w V (G) w i l l be the Orlicz-Sobolev space obtained using the N-function A and open set G c R . I t has n norm || • || where i f u £ W™ L (G) , then m, A, G A ' ' (v) W E (G) m A set |a|<_m w i l l be the closure i n W L^(G) of the m of bounded functions w i l l compact OO rn (vi) (vii) support. m W L.(G) w i l l be the closure of C (G) i n W L (G) Let u be a p o s i t i v e , continuous, increasing function on the i n t e r v a l (0, °°) , and l e t y(t)^-0 as t ->• 0; then the space C (G) w i l l consist of functions u e C(G) for which the norm ||u, y g , || ^ 1 ^ 0 ( 0 ) 1 1 , ^ ^ 1 ^ x^y is finite. m\ — (viii) If 0 < X £ 1 we define C ' (G) to be the subspace of C (G) consisting of those functions (J) for which, m for 0 <^ | ot| <_ m, D cf) a s a t i s f i e s i n G a Holder condition of exponent X; that i s , there exists a constant K such that D%(x) - D%(.y)|£ K | x-y :,yeG. C ' (G) has the norm || <J>; C m A I t>; C (G) || + max m 0<|a|<m m , X ( G ) || sup l A ( x ) - D ^ ( y ) | x,yeG x^y J x-y| Imbeddings If S and T are Banach spaces, then S -»• T w i l l mean that S continuously imbedded i n T. If Sc T, then the only imbedding that we discuss i s the natural i n j e c t i o n S T. This imbedding i s said to be compact i f the unit b a l l i n S maps onto a precompact set i n T. Measure If E i s a Lebesgue measurable its set i n R , we w i l l denote n Lebesgue measure by u ( E ) . Domains By a "domain G" we w i l l mean an open set G i n R . n G r For a domain G we define G r = '{x£ G x > r } — . If G i s a domain then 8G w i l l denote the boundary of G. Cone Condition (Adams [4]) A domain G i s said to s a t i s f y the cone condition i f there exists a f i n i t e cone C (the i n t e r s e c t i o n of an open b a l l in R n centered at the o r i g i n with a set of the form 4 {Ax|x£B, A>0} where B i s an open b a l l not containing the origin) with the property that each x eG a f i n i t e cone C (i) x i s the vertex of contained i n G and congruent to C. By a strongly L i p s h i t z domain G we w i l l a domain G where each x £ 9 G has a understand neighbourhood such that there exists a rectangular coordinate system, with o r i g i n at x, and a L i p s h i t z continuous function F such that G n u 2. x = { q , ? n ) e U x U < n F ( V — , > . O r l i c z Spaces and Orlicz-Sobolev Spaces In this section we l i s t the d e f i n i t i o n s and a few of the prop- e r t i e s of O r l i c z spaces and Orlicz-Sobolev spaces. More d e t a i l s can be found i n Krasnosel'skii and R u t i c k i i [ 9 ] , Adams [ 4 ] , or Fournier [ 7 ] . An N-function i s a continuous, convex, even function A : R"*" lim t-K) satisfying = t 0 and lim t-*» = . . . . (1) » . t The O r l i c z class K. ( G ) consists of the functions u from G to R"*" such A that A( u(x)|)dx < °°. The O r l i c z space L ( G ) i s the linear h u l l JG of K ( G ) with the Luxembure norm. It can be shown that E ( G ) c K ( G ) c L ( G ) A A A A E (G) i s the maximal vector subspace of the class K.(G), (Adams, [4, p. 236]). There are some ways of ordering N-functions. We say A^ i f there exists X Q > x and k > 0 such that for a l l x ^> X Q , A^(x) _< A(kx) A (X ) If A -< A and A -< A , then we say A ~ A . If lim . . = for 2 1 1 1 every A > 0 1 1 then we say ^ " ^ A A 2' y^co Z N o t ; i - c e X 00 i^ > A x that, for bounded domains G, A -< A „ implies that L (G) c L (G) and A . . - ^ A „ implies that 1 Z A ^ A^ 1 Z 1 L (G) A 2 L C ^ (G) . A X Every N-function A can be represented : A(x) i n the form M = a(t) dt J 0 where a(t) i s a nondecreasing, right-continuous function. This i s proved, from the d e f i n i t i o n of an N-function, by Krasnosel'skii and R u t i c k i i [6]. Let a(t) = sup { s|a(s) <_ t} . A(x) We then define a(t) dt = 0 A and A are related by y < A(x) + A(y) (Young's inequality) By applying Young's inequality to u(x) -N U J L HA,G and , v(x) — —^—^ v , and 1 I I A ,G 6 integrating over G we get juvdx < 2 ||; u || || v ||~ A (Holder's inequality) for u e L (G) A and v e L ( G ) A We say that an N-function A s a t i s f i e s the condition i f there exists x^ > 0 and k > 0 such that A(2x) _< kA(x) say that i t s a t i s f i e s the global f o r x >_ x^. We condition i f there exists k _> 0 such that A(2x) < kA(x) for a l l x > 0. Example 0.1 Let ,. 2 " a(x) 2n when 2 < x < 2~ _ n _ 1 n = x when x > 1 Then A(x) = a(t) dt s a t i s f i e s the condition but not the global A condition . These r e s t r i c t i o n s w i l l be needed i n some of the theorems of this thesis. In general, the condition A( | u (x) | ) dx -»- 0 , n G 1 1 as n -> 00 7 does not imply that u | | ^ 0 as n->-°°. I f A and G have f u r t h e r p r o p e r t i e s , however, then the i m p l i c a t i o n does hold. Lemma 0.1 Suppose t h a t A s a t i s f i e s y(G) i s f i n i t e ; condition. o r suppose merely the c o n d i t i o n and t h a t t h a t A s a t i s f i e s the g l o b a l Then the c o n d i t i o n ' A(|u (x) |) dx ->- 0 n G as n -> implies u Proof: n 0 || , A,U Suppose t h a t as A(|u (x) Ja the g l o b a l n -»• » 0 as n -> . If A 00 satisfies n c o n d i t i o n i t i s c l e a r that | A(2|u (x)|) dx + 0 n as n since t A(2|u(x)|)dx <_ (constant) G I f u(G) < follows: 0 0 and A ( 2 x ) <_ M A ( x ) A ( | u ( x ) | ) dx G for a l l x >_ x Q proceed as Suppose A(|u(x) |) dx <_ £ Choose x^ so t h a t A(x ) 1 = £ 1/2 Let ; x = {y £ G | |u(y)| < x G 2 = {y £ G | x G 3 = {y £ G | |u(y)| > x ± } < |u(y)| < x ± Q Q } } Then [ A(2|u(x)|) dx J G ' < [ " J C +[ 1 J G + [ 2 S A(2|u(x)|)dx Now [ [ J But, as a A(2|u(x)|)dx < A(2x ) y(G ) < A ( 2 A(2|u(x)|) dx < A(2x ) y(G ) A(2|u(x)|) dx < M| £ -»- 0, x.^ Q J a X ; L ) y(G) 2 A ( | u ( x ) | ) dx 0; hence A(2x^) ->• 0. < M£ Also f 1/2 e > ' G -i Thus U(G ) 2 A(|u(x)|)dx >_ A ( X;L ) y(G ) = 2 £ u(G ) 2 2 /o <_ e . Combining these estimates we have | A(2|u(x) [ ) dx <_ A ( 2 x 1 ) u ( G ) + A(2x ) e ^ + Me , 1 2 Q G which i s independent of u and tends to 0 with e. Thus, i f A s a t i s f i e s the global A and A s a t i s f i e s the A 2 2 condition, or i f | G | < condition, then | A(|u (x) | ) dx -> 0 n as n + as n-^ G => | A(2|u (x) |) dx ^ 0 n By induction oil k, j A(2 |u (x) |) dx + 0 k n In p a r t i c u l a r , for fixed as n -> k and a l l s u f f i c i e n t l y large n, | A(2 |u (x)|) dx £ 1 k n For such n, II u II . „ < •—" n " A,G — „k Since k can be taken as large as we l i k e this completes the proof. Some other relations between the mean and the norm w i l l be useful. Remark 0.1 We note that with no r e s t r i c t i o n s on A and G the condition ti II n A,G as 0 11 n -»- implies that | A(|u (x) | ) dx •*• 0 as n n -»• G Indeed, suppose II u H; AjG < e < i . Then | A(|u(x)/e|) dx < 1. G But.A i s convex and A(0) = 0. A(|u(x)|) < e Therefore A(i^lL). and A( |u(x) | ) dx <_ e G Also, i f the sequence '{ A(|u (x)|) dx } J a so i s the sequence {||u || ^} A n i s bounded away from 0 then 11 Remark 0.2 If {j A(|u (x)|) dx} n i s bounded then so i s the sequence To see this suppose that A(|u(x) | ) dx _< K ; without loss of generality assume K > !• Then A(|u(x)|/K) < (1/K) j A(|u(x)|) dx < 1 Hence U 3. HA,G 1 K Imbedding Theory The material of this thesis has i t s origins i n the imbedding theory of Sobolev spaces. The key result i n this theory i s a c o l l e c t i o n of imbedding properties c a l l e d the Sobolev Imbedding Theorem. The basic case of this theorem says that i f G i s a domain s a t i s f y i n g the cone condition then 12 (a) i f p < n, W 1 , P ( G ) -> L * ( G ) , P where p* = np/(n-p) (b) i f p > n, (c) ifp > n W 1 , P W 1 , P oo and ( G ) -»• C 1 , X ( G ) -»• L ( G ) n c ( G ) G i s a l s o s t r o n g l y L i p s h i t z and bounded (G) f o r 0 < X < 1 - n/p . T h i s r e s u l t i s due l a r g e l y to S.L. Sobolev b u t i n c l u d e s to C.B. Morrey and E. G a g l i a r d o . then r e f i n e m e n t s due See Adams [4] f o r a d e t a i l e d proof. Compactness o f some o f these imbeddings i s shown i n the R e l l i c h - K o n d r a c h o v compactness theorem. case. We w i l l now s t a t e the b a s i c Suppose G i s a bounded domain s a t i s f y i n g the cone c o n d i t i o n . Then (a) i f p < n, the imbedding W ' ( G ) -* L ( G ) i s compact f o r P q a l l q < p* (b) i f p > n and G i s s t r o n g l y L i p s h i t z , W This 1 , P (G) C '^(G) 1 t h e imbedding i s compact f o r a l l X < 1 - n/p. theorem i s a l s o proved i n Adams [4]. The above r e s u l t s were extended t o O r l i c z - S o b o l e v Donaldson and T r u d i n g e r [6]. spaces by F o r a bounded domain G s a t i s f y i n g the cone c o n d i t i o n we have the f o l l o w i n g r e s u l t s : 13 (a) If oo _1 dt A (t) (n+l)/n X x t then W L (G) + L *(G) A A where we define the Sobolev conjugate A* by setting A-V) f t A^Ct) 0 T (n+l)/n dx (In the proof i t i s assumed without loss of generality that A (t) dt (n+l)/n 0 t < 00 This i s j u s t i f i e d since an equivalent N-function s a t i s f y i n g this can be chosen and, since G i s bounded, the space w V (G) w i l l not change.) Also, i f B • « A* then the imbedding compact. (b) I f ^ A-^t) dt (n+l)/n 1 t < 00 then W L (G) A -* L^G) fl C(G) W^L^G) -* L (G) i s g 14 (c) If G i s strongly L i p s h i t z , and i f r f J .-1. 00 A (t) dt (n+l)/n X 1 t < 0 0 , then W^CG) -»• C (G) , where .00 ' -n t Moreover i f t > 0 V is a positive, such that 1 W L (G) A lim ^rpy continuous, = 0 increasing function then -»• C (G) v i s compact. Clark, Adams, and Fournier [1], [2], [3], and [5] have generalized the Rellich-Kondrachov compactness theorem to certain unbounded domains. I t i s the purpose of this thesis to extend some of the compactness r e s u l t s of Donaldson and Trudinger [6] to c e r t a i n unbounded domains, using many of the techniques of Clark, Adams, and Fournier. We w i l l now give a condition which i s s u f f i c i e n t f o r the compactness of the imbedding space such that S c L^(CJ) . S -»• L ^ ( G ) where S i s a Banach The condition i s also necessary but this w i l l not be proved since i t w i l l not be needed later. of 15 Lemma 0.2 (Fournier [7]) Let G be a domain i n R . n S -+ L (G) . Let S be a Banach space such that For u £L (G) and r > 0 l e t A A P u(x) r = u(x) if 0 if { |x| > r I I "J } I xI < r • Let B = { u e S | || u || < 1 } We then have that B i s precompact i n L^(G) i f the following two conditions hold: (a) (I-P )B TO (b) || P i s precompact i n L (G) for a l l r . A || (|| P || 0 as r + °° i s the norm of P^ as an operator from S to L (G)) A Proof: I f (a) and (b) hold then I : S compact operators I - P L (G) i s a norm l i m i t of A and i s therefore compact. • 16 We w i l l now give a technique for proving compactness of imbeddings which can be used when the N-function A s a t i s f i e s the global A 2 - condition. Lemma 0.3 Let A be an N-function satisfying the global condition. Suppose that for every £ > 0 there exists r > 0 such that A(|u(x) | )dx < £ || u G for every u £ S and suppose that condition (a) of lemma 0.2 i s s a t i s f i e d . Then ,S Proof: L^CG) i s compact. If we pick a sequence {e^} can f i n d a sequence | such that £ n -* 0 as n -»• °°, then we such that A(|u(x)|)dx < £ || u || , n G r s n that i s f A(|P u(x)|) dx G ^ < u £ n Hs Since we are concerned with the norm of P we w i l l consider only u r such that || u || = 1. We then get A(|P u(x) | ) dx < £ G n r n 17 By examining the proof of Lemma 0.1 we see that because A s a t i s f i e s the global A„ condition there exists a sequence {£"*"} with 2 n n -*• 0 as n -> 0 0 such that P r u II A,G 11 n < e n 1 which means (b) of Lemma 0.2 i s s a t i s f i e d . We have assumed (a) to be s a t i s f i e d ; so, by Lemma 0.2, S -*• ( ) i s compact. L G A • We f i n i s h this section with a r e s u l t on i n t e r p o l a t i o n . Lemma 0.4 (Adams [4, p. 241]) Let G have f i n i t e volume and suppose that A and B are N- functions such that B "^"^ A. If the sequence ^ j } i u s bounded i n L.(G) and convergent i n measure on G then i t i s convergent i n A L (G). B Theorem 0.1 Let y(G) < °°. that A-< B -4"^ C. S Let A, B, and C be N-functions such Let S be a Banach space such that •* L (G) e and Suppose S ->• L (G) and S -»• L.(G) compact. S -> L (G) . i s compact. Then S -»• L _(G) i s T 18 Proof: Since y(G) < we have 00 S. + L (G) -» S •* L (G) . C L (G) B so that B To see that this i s compact notice that i f {u^} i s a bounded sequence i n S then i t i s bounded i n L (G) Since S + L (G). Since S + ^ (G) A i s compact we w i l l have a subsequence {u^ } convergent i n measure. i i l (Here we have convergence i n norm => convergence i n mean => convergence i n measure). compact. Therefore by Lemma 0.4 we have that S L (G) i s p—i 19 CHAPTER 1 IMBEDDING THEOREMS FOR W L^(G) m In this chapter we w i l l present results due to Fournier [7] on the compactness of the imbeddings between spaces W L (G) and m A various O r l i c z spaces over a domain G which i s not necessarily bounded. The results of Donaldson and Trudinger [6] are extended to unbounded domains i n much the same way as the results f o r Sobolev spaces are extended by Adams and Fournier [3]. 1. Necessary Conditions f o r the Compactness of W E (G) m A - L (G) B where A •< B . For our f i r s t condition we need some preliminary d e f i n i t i o n s . Consider a tesselation of R by n-cubes of side h. H w i l l always denote a cube i n the tesselation under discussion. N(H), c a l l e d the n neighbourhood of H, w i l l be the cube of side 3h concentric with H and having i t s faces p a r a l l e l to those of H. F(H), c a l l e d the fringe of H, w i l l be the s h e l l N(H) - H. D e f i n i t i o n 1.1 (Adams and Fournier [3]) Suppose G i s a domain. y(H Let X > 0. n G) > X y(F(H) n G) . H w i l l be c a l l e d A-fat i f 20 I f H i s n o t X - f a t i t w i l l be c a l l e d Theorem 1.1 • Suppose t h a t A < Then f o r any t e s s e l a t i o n o f R n X-thin. B and W E.(G) -»• L ^ G ) A B i s compact. m and any X > 0 t h e r e a r e o n l y finitely many X - f a t cubes. Proof: A -< B for F i x a t e s s e l a t i o n of R , u s i n g n Q R e c a l l that means t h a t t h e r e a r e numbers k and x^ SO t h a t A ( x ) <^ B(kx) x > XQ. all d >_ x cubes o f s i d e h. Without l o s s of g e n e r a l i t y assume t h a t k _> 1. so t h a t B(kd) h Choose > 1 . n oo Now c o n s i d e r by N(k") w i t h cp = 1 a X - f a t cube H and a f u n c t i o n (j) £ CQ supported on H on h and m but not on H.) B(kc)y(HflG) = and f D^cp | <_ M Choose c so t h a t B(kd)h n Then c > d (because y(H flG) < h") . B(k|u(x)|)dx f o r a l l a _< m. = (constant) L e t u = c$. > 1 . Then :> B ( k c ) y ( H H G) > 1 JG so that I ..8 > 1 (see K r a s n o s e l ' s k i i and R u t i c k i i li» I I . . 0 i But i f \ a |. < m, [9, p. 73, Lemma 9.1]) and I• then A(|D u(x)|/M) < A(c)y(N(H) flG) < B ( k c ) y (H fl G) • (1+ j) a G = (constant)(1+ 1 y) . (M depends 21 Thus, by the above observations, II D u|| <_ (constant) A ^ LT for a l l d < m. Now suppose that the t e s s e l a t i o n contains an i n f i n i t e sequence, {H^}, of d i s t i n c t A-fat cubes. Passing to a subsequence we can arrange that the sets N(H ) are d i s i o i n t . Let {u } be the function n n corresponding to {H }. Then {u } i s bounded i n W V ( G ) and n n A J u Since the IL n "B,G > l/k — > 0 f o r a l l n. have d i s j o i n t supports the sequence {u^} i s not precompact i n L_(G). • Corollary: The conclusions of Theorem 2 and Theorem 4 i n Adams and Fournier [3] are v a l i d . W E (G) - L (G) m A B compactly with A-< B, be the surface then |i(G) < °°. {xeG | |x| = r}. surface area of S . r (a) Given e , 6 r >R That i s , i f Also, f o r each r > 0, l e t S Let A r denote the (n-1) - dimensional Then > 0 there exists an R so that for :. u(G ) < 6 u{x e G | r-e <_ | x | < r r r} . 22 (b) If i s p o s i t i v e and ultimately nonincreasing as r tends to i n f i n i t y then, for each e > 0, ' ' A . /A r+£ r 3 as r tends to i n f i n i t y . -kr rapidly than e Proof: tends to zero Note that y(G )tends to zero more r for any r . The proof i s exactly the same as that i n Adam's and Fournier [3] and i s based on the fact that there are only f i n i t e l y many X-fat cubes. Remark 1.1: The proof of Theorem 1.1 also shows that i f W E (G) -»- Lg(G) m A for some B with for which A <•< B, then there i s no sequence {K^} AG) "*" 0 as j y(HnG) = 00 of X-fat cubes For, given a X-fat cube H choose c . so that B(c) Let u = c<j> 1 . i s the function defined i n the proof of Theorem 1.2). then have || u || ^ ^ > 1 B(|u(x)| dx f because > B(c) u(HflG) = 1 . G Now f i x e > 0. For |a| <_ m [ A( | D u(x) | ,/ £M) dx < A(c/e)u(N(H) n G) a JG < (1 + 1/X) • y(H QG) • A(c/e) = (1+1/X) i°/^ A B ( c ) , because y(HflG) = , ^ ^ u a — B ( c ) We Since A « as A(c/e) B, B(c) -V 00 c But as -»• 0 , c ^ - o o u(Hfl.G) Thus, f o r u(HflG) s u f f i c i e n t l y small, A(|D u(x)|/eM) dx a < 1 for a l l a < m Then D u/ M|| a £ AjG < 1 for a l l a < m Thus u . < "m,A,G eM 0 and U HB,G > 1 Since £ i s a r b i t r a r y this contradicts the continuity c tft^CG) + L (G) B 24 Thus i f W^CG) Lg(G) for some B>"V-A, then either (a) y(G) = 0 0 (b) y(G) < 0 0 and y(G^ - G and u(G^) ^) does not tend to zero or tends to zero very r a p i d l y . We w i l l now present a theorem which provides a necessary condition f o r compactness i n the case that the N-function A does not s a t i s f y the A^condition. Theorem 1.2: Suppose there exists a function u i n W L (G) such m A that A( |u(x) J yet D u a I) dx = 00 , G i s bounded on every bounded subset of G for a l l a with |a| _< m. Then the imbedding -y L (G) W E (G) m A A i s not compact. Proof: From the hypothesis i t i s clear that we can form a d i s j o i n t sequence of s h e l l s { Q > n n=l n with Q* n. such that = G * + 2 < b a" K b (a n-1 n n G n <na ) n l A(|u(x)|dx > 1 for every ri. Q flG n 00 function <f> so that n 1 i f x£ Q n *n 0 i f Ixl < b -1 or n Choose a C ( x ) { 1 1 1 | x |' > a n 1 +1 , and so that, f o r a constant K independent of n, we have D <b r n 00 < — K for a l l n and a l l a with |a| <_ m. JL,A,G Hence ty e w V ( G ) n A r - K n Then n ' I l - ' lm,A,G u for a l l n. | A(|^ (x)|) dx > Let TJJ = u • <{> . However 1 , so that < M*JIA.G> i s bounded below. n^n' Since , and ty^, have d i s j o i n t supports f o r ||* - V " I A , G N > M* ll , n A > 1 G Now we have ity } bounded i n W E (G) but not precompact i n m A thus L ( ) G A W E(G) + m L (G) A • i s not compact. An example of a domain G and an N-function A where the s i t u a t i o n of Theorem 1.2 arises has been given by Fournier [7]. Example 1.1: Let 7 G = {(x,y) eR _ 2 | x > 0, 0 < y < e } x . 2 If we l e t A(x) = e -1, the s i t u a t i o n i s that of Theorem 1.2 and the imbedding i s not compact. To see t h i s , l e t u : (x,y) x. Note that ueL^CG), although A( | u(x) | ) dx = °° . G J t r°° i i . , u(x)L j A( —^ dx G = J J 2 2 -3x /4 -x , . [e - e ] dx 0 r J < 1 so that - I' II A,G 1 U 2 • S i m i l a r l y , D°u £ ( G ) for a l l a . L A Clearly, D u i s bounded on every bounded subset of G, for a l l a . a 27 The l a s t necessary condition we give i s a simple geometric one. Theorem 1.3: If W E (G) -> L (G) m A A i s compact then G has only f i n i t e l y many components, Proof: Suppose G has a sequence of components Define u n -1 1 A ( *~ v ) U(Q ) = n on Q n and u n =0 elsewhere on G. This means that A(|u (x)|)dx G n = • • = y(Q ) A(A ( 7 ^ - y ) ) n 1 n = l We now have a sequence, { u } . n n=1 bounded i n W™E (G) yet not precompact i n ( G ) . L A A • 28 2. A S u f f i c i e n t Condition for the Compactness of W^Y^CG) L^(G) In this section we give a s u f f i c i e n t condition f o r the compactness of imbeddings W L (G) - m A L (G) , A with the r e s t r i c t i o n that A must s a t i s f y the condition. F i r s t we need a lemma. Lemma (Adams 1.1: [4, p. 248, Lemma f s a t i s f y a L i p s c h i t z condition on R. 8 £ W loc ( G ) ' a n = we mean a C^ map Gx{0}, and where set i n R (a) x n Let g(x) = f ( | u ( x ) | ) . and l e t Then f'(|u(x)|) • sgn u(x) • D u(x) . (Adams and Fournier 1.2: Theorem 1 . 3 : U E W | ^ ( G ) d Djg(x) Definition Let 8.31]) [3, p. 526]). By a flow on G < J > : U -> G, where U i s an open set i n G x R"*" containing <J>(x, 0) = x for a l l x i n G. Suppose A s a t i s f i e s the condition. Let G be an open for which: there i s a sequence ( H i , . ^ N N=l of open subsets of G such that for a l l N the imbeddings w l L are (b) A V ( A<V L compact. there i s a flow <J> : U -»• G such that i f G„ = G - H then: N N 29 (i) G x [ 0 , l ] c U f o r each N N (ii) the f u n c t i o n <j) :x (j)(x, t) i s one to one f o r a l l t <\ (iii) (c) <()(x,t) | <_ M f o r a l l ( x , t ) i n U; the f u n c t i o n s d ( t ) = sup X £ G | det (J) (x) | ^ 1 N satisfy: (i) d ( l ) N -> 0 as N -»• I d ( t ) d t -»- 0 (ii) as 0 0 and N -* 0 0 ; 0 (d) y(G) < » Then the imbedding W^"L (G) A Proof: E S i n c e y(G) < A ( G ) = L A ( G 0 0 L A ( and A s a t i s f i e s G ) i s the compact. condition, • ) Thus W L (G) 1 A and = W E (G) 1 A i t i s enough to prove t h a t W E (G) 1 A i s compact. We want -> L ( G ) A We proceed as i n Adams and F o u r n i e r to e s t i m a t e N>(x)| dx . G For , N each x i n G„ we have N T [3], L e t ii;eW?"'"'"(G) loc 30 •g- iK<f> (x)) o 9 t dt 11 Now | |i|)(ct) (x))| dx < i 1 d < N d (l) | |^(<}) (x))| |detcf>|(x)| dx N 1 k ( y ) | dy ( 1 ) d ( l ) | |^(y)| dy N Also ^ G N J ijj((j) (x)) dt dx t ° dx V (<|> (x))| | ^ t * ( x ) | dt t N J < | dt 0 < M < M { | Vi|> (* (x)) | Mdx t Vij; (4> (x)> | | det <J>J.(x) | dx d ( t ) dt t N d ( t ) dt} {J N |V^(y)|dy } Letting 6 we have N = max(d (l) , N M | d ( t ) dt ) N . 31 i|Kx)|dx<6 " •[ JxpCx) | + [ Vip(x) | > dx <_ 6 N" " 1,1,G Y and 6„ + 0 N as N ->• || u II •]_ Now suppose uEC^(G) i s a bounded function, and Let A G 1/2 lp(x) = A(u(x)) (extend A to be even on (- °°, °°)) . Then f f A(|u(x)|) dx J a. N I < 6 „ { [ ( A ( | (x)|) + W |D A(u(x))|) dx } a lct|=l JG Since U "A,G K 1 • we have A(|u(x)|)dx Fix a with |oc[ =1. < 1 Lemma 1.1 applies, with f = A, because u i s bounded and A s a t i s f i e s the L i p s c h i t z condition i n every f i n i t e i n t e r v a l ot ot [9, p. 5]; therefore D A(u) = A'(u)«D u . complementary to A. ^ Let A be the function Then by Holder's inequality D A(u(x))|dx = I | A' (u( )y| |D u(x)| dx a a x < I | A - ( U ) | | X G ,ot . | | D U | | A > G 32 < II A'(u) lfy >G , since B A " IIA.G i To estimate II»III.A.G i || A'(u) |K [9, p. 73, lemma 9.1]. 1 / 2 we use K r a s n o s e l s k i i and R u t i c k i i 1 Recall that their norm i s at most twice the Luxemburg norm; thus " « A,G i 1 By their lemma 9.1, A(A'(u(x))dx < A'(u) | | ^ 1 1 Therefore G < Combining these estimates, we have that, i f u i s r e a l valued, c \ and U II 1,A,G 1 1/2 > then A(|u(x)|) dx < (n + 1) 6 N , which tends to zero as N tends to i n f i n i t y . Let 33 u V X ) = { 0 if xeG if X E H N } N Then | A d u ^ x ) ! ) dx < ( n + 1) 6 N • G It follows, by the argument of Lemma 0 . 1 , that, with the above assumptions on u, " A . G " - £ ' N where . e -> 0 N as XT N 0 0 Thus f o r a r b i t r a r y real-valued u i n IK"A,G i 2 we have S « « II 1.A.G • F i n a l l y , since c (G) fl{set of bounded functions) 1 ( 1 > i s dense i n W*E (G) , A we have ( 1 ) f o r a l l u i n W^E^G) . Now by ( 1 ) , an argument similar to Lemma 0 . 2 , and the assumption W \ V ( * A<V L i s compact f o r a l l N, we have that 1 W E (G) A i s compact. -»• L (G) A • Remark 1.2 I t i s c l e a r that i f wh^CG) i s compact m A - L (G) A compact. Remark 1.3 The h y p o t h e s i s o f theorem 1.3 i s t h e flow h y p o t h e s i s i n Adams and F o u r n i e r ing A then W L (G) is •+ L ( G ) [3], N o t i c e t h a t the examples of domains s a t i s f y - the f l o w h y p o t h e s i s g i v e n by Adams and F o u r n i e r apply Orlicz space c a s e . W L (G) m A t o the F o r example - L (G) A i s compact f o r t h e horn of t h e i r example 2, p r o v i d e d A s a t i s f i e s the &2 c o n d i t i o n . Remark 1.4 The domain d e s c r i b e d i n Example 1 . 1 . s a t i s f i e s the h y p o t h e s i s o f Theorem 1.3. The noncompactness of the imbedding does not c o n t r a d i c t the theorem, because the N - f u n c t i o n c o n s i d e r e d i n Example 1.1 does not s a t i s f y 3. the condition. A p p l i c a t i o n s t o Bounded Domains The f l o w c o n d i t i o n can a l s o be used t o show t h e compactness of W^G) + L (G) A f o r some bounded domains not s a t i s f y i n g the cone c o n d i t i o n . 35 Example 1.2: (Given i n Adams [4, p. 172, Ex. 6.49]) Let G = {(x,y) e R : 0 < x < 2, 0 < y < f(x) } 2 where f e C (0,1) 1 i s p o s i t i v e , nondecreasing, has bounded derivative f , and s a t i s f i e s lim f ( x ) = x+0 0 . + Let U = {(x,y,t) e R 3 : (x,y) EG, -x < t < 2-x } and define the flow <f> : U -* G by cKx.y.t) = { x+t, ^f^-y} Thus j . i|/ , det <J) (x,y) t f(x+t) = -j^r- . If G* = {(x,y) eG | x > 1/N } then d N N ( t ) S U P = 1 0<x<l/N ffx+tY *lx+t; satisfies d (t) N and < 1 for 0 < t < 1 36 lim d ( t ) N = 0 i f t > 0 . N+co Hence a l s o lim [ d (t) dt = by dominated convergence. and 0 S i n c e G* i s bounded and has the cone property, s i n c e the boundedness o f -TT- i s assured by t h a t of f , we have, ot by Theorem 1.6, the compactness o f W L (G) m A + whenever A s a t i s f i e s L (G) A the A n condition. 37 CHAPTER 2 IMBEDDINGS OF W™L (G) A In this chapter we w i l l give some necessary and s u f f i c i e n t conditions f o r the compactness of the imbeddings W^L^G) The results are analogous L^CG) . to those given for Sobolev spaces by Clark [5] and Adams [1], [2], and [3]. 1. Necessary Conditions f o r the compactness of W^L^CG) -*• We begin with a simple geometric condition on a domain G given by Clark [5]. We say that a domain G i s quasibounded i f d i s t ( x , 3G) -»- 0 Clearly G i s not quasibounded as - |x| + °° in ' G . i f and only i f i t contains an i n f i n i t e sequence of d i s j o i n t congruent b a l l s {B j • Note that an n=l unbounded domain s a t i s f y i n g the cone condition i s not quasibounded. oo Let <j) £ CQ (G) with support i n B^ for one of these b a l l s and with r A(<|>(x))dx > 0 . B, Consider the sequence in B . n {(j) } of translates of <j> with <p having support n n {d> } i s bounded i n W^L. (G) but bounded away from zero i n n UA L.(G) . Since the supports are d i s j o i n t this means {(J) } i i s not pre- compact i n L (G) which means A 38 oV w G) i s not compact. for r A< > L e Quasiboundedness i s , t h e r e f o r e , a n e c e s s a r y condition compactness. We w i l l now g i v e a n e c e s s a r y a n a l y t i c c o n d i t i o n on G f o r the compactness o f ^ A L ( G > * V where A s a t i s f i e s the G ) condition. T h i s i s analogous t o C ™ ' P defined i n Adams [1]. F i r s t we w i l l need some d e f i n i t i o n s . I f H i s an n - d i m e n s i o n a l n cube o f s i d e h i n R and E a c l o s e d subset o f H we denote by C (H,E) 0 0 the c l a s s o f a l l i n f i n i t e l y identically X For d i f f e r e n t i a b l e f u n c t i o n s on H which v a n i s h i n a neighbourhood o f E. A,H ( U A( I h H l.<|a|<m " ) We d e f i n e the f u n c t i o n a l | a | | D ° u ( x ) | ) dx each 6 > 0 , I» C . (H,E) A,6 (2n 1 / 2 u) inf m n fi over a l l u i n 00 C (H,E) with A(|u(x)|)dx H = 6h n 39 D e f i n i t i o n 2.1: if A domain G w i l l be s a i d to s a t i s f y the C"' c o n d i t i o n f o r every e > 0 and 5 > 0 t h e r e e x i s t s h ^ > 0 such t h a t f o r every h, 0 < h < hp t h e r e e x i s t s r > 0 such t h a t f o r every n-cube H of s i d e h meeting G^ we have C™ r. (H,H-G ) A,o ' r > v h/e We w i l l a l s o need a lemma due to F o u r n i e r [7] which i s analogous t o lemma 1 i n Adams [ 1 ] . Lemma 2.1 Suppose t h a t DCH has p o s i t i v e measure and t h a t u e C (H) Then r A ( | u ( x ) | ) dx H y(H) I A(2n {] A(2|u(x) |) dx + C JD Proof: J Let x £ H and u(x) u(y) + y £D . 1 / 2 h | g r a d u(x) | ) dx } H Then x-y = v • grad u ds where v and = (x-y)/|x-y s measures d i s t a n c e a l o n g t h e l i n e from y to x. L e t ( r , a ) denote p o l a r c o o r d i n a t e s c e n t r e d a t y so t h a t the boundary o f H i s g i v e n by r By = f (a), a e Z the c o n v e x i t y o f A we have . AO|u(x)|) = f A(|u(y) + i i I x-y v • grad u d s ) \ I~yI x < Y A(2|u(y)|) + \ A(2|J v • grad u d s |) Integrating the inequality over H we have A(|u(x)|)dx 1/2 < H J r ( A(2|u(y)|)dx H r|x-y| + A(2| v • grad u ds | ) dx H JO < 1/2 h + < to f A(2|u(y).|) n ,t(0) £<0> J 0 - dr r rr {A(2 J 0 | grad u | ds ) } Using Jensen's i n t e g r a l inequality we get A(|u(x)|)dx r da• E r f ( a ) Jo < n-i. r" "dr h A(2|u(y)|) n { A(r2 j grad uj ) ds } < h A(2|u(y)|) n f(a) da + r n-2 , r dr { ( A ( n J since r < n 1 / 2 h 0 1/2 h 2 | g r a d u|)ds} F i r s t note that r < f ( a ) f(a) i n the l a s t inside i n t e g r a l and then that 1/2 < n h . We then have A(|u(x)|)dx < h A(2|u(y)|) n H f + = , 1/2, n-1 (n_iO Z n-1 , ( a ) A(n 1/2 h|grad u|) dr JO h A(2|u(y)|) n . 1/2. .n-1 + rf d a (n h) n-1 A(n f 1/2 h | g r a d uj) dx i n-1 x-y H i f we now integrate over D and use the fact that C (n){y(D)} 1/n 1 D | x-y n-1 (see Hellwig [ 8 , p. 5 7 ] , we get A(|u(x)|)dx y(D) < h 11 H , 1/2. .n-1 + is_Jy C .(n){y(D)} n-1 Since . {y(D)} this becomes 1/n < (h ) n 1 / n = h A(2|u(y)|)dy D 1/n A(2n h|grad u|) dx 1/2 H 42 y(D) A(|u(x)|)dx H < h n A(2|u(y)|)dy n-1 + h ( n C^n)) n _ 7 n-1 j A(2n 1/2 h|grad u|)dx Now, l e t t i n g n-1 2 C (n) n-1 we have the lemma. • Theorem 2.1 I f W^L.CG) -»• L.(G) OA A m s a t i s f i e s the condition C. i s compact then the domain G A ,m . Suppose that G does not s a t i s f y the condition C~ Proof: there exists k = — < e 0 < h^ < 1 1.H.Jj=l 0 0 and 6 > 0 Then such that for every h. with J 0 there exists h < h^ such that there exists a sequence of mutually d i s j o i n t cubes of side h meeting G such that J •^m C .(H.,H.-G) A,6 2 3 A < kh . By the d e f i n i t i o n of capacity for each cube H^. there exists a function 00 u. £ C (H.,H.-G) J 3 3 such that r A(|u.(x)|)dx H. 3 2 and = 6h n 43 A< I h>l l.<Ja|<m | D 2n a 6h H. 1 / 2 u(x)|) dx < kh n Certainly [ A(2n 1 / 2 h | g r a d u ( x ) | ) dx < k6 H. Also Now l e t D. be the s e t J {x|A(2|u (x)|) < .6/2} . By lemma 2.1 we have 6 h - n 1 <f ( j ) u jf.n oh D + 6C.kh 1 + c i that C.kh 2n+l y(o.) 6 2n+l y(o.) which i m p l i e s k h I 1 + 1 > h n + 1 44 or y(D..) < C h n + 1 2 Choose h small enough that C h < 2 1/3 We then have y(D..) < 1/3 y (H ) Choose functions a subset of w. £ C_ (H.) 3 0 3 such that w.(x) having measure no less than (2/3)y(H_.) that sup j max |a[<m sup x£H_ |D w.(x) | = a k* 3 = u. w. £ C.(H.n G) C C.(G) 3 3 0 3 0 and A(2|v (x)|) > 6/2 on S.fl(H. - D.) 3 3 3 < 00 2 for a constant k* that i s independent of j . v. , a set of measure not less than ( l / 3 ) h n 1 3 Then , and such on 45 Then A(|v (x)|) > C > 0 on S. fl (H. - D.) 3 3 for some constant 3 C^, so that A( v.(x) ) dx _> H. J h C By remark 0.1 (b). J We have A( | D V (x) j • I Dw_. (x) |) dx < C**(h) 3 for a| , |@| < m and by remark 0.2 i t follows that W L (G). (J A Since the functions j k " A,G so that the sequence ^ j ^ v i s a bounded sequence i n {v.} j — 3 is n o t have d i s j o i n t supports we have n precompact i n 46 L (G) and A W L^G) Q •+ L^G) cannot be compact. 2. • S u f f i c i e n t Analytic Conditions VA ( G ) ^ L A ( G for Compactness of ) We give an a n a l y t i c condition on G which i s , again, analogous to C ' n m P i n Adams [1]. Definitions 2.2 If H i s an n-dimensional cube of side h i n R define the functional A(|D u(x)|) dx a max l<Ja|<m J H me* and V £ > 0 define the (C ' ) capacity of a closed set E i n H by inf (C™' ) (H,E) oo u e C (H,H-E) H A domain G w i l l be said to s a t i s f y the (C™) a constant condition i f there exist C such that for every e > 0 there exists h <_ 1 and r > 6 such that, f o r every cube H of side h meeting G^, we have (C ' )*(H,H-G ) E A r > Ch ITi Theorem 2.3 " I f a domain G s a t i s f i e s the (C^) condition, then the imbedding W 0 A< > L * G L A ( G ) i s compact. Proof: By lemma 0.2 we need only show that || P Given || 0 as r 00 . e > 0, choose r and h as i n d e f i n i t i o n 2.2. Let u e W^L^CG) such that || u || "m,A < — 11 1 . TO ^ By the d e f i n i t i o n of the (C.) condition A Tesselate G with cubes H.. r l 3 we then have A( Ju&LL ) G dx e ( 00 < Z 1=1 1 • H. A( 1?J£1 ) £ 1 00 < 1/C . Z max A(|D u(x)I) dx i = l l<|:a|<m ' H . —' -•— 1 a 00 < C, max l<|a|:<m < C — 2 n dx r Z A(|D u(x)|) dx i = l -"H. because a || u || . < ' m,A — 1 1 48 Hence P r llA,G ± U £ C 3 » f o r a l l such u, and P < — r 2.3 Definitions e C, : I f H i s an n - d i m e n s i o n a l cube of s i d e h i n R , d e f i n e the f u n c t i o n a l - r t ^ m ^ H ( U ) l<|a|<m = A ( h ' ' | D u ( x ) | ) dx a a H e > 0 , and, g i v e n d e f i n e the ,m,£ C capacity of a closed s e t E i n H by C™' (H,E) £ inf u £ C°°(H,H-E) A domain w i l l be s a i d t o s a t i s f y the T A . H ( U ) dx A( u(x) /£) H c o n d i t i o n i f , f o r every £ > 0, there e x i s t s h < 1 and r ^> 0 such t h a t , f o r every n-cube o f s i d e h meeting G , r C ' A m £ (H,H-G ) r > h 49 Remarks 2.1 (a) I t i s easy to see that, f o r a domain G, so that C™ i s a s u f f i c i e n t condition f o r compactness. (b) Like Adams' C ' n m , ' P and A C. ~ have been constructed so A,0 that the capacity of a closed set E c A i s invariant under magnification of H and E by the same factor. (C™ ) ,£ i s not invariant. (c) ^m The term h on the right side of the C„ M condition can be replaced by <j)(h) for any function (J) such that <J)(h) 0 as h -»- 0. (d) I f there exists a constant D such that f o r every a > 0, b > 0, and c > 0 A(cb) A(b) then i t can be shown that, for a domain G, <=> a <=> ( y . m m C C A A In this case of W 0 A L ( G ) A i s necessary and s u f f i c i e n t f o r compactness - L A< G ) In p a r t i c u l a r , when t A(t) P J L . , = the conditions C™, C™ to the condition C ' m 3. , and (C™) are a l l equivalent given by Adams [ 1 ] . P S u f f i c i e n t Geometric Conditions f o r the Compactness of i W L (G) •* L ( G ) . m A A Probably the simplest s u f f i c i e n t geometric condition on G i s f i n i t e volume. By examining the proof of compactness given by Donaldson and Trudinger 0 0 , then i s compact. W Q L ( G ) -> L ( G ) A [6] one can see that i f ]i ( G ) < A Next we give two geometric conditions due to Clark and Adams which w i l l be shown l a t e r to imply another that i s s u f f i c i e n t for compactness. D e f i n i t i o n 2.4 (Clark [5]) A domain G w i l l be said to s a t i s f y Clark's Condition 1 i f to each R ^> 0 there corresponds positive numbers d(R) and <5(R) s a t i s f y i n g (a) d(R) + 6(R) 0>> M | I (c) for each x e G 6(R) < — | x-y | M < -> 0 < - as f with o r R a |x| 00 n R > R d(R) and Gfl{z||z-y| there exists y such that < 6(R)} = < | > . 51 D e f i n i t i o n 2.5 (Adams [2]) A domain G w i l l be said to s a t i s f y Adams condition 2 i f there exists RQ > 0 such that to each R > R^ there correspond numbers d ( R ) , 6 ( R ) > 0 (a) d ( R ) + 6 ( R ) -> 0 (b) d(R)/S(R) < M < (c) for each x e G as 0° R ->• °° such that , for a l l R > R fl such that |x| >R , >_ R Q the b a l l B ( ) () x 3 d R i s disconnected into two open components C^, and C^ by an n-1 dimensional manifold forming part of the boundary of G i n such a way c -fl j^\W x d (R) B D e f i n i t i o n 2.6 F i = that each of the two open sets 1>2 contains a b a l l of radius 6 (R) A domain G w i l l be said to s a t i s f y the condition i f there exists a 6 > 0 such that, to every e > 0, there correspond numbers h and R, with 0 < h <_ e and R > 0, such that for every n-cube H of side h meeting G , K U , n-l (H,G) (3) where u n-l (H,G) i s the maximum, taken over a l l projections P onto the n-1 dimensional face of H, of the n-1 measure of P(H-G). 52 Theorem I f a domain G s a t i s f i e s 2.4 W SV * G ) the c o n d i t i o n F A' then V > G i s compact f o r m >_ 1. Proof: We Suppose we a r e g i v e n £ > 0. Then c o n s i d e r £^ = e/2n. then have numbers h and R depending on £^ as d e s c r i b e d i n d e f i n i t i o n 2.6. P i c k a cube H meeting G . K L e t P be the maximal p r o j e c t i o n r e f e r r e d to i n the d e f i n i t i o n of y _^(H,G) and l e t n E = P(H-G). dimensional Without l o s s of g e n e r a l i t y we may f a c e F of H which c o n t a i n s E i s p a r a l l e l coordinate plane. and x " is assume t h a t the (n-1) F o r each p o i n t x = ( x , x " ) 1 = (x , x ) letH have This implies X I! i n E where x' = x, be the segment of l e n g t h h i n H which c o n t a i n e d i n the l i n e through x normal to F. Suppose u £ Co„o( G ) . P t h e r e e x i s t s y £ H „ - G. to the x^, ' " , x^ By the d e f i n i t i o n of We then A(|D u(x' , x") dx') H by Jensen's inequality. Integrating over H^,, A(|u(x',x") |/e')dx < we get A(|D (x',x") | )dx' lU H H „ x Now notice that Du(x',x") | < |grad u(x',x") so that we have A( Iu(x',x ") |/£')dx' < A(|grad u(x',x") | ) dx' H „ x Integrating this inequality over E and denoting {x'|x = (x',x") £ H A(|u(x)|/e') dx H xE < C f o r some x" } we A(|grad u(x)|)dx -"H obtain 53 Now r e c a l l lemma 2.1 which says that A(|u|/e). < h ^ { f A(2|u|/e) + C f A ( 2 n „l „ H H xE U(E) J 1 / 2 h (.|gradu |/e)} J 1 Recall that _ 1/2,.2n h/e ,2nh, , 1-2 , 1/2 , (——) / n = h/(n e') = . <_ v 1/n 1/2 and 2|u|/e = 2|u|/2ne' < |u|/n e' , 1/2 so that we have n-1 <H'> i A H e V7(H^) Now we use inequality (3) ( (M/£> < A < C Note that u { A(|u|/n xE 1/2 e') + c f A([grad u j / n H J 1/2 and ( 4 ) , ^(H,G) jH replacing u by u/n { C 2 { 1/6 f A(|grad u | / n A ( H 1/2 ) ' g r a d U ' / n l / 2 ) , to get + C { A ( | S ^1/2 U | ) } (5) 1 7 grad u| •\ i<) • , £ | <x I u| I m a x K-|a|<m =•>' A(|grad u | / n 1/2 D ) <_ A( max |D°\I|) l<Ja|<m. max A(|D u|) 1<I a I<m a => f A(|grad u | / n _< (constant) < ~~ 1/2 ) < f Z 1 <|a|<m (constant)' max A(|D | J H l<|a[<m a U A(|D u|) a H max A(|D u|) l<|a|<m •'H a so that we have, from (5), A(|u(x)|/e ) dx A(|D u(x)|) dx a l<[a <m This means > A(|u(x)|/e) dx. t h 55 m * the (C^) condition and Thus the domain G s a t i s f i e s W 0 A L + ( G ) L A ( G ) i s compact for m > 1. Remark 2.2 • Evidently, for a domain G, each condition we have given i s weaker than the proceeding condition. F i n a l l y we give a geometric condition which ensures the compactness of W 0 A L -* ( G ) L A and which varies with m. ( G ) When m > 1 we w i l l assume, as i n Adams [4], that 1 0 a _ T 1 ( T ) - (n+l)/n dT < -. We w i l l need some notation. a sequence of N-functions BQ, B ^ , B (t) Q = A(t) . Given an N-function *'* A we define as i n Adams [4, p. 258]. 56 At each s t a g e we assume t h a t (B ) f l V ) k 6 dx (n+l)/n r e p l a c i n g B^, (6) i f n e c e s s a r y , by infinity and satisfying negative i n t e g e r such t h a t , - r (6). B (T) 1 (2n+l)/n d another N - f u n c t i o n Now < let 0) = CO (n,A) equivalent be to i t near the s m a l l e s t non- oo T T I t can be shown t h a t D e f i n i t i o n 2.6 We i f m > co(n,A) and { k(r) w(n, A) < w i l l say n. t h a t a domain G s a t i s f i e s the K™ t h e r e e x i s t s a sequence of p o s i t i v e i n t e g e r s | r = 1, 2, 3, ••• } w i t h the f o l l o w i n g p r o p e r t i e s : (a) S r r=l 1 1 - 1 (k(r)) ? _ 1 / n f o r some £ such t h a t (b) f o r every x £ G^ d(x,3G) < we ^ < « 0 < E, < 1/n have , condition where d(x,8G) i s the distance between x and the boundary of G. Theorem 2.5 W If a domain G s a t i s f i e s the K™ condition then A 0 A L i s compact. ^ ( G ) L A ( G ) In the proof of theorem 2.5 we w i l l use the following lemma. Lemma 2.2 Let e > 0 and 0 < £ < 1/n. Suppose B i s an N-function such that •1 T (2n l)/n d < T + and A i s any N-function. ^(n/(n+l))£ < £. 00 Let H be a cube of side h such that Then there exist positive constants c^ and C2> independent of h, such that f J M ^ ! ) d x H < c {h n f B(|grad u(x)|)dx + n + 1 h / n _ ? } CO for a l l functions u i n C (H) which s a t i s f y the inequality • B(|grad u(x) | ) dx <_ c and vanish i n a neighbourhood of some point y £ H. 58 Proof: Suppose H i s a cube o f s i d e h such t h a t ^ tH be a c o n c e n t r i c cube of s i d e t h . n ^ n ^ ^ + < £ i L e t For every x e H we have, from Adams [4, p. 255], that r u(x) - h -n < n 1/2 r •1 _ dt (n/n+l)£. J We n h | grad u ( z ) | t 0 Let X u(z) dz | • — tH — 1 _ n dz £ then have h^ < e or ~ < h"*" n (7) X Now 1-n |grad u ( z ) | t < ^ n B(|grad u ( z ) | ) + B ( t V n X ) by Young's i n e q u a l i t y and (7) . Thus |u(x) - h < n n J 1 / 2 u(z)dz | • — e H ^ d t f {B(|grad 0 J tH u(z)|) + B ( t ~ h " 1 <_ n 1 / 2 + h { 11 B(|grad u ( z ) | ) dz f1 0 J ^ ^ ( t ^ " 1 1 1 ^ ) dt } n 1 n _ X ) } dz 59 < n 1/2 B( grad u(z) ) dz H B(T) ^ . ..-1/2. 1/n-X/n-X + n n .(2n+l)/n dx 1-n-X < n B(|grad u(z)|) dz 1 / 2 + n -l/2 l/n-C oo % B(X) h 1 X Now i f x, y e H and x-y dx (2h+l)/n < h < 1 then u(x) - u(y) < 2n 1/2 B( grad u(z)| dz H .00 + 2 n 'Xi B(X) -l/2 l/n-e h 1 X I f we choose y such that u(y) = t |u(x)|/e <_ 2n 1 / 2 + 2 n dx (2n+l)/n 0 then 00 B(|grad u(z) |) dz -l/2 l/n-g .CO <\j h 1 x B^X) (2n+l)/n d T 60 < c { [ B(|grad u(z)|) dz + h 1/n ~^} In Using Jensen's inequality we get A( Jiiool) < 1/2 {A(2c, [ B(|grad u(z)|) dz)+ A ( 2 c „ h H < 1/2 A(2c -c ) • — C + 1/2 h < Now c. { j 1 / n ? 1/m ? )} B(|grad u ( z ) | ) d z l J H A(2c ) 3 B(|grad u(z)|) dz + h 1 / n } ? integrate over H to get A(|u(x)|/£) dx H 1 c 4 (h B(|grad u(z)|) dz r + n h + 1 / n _ ? } H • 61 Proof of theorem 2.5: Suppose we are given £ > 0. Choose R > 0 such that (k(r))" < ( n / n + 1 ) ? £ for every r > R. Consider a unit cube I centered at the o r i g i n . with unit cubes by adding successive layers of cubes T . n 2 1 / r>R 2 R Tesselate R Notice that r R where r takes on only integer values. Now subdivide each unit cube i n T with cubes of side r h(r) Let H = H(r) k(r) ' be one of these cubes of side h = h(r) where r > R. Since d(x,9G) < ^ ^ b r there exists y £ H - G . CO u E C (H, H-G) and U llm,A,G ± Then u £ C (H,y) . Let £' n = / 1/2 £/n 1 Suppose 62 Applying lemma 2.2 with B = B^, £ replaced by £', and u replaced by u/n , we obtain f A(|u(x)|/n e') dx 1/2 , [ grad u(x) ] > i =6 ° t h V u 1/1 H } d x n+l/n-£ + h } n or A(|u(x) | /e) dx < c fi {h [ B (|grad u ( x ) | / n 1/2 ) dx (9) + Now h n + 1 / n " } ? . suppose that I i s a unit cube i n T . (9) over I to get A ( < Ii(x)l c + Now sum this over ? ) {h h d x n 1 / n I "^ B (|grad w } to get A(|u(x)|/e) dx u(x)|n 1 / 2 )dx Sum the inequality + c g r (h(r)) " } F i n a l l y we have A ( | u ( x ) | / e ) dx T R 1 A ( | u ( x ) | / e ) dx 1 r>R 1 c 10 {h ( l ^ r a d " ( r ^ C h C r ) ) 1 7 1 1 ^ f n Z B n x ) } r >R Now suppose t h a t U I' m,A,G < c 11 Then grad u n Hence 1/2 B ,G to' < l ) 1 dx . Jgrad u(x)l H n W < L 1 / 2 Notice also that I r r>R (h(r)) ^ < c Thus we have | A(|u(x)|/e) dx < c 1 3 1/2 G This implies that || P nl/2 2 We then have W || P 0 A L ( G ) .|| < (a) e . R || ->• 0 * L A as r -* 00 so that, by lemma 0.2, ( G ) i s compact. Remark 2.3 c • In general we can weaken the condition K™ by replacing i n the d e f i n i t i o n by 65 00 i i / c- , r B 00 ( h ( r ) ) 1 _ n " W e (T) REFERENCES R.A. ADAMS, Capacity and compact imbedding. J . Math. Mech., 19 (1970), 923-929. R.A. ADAMS, Compact Sobolev imbeddings for unbounded domains, P a c i f i c J . Math., 32 (1970), 1-7. R.A. ADAMS and JOHN FOURNIER, Some imbedding theorems for Sobol Spaces, Canad. J. Math., 23(1971), 517-530. R.A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975. C.W. CLARK, An imbedding theorem for function spaces, P a c i f i c J . Math, 19 (1966), 243-25. T.K. DONALDSON and N.S. TRUDINGER, Orlicz-Sobolev spaces and imbedding theorems, J . Functional Anal. 8 (1971), 52-75. J.J. FOURNIER, Compact imbeddings Unpublished notes, U.B.C., 1971. for Orlicz-Sobolev spaces, G. HELLWIG. D i f f e r e n t i a l Operators of Mathematical Physics, Addison Wesley, Reading, Mass., 1967. M.A. KRASNOSEL'SKII and Ya. B. RUTICKII, Convex Functions and O r l i c z Spaces, Noodhoff, Groningen, the Netherlands, 1961.
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Compactness of Orlicz-Sobolev space imbeddings for unbounded domains Cahill, Ian Graham 1975
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Title | Compactness of Orlicz-Sobolev space imbeddings for unbounded domains |
Creator |
Cahill, Ian Graham |
Date Issued | 1975 |
Description | In this thesis we are concerned with the compactness of im-beddings, for unbounded domains, of Orlicz-Sobolev spaces. Donaldson and Trudinger [6] have given a generalization of the Sobolev imbedding theorem and the Rellich-Kondrachov compactness theorem for Sobolev spaces to the case of Orlicz-Sobolev spaces on bounded domains. Clark, Adams, and Fournier [1], [2], [3], and [5] have given extensions of the Sobolev compactness results to certain unbounded domains. In this thesis we consider similar extensions for Orlicz-Sobolev spaces. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080117 |
URI | http://hdl.handle.net/2429/19247 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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