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M athematical I nequalities & A pplications Volume 5, Number 2 (2002), 181195

IMBEDDINGS OF ANISOTROPIC ORLICZSOBOLEV SPACES AND APPLICATIONS PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON

(communicated

AND

NILS SVANSTEDT

by A. M. Fink)

1. Introduction Generalized Young functions (Young functions of several variables) were introduced and studied by S. Wang 26] and M. S. Skaff 20], 21]. The so called G functions were studied by N. S. Trudinger 25] who introduced the space L G () of such G -functions of n variables (n 2 Z+ ) . This type of construction is very important as it enables us to describe the different integral behaviour of the derivatives in different directions. In 25], an imbedding theorem was proved for the completion 1 of C0 () with respect to the norm kDukG : A variant of this theorem is given in 11] for the space W 1 LG () of weakly differentiable functions u of (n + 1) variables with the norm k(u Du)kG : In this paper we generalize this result to the traces on k ( k 6 n ; 1) , where k is the k -dimensional hyperplane. This means that we prove some new inequalities involving Orlicz-Sobolev norms. Moreover, we present an application of these inequalities to variational problems. The paper is organized as follows. In Section 2, we give some notations and terminology which we shall be using in the sequel. In Section 3 we discuss some imbedding properties of Orlicz-Sobolev spaces. Section 4 contains the continuous imbedding and two compact imbeddings. Some applications are given in Section 4. Finally, Section 5 is left for some concluding remarks. 2. Preliminaries A Young function A : 0

1) !

0

1] is a function defined by

Z

t

A(t ) =

0

a(x )dx

where a : 0 1) ! 0 1 ] is an increasing, left continuous function which is neither identically zero nor identically infinity on (0 1):

Mathematics subject classification (2000): 26D10 , 26D15. Key words and phrases: Inequalities, variational inequalities, imbeddings, Orlicz-Sobolev spaces, Young functions, applications.

c

D l

181

, Zagreb

182

PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON AND NILS SVANSTEDT

The Orlicz space LA () Rn is defined as the set of all (equivalance classes of) measurable functions f on such that k f kA < 1 , where k kA denotes the Luxemburg norm on L A () given by

k kA = inf

Z

>0:

A

j f (x )j

dx

61

:

A G -function of n variables G : Rn following properties :

(i) (ii) (iii)

!

0

1]

;Pn

is a function satisfying the

i

G(0) = 0;

limjxj!1 G(x ) = 1 G is convex i.e.

h

x

2 R n : jx j =

i=1

x2 i

1=2

;

G( x + (1 ; )y ) 6 G(x ) + (1 ; )G(y ) for all 0 6 6 1 x y 2 Rn ; n (iv) G is symmetric i.e. G(;x ) = G(x ) x 2 R ; ;1 (1) = fx 2 Rn ; G(x ) = 1g is separated from 0; (v) the set G (vi) G is lower semi-continuous. Clearly, G function of 1 variable is a Young function. We shall be assuming, in addition, that (vii) G is monotonically increasing in each variable separately. The vector valued Orlicz-space LG () is defined as follows : Let G be a G -function and let be a domain in R n : Further, let u1 u2 ::: un be real valued measurable functions defined on and let u = (u 1 u2 ::: un ) be a vector valued function. Then, u is said to belong to L G () if there exists a > 0 such that

Z

G( u(x )) < 1:

The space LG () is equipped with a norm corresponding to the Luxemburg norm given by

kukG = inf

Z

>0:

G

juj

dx

61

:

It is noted that the space LG () so defined is a Banach space. Let us point out that there should not be any ambiguity for the same notations L A () and LG () (also k kA and k kG ) used, respectively, for Young function and G -function. Moreover, we have used the symbols A B C for Young functions and G H for G -functions. For a G -function G the complementary function G + is defined by G+ (u) = sup

vi 0 i=1 2 ::: n

>

(u:v

; G(v))

IMBEDDINGS OF ANISOTROPIC ORLICZ-SOBOLEV SPACES AND APPLICATIONS

183

where u:v = i=1 ui vi . For u inequality holds: Z

Pn

2 LG()

and v

2 LG+ ()

the following H¨ lder's o

(1)

u:vdx

Let G : Rn+1 ! 0 1 ] be a G -function. The anisotropic Orlicz-Sobolev space, denoted by W 1 LG () is defined to be the space of weakly differentiable functions u for which (u Du) = (u D1 u D2 u ::: Dn u) belongs to LG (): A norm for the space W 1 LG () is given by

6 2 kukG kvkG+ :

kuk1 G = k(u

Du)kG :

A domain Rn is said to be admissible if there exists a constant (depending only upon n ) such that

where

n 1( n

property if there exists a finite cone K such that each point x 2 is the vertex of a finite cone Kx contained in and congruent to K: Finally, we shall be using the symbols ,! and ,!,! for, respectively, continuous and compact imbeddings. For further details regarding the concepts given in this section, one may refer to the monographs 1] and 17]. 3. On some anisotropic Orlicz-Sobolev spaces

L ; ) and in the Sobolev space W 1 1 (): A domain is said to have the cone

kk ;

n n 1

kuk ; 6 kuk1 1 u 2 W 1 1 () and k k1 1 denote, respectively, the norms in the Lebesgue space

n n 1

Let be a domain in Rn and let G be a G -function of n variables (equal to the 1 dimension of the space). For u 2 C0 () define the space H 0 (G ) as the completion 1 of the space C0 () with respect to the norm

kukH

0 (G

) =

kDukG

where Du is treated as a vector valued function in L G (): This gives the anisotropic character of the function u 2 H 0 (G ): The space H 0 (G ) was introduced by Trudinger 25] where he proved the following famous imbedding theorem : THEOREM A. Let Rn be a domain, f 1 f2 ..., fn be continuous non-negative non-decreasing functions on 0 1) and let Let G : R n+1 ! 0 1 ] be a G -function of n + 1 variables such that G+ (0 f1 (s) f2 (s) ::: fn (s)) Also assume that

R1

ds 0 m(s) <

6 s:

1

where m(s)

=

n Y i=1

1 !n

s

f i (s)

:

184

PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON AND NILS SVANSTEDT

Then, the continuous imbedding H 0 (G ) ,! LA () holds for any Young function A satisfying

Z

0 t

ds m(s)

6 kA;1(t ):

Let us mention that following Trudinger, people have worked with Orlicz-Sobolev spaces of anisotropic nature e.g. one can see 13], 16] and very recently Cianchi 6] derived an anisotropic Sobolev inequality which is more general than those discussed above. In the literature, a different kind of anisotropy has been considered which is in terms of so called "mixed norms". These mixed norms in Lebesgue spaces were first considered in 5 ] and then many people followed e.g. see 14], 18] and 19]. In the setting of Orlicz spaces, mixed norms were initiated by Firlej and Matuszewska 10] (see also 9 ]). Our results have no concern with mixed norms but since this norm gives rise an anisotropic space, a few lines have been mentioned. In 11], the following theorem was proved which is a variant of the Trudinger's Theorem A : THEOREM B. Let be a bounded admissible domain in Rn f be a continuous non-negative function on [ 0 1) and G be a G -function of (n + 1) variables on 0 1) such that G+ (0 f (s) f (s) ::: f (s)) Further, let A be a Young function given by 1 A;1 (jt j) =

Z

0

6 s:

jtj

ds ds s1=n f (s)

for some constant > 0: Then, the continuous imbedding W 1 LG () ,! LA () holds. In Theorems A and B (and also in all the results obtained by others mentioned above in this section), the spaces are considered in which the functions are defined on : From the application point of view it seems also reasonable to deal with spaces where the functions are defined on the boundary @ of : Such boundary values (or traces) can even be defined on the intersection of a k; dimensional hyperplane with (this intersection is denoted by k ): A good account of results concerning Lebesgue spaces and Sobolev spaces with traces is given in 17]. Our aim, in this paper, is to estabilish the imbedding in Theorem B for traces on k : We also give compactness of this imbedding and also for the imbedding in Theorem B.

IMBEDDINGS OF ANISOTROPIC ORLICZ-SOBOLEV SPACES AND APPLICATIONS

185

4. Imbedding properties of Orlicz-Sobolev spaces We begin with the following theorem : THEOREM 1. Let be a bounded domain in Rn having the cone property and let k ( 1 6 k 6 n) denotes the intersection of with a k -dimensional hyperplane in Rn . Let G : Rn+1 ! 0 1 ] be a G; function and suppose that f is a continuous, non-negative function on 0 1) such that G+ (0 f (s) f (s) ::: f (s)) holds. Let A be a Young function such that 1 A;1 (jt j) =

Z

0

6s

(2)

jtj

1 s n ; p +1 f (s) 1

ds

(3)

for some constant > 0 and p 2 1 n) where p is such that is also a G -function. If either n ; p < k 6 n or p = 1 and n ; 1 6 k imbedding W 1 LG () ,! LAk=n (k ) H (t 1 t 2 ::: t n+1 ) = G(t 1

1=p

t2

1=p

::: t n+1 )

1=p

6n

(4)

then the

holds, where Ak=n (t ) = A(t )]k=n REMARK. When has the cone property Theorem 1 contains Theorem B which can be obtained by taking p = 1 and k = n and using the fact that a domain having cone property is admissible (see 8]). For proving Theorem 1, we need the following: LEMMA 1. 1, Lemma 5.19 ] Let be a domain in R n having the cone property and let k denote the intersection of with some k -dimensional plane in Rn where 1 6 k 6 n (n ) : If n > mp and n ; mp < k 6 n then the imbedding holds for p 6 q 6 kp=(n ; mp) if n > mp or for p 6 q < 1 if n = mp: If p = 1 n > m and n ; m 6 k 6 n then the above imbedding holds for 1 6 q 6 k=(n ; m): Proof of Theorem 1. It can be verified that Ak=n is a Young function. We shall first prove the assertion for a bounded function u 2 W 1 LG (): If we take = kukAk=n k , then

Z

k

W m p () ,! Lq (k )

Ak=n

ju(t )j

1

dt

=

1:

(5)

Set h(t ) = A(t )] p ; n :

1

186

PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON AND NILS SVANSTEDT

Then, (5) and Lemma 1 give

Z

1

=

k

A

ju(t )j

p

k=n

! n;p

k

dt

=

h

"

ju(t )j

6

=

K1 K1 p

n XZ

k

; p k

kp

i=1 n XZ i=1

Di h h0

ju(t )j

p

p

dx + h

ju(t )j

p p p

#

juj

Di u dx + K1 h

ju(t )j

(6)

p

for some constant K1 : In view of the H¨ lder's inequality (1), we have o

n XZ i=1

h0

juj

p

Di u

62

0 h0

juj

:::

h0

juj

p

(u

H+

D1 u ::: Dn u)p

(7)

H

where we use the symbol

(t 1

t 2 ::: t n ) p

=

;

p p p t 1 t 2 ::: t n :

Now, in view of (4), we note that

(u

D1 u ::: Dn u) p

H

6 k(u

p

D1 u ::: Dn u)kG

p

=

p kuk1 G :

(8)

Thus, from (6), (7) and (8), we obtain 16 2K1 p 0 h0

juj

:::

h0

juj

H+

p kuk1 G + K1

h

ju(t )j

p

(9)

p

Further, by the definition of h and (3), we have h0 (y ) = A p ; n (y )

1 1

h

i0

=

1 p

;1 n

p G+

f (A):

Using this along with (4), we obtain from (9 ) 16 2K1 1 1 ( ; ) 0 f A p p n

=

juj

k=n

:::

f A

juj

p kuk1 G +K1

h

ju(t )j

p

:

p

Now, recall that such that

kukA

k :

(10) The aim is to show that there exists a constant K2 > 0

6 K2 kuk1 G

IMBEDDINGS OF ANISOTROPIC ORLICZ-SOBOLEV SPACES AND APPLICATIONS

187

but in view of Theorem B, the last estimate holds for the special case k and thus without any loss of genarality, we may assume that

=

n and p = 1

(11)

kukA

From (2) and (11), we get

Z

n=n n

=

kukA 6 : juj

G+ 0 f A

juj

:::

f A

dx

6

p

Z

A

juj

dx

61

and so 0 f A Using this in (10), we obtain 16 Setting (t ) = 2K1 1 ( p p

juj

:::

f A

juj

G+

6 1:

p

p ; 1 ) kuk1 G + K1 n p

h

ju(t )j

:

p

(12)

A(t ) tp

and (t ) =

(t ) (t )

=

h(t ) t

we observe that

A p=n (t ) ! 1 as t

!1

) such

and therefore for each that or

>

0 , there exists a constant K 3 (depending only upon

(t )

(h(t ))

p

6 6

A

(t ) + K3

A(t ) + K3 t p

which along with (11) gives h

juj

p p

6 6

Z

juj

dx +

K3 p

Z

jujp dx

(13)

K3 jujp 1 : p An application of H¨ lder's inequality (1) and (4) yield o

+

jujp

1

6 2 k(1 0 ::: 0)kH+ (juj jD1uj ::: jDnuj) p p p 6 2 k(juj jD1uj ::: jDnuj)kG = 2 kuk1 G :

2K1 1 ( p p

p ; 1 ) kuk1 G + n

H

Now, using the last estimate and (13) in (12), we get 16 Choosing

= 2K 1

1

K1 +

2K1 K3 p kuk1 G : p

and using the definition of we obtain

p kukA

k=n k

p 6 K2 kuk1 G

188 where K2

PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON AND NILS SVANSTEDT

=

; 1 + K3 which depends only upon n theorem for bounded u 2 W 1 LG (): In the case of arbitrary function u 2 W 1 LG () define u(t ) ju(t )j 6 u (t ) = sgn u(t ) ju(t )j > :

4K1

1 p

n: This establishes the

Then, u is bounded and by so called "Chain Rule" ( 1] Lemma 8.31) belongs to W 1 LG (): Also, u A increases with but bounded by K2 kuk1 G and therefore lim !1 u A = exists. By Fatou's lemma

Z

A

juj

dt

6 lim !1

Z

A

u

!

dt

61

and consequently u 2 LA (): Thus , the theorem is proved for arbitrary u too. Now, we proceed to establish the compactness of the imbeddings given in Theorems B and 1. For that we need a notation and a lemma which we give below: A notation: For two functions A and B we shall write A B if for every >0 A(t ) =0 lim t !1 B( t ) and for this situation, we usually say that A increases essentially more slowly than B near infinity. LEMMA 2. ( 1], Theorem 8.23.) Let be a domain in R n with finite volume. Let A and B be Young functions such that B A: Then, any bounded subset S of L A () which is precompact in L1 () is also precompact in LB (): THEOREM 2. Assume that all the hypothesis in Theorem B hold. If B is a Young function such that B A then the compact imbedding W 1 LG () ,!,! LB () holds. o Proof. Let u 2 W 1 LG (): By H¨ lder's inequality (1), we have

kuk1 6 2 k(1

0 ::: 0)kG

(u + (u +

k

D1 u ::: Dn u)kG D1 u ::: Dn u)kG :

(14)

and

kDuk1 6 2 k(0

1 ::: 1)kG

k

(15)

Now, (14) and (15) give the imbedding W 1 LG () ,! W 1 1 ():

IMBEDDINGS OF ANISOTROPIC ORLICZ-SOBOLEV SPACES AND APPLICATIONS

189

Further, we have the trivial imbedding W 1 1 () ,! L1 () which, by Rellich-Kondrachov Theorem, is compact. Hence, if S is any bounded subset in W 1 LG () it is bounded in LA () and precompact in LB () by Lemma 1. This proves the result. THEOREM 3. Let all the assumptions from Theorem 1 hold. If p > 1 and C is any Young function such that C A k=n then we have the imbedding W 1 LG () ,!,! LC (): Proof. Since H (t 1 t 2 ::: t n ) = G(t 1 t 2 ::: t n ) is a Young function, we can apply H¨ lder's inequality as in Theorem 2 to get the imbedding o W 1 LG () ,! W 1 p () being bounded. Already, the imbedding W 1 p () ,! L1 (k ) is known to exists which is compact as well, again, by the Rellich-Kondrachov Theorem. By using the same argument as in Theorem 2 the result now follows. 5. Application to variational problems Let all the hypotesis of Theorem 2 hold with the following additional assumption G(u x 1 x 2 ::: x n ) = B(juj) + G0 (x 1 x 2 ::: x n ) where G0 (x 1 x 2 ::: x n ) = G(0 x 1 x 2 ::: x n ): Moreover we assume that G and G are continuous and satisfy the 2 condition (so that W 1 LG () is reflexive). Let f : Rn ! R be a Caratheodory function, i.e. f (x ) is continuous for a.e. x 2 f ( ) is Lebesgue measurable for each

1=p 1=p 1=p

2 Rn

:

We assume that 1. f is convex in the second variable 2. There exist a f 2 L1 () and a constant c f > 0 such that

for a.e. x 2 and every 2 R n . Let g : R ! R be a Caratheodory function and assume that 1. g is lower semicontinuous in the second variable 2. There exist ag 2 L1 () and a constant cg > 0 such that g(x u) > cg B(juj) ; ag (x ):

f (x ) > c f G(0 1 ::: n ) ; a f (x )

190

PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON AND NILS SVANSTEDT

PROPOSITION 1. The functional F defined by F (u) = f (x Du(x ))dx is lower semicontinuous in the weak topology of W 1 LG () i.e. if uh * u weakly in W 1 LG () then the following inequality holds F (u) 6 lim inf F (uh ):

h

R

!1

Proof. We first prove that the functional F1 (w ) = f (x w (x ))dx is lower semicontinuous in LG0 (): Let wh converge to w in LG0 () such that limh!1 F1 (wh ) exists. If we can prove that F1 (w ) 6 limh!1 F1 (wh ) we are done. It is possible to prove that there exists R subsequence (still denoted (w h ) ) such that w h ! w and that a R (G0 (w h (x ))) dx ! (G0 (w (x ))) dx : Indeed, by Fatou's Lemma and the definition of the norm in LG0 () it follows that

Z

R

G0 ( k

u kuk )dx

6 1:

(choose a decreasing sequence (k) obtain that Z

h

! kuk): Thus, using Fatou's Lemma again we wh ; w lim inf G ( )dx 6 1

!1

0

where h

=

kwh ; wk : Hence by the lower semicontinuity of G0 it follows that wh ; w wh ; w G (lim inf ) 6 lim inf G ( ) < 1 a.e.,

0 h

h

!1

h

h

!1

0

h

and by property (ii) and (iv) of the G -function we obtain that lim inf

h

!1

jwh ; wj < 1 a.e.

h

This shows that there exists a subsequence (still denoted (wh ) ) such that w h By the convexity we have wh (x ) ; w (x ) h wh (x ) ; w (x ) h

( ; h) 1w;x) )

h

! w a.e..

G0 (wh (x ))

=

G0 (h

h G0

+ (1

6

i.e.

Z

w (x ) + (1 ; h ) G0 ( ) 1 ; h

Z

G0 (wh (x ))dx

6 h + (1 ; h )

1 2

G0 (

Moreover, by assuming that h < 2 -condition yields that G0 (

and by letting m be such that kw k

w (x ) )dx : 1 ; h

(16)

6 2m

the

w (x ) ) 6 G0 (2w (x )) 6 kG0 (w (x )) 1 ; h

IMBEDDINGS OF ANISOTROPIC ORLICZ-SOBOLEV SPACES AND APPLICATIONS

191

where

Z

Z

G0 (w )dx

=

6

Z

G0 (kw k G0 (2m

kwk

w kwk

w

)dx )dx

6 km

Z

=

Z

G0 (

kwk )dx 6 k

w

m

<

1:

Thus the Lebesgue Dominated convergence theorem gives that

Z

h

lim

!1

G0 (

w (x ) )dx 1 ; h

G0 (w (x ))dx

and therefore, by (16), we have that

Z

lim sup

h

!1

G0 (wh (x ))dx

6

Z

Z

G0 (w )dx :

Moreover, Fatou's Lemma yields

Z

G0 (wh (x ))dx

6 lim inf h!1 !

Z

G0 (wh (x ))dx

so we obtain the convergence

Z

(G0 (w h (x ))) dx

(G0 (w (x ))) dx :

Because f (x

)

and G0 are continuous, it holds that

h

f (x w (x )) ; c f G0 (w (x )) + a f (x ) = lim ( f (x wh (x )) ; c f G0 (wh (x )) + a f (x )) a.e.

!1

Thus, Fatou's Lemma, applied on the sequence f (x wh (x )) ; c f G0 (wh (x )) + a f (x ) > 0 gives that

Z

f (x w (x ))dx

R

6 hlim !1

Z

f (x wh (x ))dx

and it follows that F1 (w ) = f (x w (x ))dx is lower semicontinuous in LG0 (): R This implies lower semicontinuity of the functional F (u) = f (x Du(x ))dx in W 1 LG ()) (only using the fact that uh ! u in W 1 LG () implies that Duh ! Du in LG0 () ). Hence, since F trivially is convex and since W 1 LG () is a locally convex Hausdorff topological vector space it follows that F is lower semicontinuous in the weak topology (this result is classical, see e.g. 7] p. 14). PROPOSITION 2. The functional F2 defined by F2 (u) = g(x u(x ))dx is sequentially lower semicontinuous in the weak topology of W 1 LG () .

R

192

PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON AND NILS SVANSTEDT

Proof. If uh * u weakly in W 1 LG () then uh is norm bounded by the Banach Steinhaus Theorem. Thus Theorem 2 yields that u h contains a subsequence (still denoted uh ) such that uh ! u strongly in LB (): Now the lower semicontinuity on W 1 LG () follows by the fact that F2 is lower semicontinuous on LB () (which is seen by using the functional F1 in the previous proof with f replaced by g and G 0 by B). THEOREM 4. Assume that K is a sequentially weakly closed subset of W 1 LG (): Then there exists a solution to the minimum problem

Z Z

min

u K

2

f (x Du)dx +

g(x u(x ))dx

(17)

Moreover, if in addition K is convex and g is strictly convex in the second variable, then the solution is unique. Proof. The minimum problem (17) can be written in the following equivalent form: min (F (u) + F2 (u) + K (u)) 1

u W LG ()

2

where F and F2 are defined as in Proposition 1 and Proposition 2 above, respectively, and where K is the indicatior function on K ( K = 0 on K and 1 elsewhere). It is easy to see that K is sequentially lower semicontinuous in the weak topology of W 1 LG () (since K is a sequentially weakly closed) and, hence, by the previous propositions, so is the sum F + F2 + K (since the sum of lower semicontinuous functions are lower semicontinuous) : By the properties of F and F 2 we find that the inequality (18) F + F2 + K > k1 ; k2 holds for some positive constants k1 and k2 , where (u) = G(u(x ))dx : The functional is sequentially coercive in the weak topology of W 1 LG () . In order to see this we observe that (u) 6 kuk if kuk 6 1 and that kuk < (u) if 1 < kuk : Thus the set fu : (u) 6 t g is bounded in W 1 LG () and, hence, fu : (u) 6 t g is sequencially compact in the weak topology of W 1 LG () since this space is reflexive. Therefore we obtain that also F + F2 + K is sequentially coercive in the weak topology of W 1 LG () . The existence of a minimizer of (18) now follows from the "direct method in the Calculus of Variation" which states that a sequentially coercive and sequentially lower semicontinuous functional on a topological vector space has a minimum ]. If K is convex and g is strictly convex in the second variable then the functional = F + F2 + K is strictly convex and thus the minimum is unique. Indeed, assume on the contrary that u 1 and u2 are minimum points and u1 6= u2 then the strictly convexity would imply the inequality which is impossible. 1 1 u1 + u2 2 2

<

R

1 1 (u1 ) + (u2 ) = (u1 ) , 2 2

IMBEDDINGS OF ANISOTROPIC ORLICZ-SOBOLEV SPACES AND APPLICATIONS

193

6. Some final comments In their famous paper 3] Alt and Luckhaus prove existence and uniqueness of variational solutions to a class of doubly non-linear parabolic problems of the form

(b (u))

0 ; div (a (x t Du)) = f in

]0

T :

(19)

Their proof is based on a new integration by parts formula and compactness arguments like the Minty lemma for monotone operators. In 12] Kacur extends the result of Alt and Luckhaus and proves existence and uniqueness for more general continuity and growth conditions in Orlicz-Sobolev spaces. The equation (19) contains many equations which are important in various applications. One example is the porous medium equation u0 ; um and its cousin, the p-parabolic equation u0 ; div jDuj p;2 Du = f :

(21) =

f

(20)

The porous medium equation in fine structures is studied widely. For linear problems Darcy law-type asymptotics is well understood. The porous medium equation and the p-parabolic equation are subject to intensive studies. Theoretically the p-parabolic equation can be seen as a natural generalization to the L p setting of the usual heat- or diffusion equation where one allows to play with the parameter p . It turns out that different regimes for the value of p corresponds to different physical situations which are described by (21). For example, the extreme case p = 1 together with u 0 0 corresponds to the equation describing mean curvature, and the case p = 2 corresponds to usual linear heat distribution or linear diffusion The case p = 1 appears e.g. in the study of growing sandpiles, see e.g. Aronsson et. al. 4]. By varying p one can also vary the physical properties in the problem of say a fluid or a fine structured composite, porous or stratified medium. In many situations it is very useful to consider a sequence of problems like e.g. u0 ; div Ah (x t )jDuh j p;2 Duh h

= f: (23)

u0 ; div Du = f :

(22)

This can be the case in homogenization, numerical analysis or e.g. control problems. It follows, by the general G-compactness results of Svanstedt 22], for nonlinear parabolic operators that the porous medium equation and the p-parabolic equation homogenize. For the p-parabolic equation there are also corrector results for the modeling error in the strong L p -topology for the gradients and numerical schemes based on augmented Lagrangians available, see 23] and 24]. In the proof of convergence one make significant use of compact embedding properties like the Rellich embedding theorem in usual Sobolev spaces and the weak lower semicontinuity of the norm in L p -spaces, 1 < p < 1 . Together with appropriate structure conditions on the problem this guarantee existence and uniqueness of solution. But it is also one of the cornerstones in the theory of variational and operator convergence

194

PANKAJ JAIN, DAG LUKKASSEN, LARS-ERIK PERSSON AND NILS SVANSTEDT

associated to these problems. By using the new compact embedding result Theorem 2 and existence result Theorem 4 one can now build up a theory for variational convergence analogous to the De Giorgi's Gamma-convergence or an operator convergence like the G-convergence for a large class of elliptic or parabolic operators now being defined on anisotropic Orlicz-Sobolev spaces. There are many advantages of such a development. Some work has already been done in this direction (see 15]). The analysis in Orlicz-Sobolev spaces uses the properties like convexity and growth ( 2 -property) in such a way that one can vary parameters with more flexibility than for usual Sobolev spaces. Therefore it is presumable that one should be able to study G- and Gamma-convergence for problems like (19) above in an Orlicz-Sobolev setting. In the periodic setting this means that one should be able to study the homogenization problem for a large class of elliptic-parabolic problems of the type (19). This would hopefully also give new important insights in simpler problems and special cases of (19) via the new results in the new function spaces exploiting their features. Acknowledgement. We thank Professor Lech Maligranda for some generous comments and advices, which have improved the final version of this paper.

REFERENCES 1 ] R. A. ADAMS, Sobolev Spaces, Academic Press Inc., New York, 1975. 2 ] R. A. ADAMS, Anisotropic Sobolev inequalities, Casopis P st. Mat. 3 (1988), 267279. e 3 ] H. W. ALT AND S. LUCKHAUS, Quasilinear elliptic-parabolic differential equations, Math. Z, 183 (1983), 311341. 4 ] G. ARONSSEN, L. C. EVANS AND Y. WU, Fast/slow diffusion and growing sandpiles, J. Differential Equations, 131 2, (196), 304335. 5 ] A. BENEDEK AND R. PANZONE, The spaces Lp with mixed norms, Duke Math. J. 28 (1961) 301324. 6 ] A. CIANCHI, A fully anisotropic Sobolev inequality, Preprint no. 15 (1998), Dip. Mat. "U. Dini", Univ. di Firenze. 7 ] G. DAL MASO, An introduction to -convergence, Birkh¨ user, Boston, 1993. a 8 ] T. K. DONALDSON AND N. S. TRUDINGER, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971), 5275. 9 ] C. E. FINOL AND L. MALIGRANDA, On a decomposition of some functions, comment., Math. Prace. Mat. 30 (1991), 285291. 10 ] B. FIRLE AND W. MATUSZEWSKA, Some remarks on spaces provided with mixed norm, Comment. Math. Prace Mat. 17 (1974), 347357. 11 ] G. H. HARDY AND H. B. THOMPSON, An imbedding theorem for anisotropic Orlicz-Sobolev spaces, Bull. Austral. Math. Soc., 51 (1995), 463467. 12 ] J. KACUR, On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev spaces I, Math. Z., 203 (1990), 153171. 13 ] V. S. KLIMOV, Imbedding theorems and geometric inequalities, Math. USSR Izvestija 10 (1976), 615638. 14 ] I. M. KOLODIJ AND S. N. KRUZHKOV, On the theory of embedding of anisotropic Sobolev spaces, Russ. Math. Sury. 38 no.2 (1983) 188189; translation from Usp. Mat. Nauk 38 no. 2 (230) (1983) 207208. 15 ] E. YA. KHRUSLOV AND L. S. PANKRATOV, Homogenization of the Dirichlet variational problems in Orlicz-Sobolev spaces, Operator Theory and its Applications (Winnipeg, MB, 1998), 345366, Fields Inst. Commun., 25, Amer. Math. Soc., Providence, RI, 2000. 16 ] A. G. KOROLEV, Embedding theorems for anisotropic Sobolev-Orlicz spaces, Vestnik Moskovskogo Universiteta, Matematika, 38 (1983), 3237. 17 ] A. KUFNER, O. JOHN AND S. FUCIK, Function Spaces, Nordhoff International Publishing, Leyden, 1977. ´ 18 ] J. RAKOSN´K, Some remarks to aniosotropic Sobolev spaces I, Beitr. Anal. 13 (1979), 5568. I ´ 19 ] J. RAKOSN´K, Some remarks to aniosotropic Sobolev spaces II, Beitr. Anal. 15 (1981), 127140. I

IMBEDDINGS OF ANISOTROPIC ORLICZ-SOBOLEV SPACES AND APPLICATIONS

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20 ] M. S. SKAFF, Vector valued Orlicz spaces, Generalized N-functions I, Pacific J. Math. 28 (1969), 193206. 21 ] M. S. SKAFF, Vector valued Orlicz spaces, II, Pacific J. Math. 28 (1969), 413430. 22 ] N. SVANSTEDT, G-convergence of parabolic operators, Nonlinear Analysis TMA, Vol. 36, no. 7, (1999), 807843. 23 ] N. SVANSTEDT, Correctors for the homogenization of monotone parabolic operators, J. Nonlinear Mathematical Physics, Vol. 7 3, (2000), (in print). 24 ] N. SVANSTEDT, N. WELLANDER AND J. WYLLER, A numerical algorithm for nonlinear parabolic problems with highly oscillating coefficients, Numerical Methods for Partial Differential Equations, Vol. 12 (1996), 423440. 25 ] N. S. TRUDINGER, An imbedding theorem for H0 (G ) spaces, Studia Math. 50 (1974), 1730. 26 ] S. WANG, Convex functions of several variables and vector valued Orlicz spaces, Bull. Acad. Pol. Sci. S´ r. Sci. Math. Astronom. Phys. 11 (1963), 279284. e

(Received

October 25, 2000)

Pankaj Jain Department of Mathematics Deshbandhu College (University of Delhi) New Delhi - 110019, India Dag Lukkassen Narvik Institute of Technology and Norut Technology Ltd., N-8505 Narvik, Norway Lars-Erik Persson Department of Mathematics Lule° University of Technology a S-971 87 Lule° , Sweden a Nils Svanstedt Department of Mathematics Chalmers University of Technology S-412 96 Gothenburg, Sweden

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