Arch. Math. 72 (1999) 293 ± 297 0003-889X/99/040293-05 $ 2.50/0 Birkhäuser Verlag, Basel, 1999
Archiv der Mathematik
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Author:
Fuchs M.

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Arch. Math. 72 (1999) 293 ± 297 0003-889X/99/040293-05 $ 2.50/0 Birkhäuser Verlag, Basel, 1999

Archiv der Mathematik

L 1 -bounds for elliptic equations on Orlicz-Sobolev spaces By MARTIN FUCHS and LI GONGBAO

Abstract. We apply a lemma of Stampacchia and Ladyzhenskaya-Uraltseva to weak solutions of elliptic boundary value problems whose principal part has nonstandard growth. Our main result is a maximum principle bounding the absolute values of the solution in terms of the supremum of the absolute values of the boundary data.

1. Introduction and results. Let W Rn denote a bounded Lipschitz domain and consider a weak solution u : W ! R of the equation 1.1

ÿdiv a ru f

on W

satisfying 1.2

uV

on @W;

with prescribed boundary datum V. The properties of u are mainly investigated under the assumption that the vectorfield a : Rn ! Rn satisfies a power growth condition which means that a h grows like jhjpÿ1 for some number p > 1 (see [8] and [7] for detailed statements). Instead of this we require a to satisfy 1.3

ja hj % Lg jhj;

a h h ^ ljhj g jhj

n

for all h 2 R , l; L > 0 denoting constants. Here g : 0; 1 ! R is a given function with the properties a) g 0 0; g t > 0 for t > 0; b) g is increasing; c) g is continuous.

lim g t 1 ;

t! 1

Following [1] we introduce the N-function t G t g s ds; 0

t ^ 0;

and its conjugate function G . From now on we assume that G satisfies a D2 -condition near infinity, i.e. G 2s % aG s Mathematics Subject Classification (1991): 35J.

294

M. FUCHS and L. GONGBAO

ARCH. MATH.

is true for all sufficiently large s and some constant a > 0. The choice G t tp ; p > 1, leads to the above mentioned power growth condition, nonstandard examples are G t t ln 1 t; t G t s1ÿa ar sinh sa ds; 0 % a % 1; 0

G t t ln 1 ln 1 t which occur in the mechanics of solids and fluids (see [2], [5], [6] and [4]). Let us briefly discuss the notion of a weak solution to equation (1.1). The defining 1 W (compare [1]) and the following function G generates the Orlicz-Sobolev space WG 1 W and calculations show that weak solutions u should belong to this class. Consider u 2 WG 1 f 2 C0 W. Then by Youngs inequality and (1.3) (G denoting the conjugate function of G) ja rujjrfj % Lg jrujjrfj % LfG g jruj G jrfjg which together with the identity G g jruj jruj g jruj ÿ G jruj implies finiteness of ja rujjrfjdx in case that W

W

jruj g jrujdx < 1 :

But this follows from convexity of G and the D2 -condition, since s g s % G 2s ÿ G s % G 2s % aG s for all large enough s. In conclusion we see that 1 W and f 2 C01 W. u 2 WG

W

a ru rfdx is well-defined for

1 D e f i n i t i o n 1 . 1. Suppose that we are given functions V 2 WG W and 1 1 f 2 L W: u 2 WG W is a weak solution to the boundary value problem (1.1), (1.2) iff

u ÿ V 2 WG1 W and a ru rfdx f fdx W

holds for all f 2

C01 W.

W

1 Here WG1 W is the closure of C01 W in WG W.

Now we can state our main result: 1 Theorem 1.1. Let u 2 WG W denote a weak solution of (1.1),(1.2) and assume that f is in L W. Then sup V < 1 implies that u is bounded from above, i.e. @W sup u % const sup V; kukL1 W ; n; jWj; G; kf kL 1 W ; l; L < 1 : 1

W

@W

R e m a r k 1 . 1. The method used in the proof of the above theorem also gives inf u > ÿ 1 W

provided that V is bounded from below. In particular, boundedness of V implies u 2 L 1 W:

Vol. 72, 1999

295

Equations on Orlicz-Sobolev spaces

R e m a r k 1 . 2. The minimisation problem 1.4 G jruj ÿ fudx V min in V WG1 W W

h : Another example was studied in jhj the paper [3]: suppose that Y : W ! R is sufficiently smooth satisfying Yj@W < 0: We then considered the obstacle problem 1.5 G jrujdx V min in fv 2 WG1 W : v ^ Ya:e:g immediately leads to equation (1.1) with a h g jhj

W

and proved for G of class C2 the existence of a function f 2 L 1 W such that any solution of (1.5) is also a solution of (1.1), (1.2) with V 0: 1 R e m a r k 1 . 3. Up to now the regularity theory for weak solutions u 2 WG W of (1.1) is very poor. Roughly speaking, in the papers [5], [6], [4] and [3] we demonstrated C1 -regularity in case n 2, whereas for n ^ 3 some partial C1 -results were established. But again G is required to be of class C2 together with some extra coercivity conditions. We hope to extend our Theorem 1.1 in the sense that under the stated hypothesis the solution satisfies an interior Hölder condition.

A c k n o w l e d g e m e n t . This note was finished during the second authors stay at the Universität des Saarlandes in June 1997. The second author was partially supported by NSFC. 2. Proof of Theorem 1.1. The following lemma due to Ladyzhenskaya and Uraltseva [7] will turn out useful. Lemma 2.1. Let k0 > 0; g > 0; " > 0 and a 2 0; 1 " denote constants and suppose that u 2 L1 W satisfies the estimate u ÿ kdx % gka jAk j1" Ak

for all k ^ k0 where Ak denotes the set of points x 2 W for which u x > k. Then sup u is W bounded by a finite constant depending on g; "; a; k0 and kukL1 Ak : 0

h . In order to verify the jhj hypothesis of Lemma 2.1 we let k0 sup V < 1 and claim (u denoting the weak solution @W of (1.1), (1.2)) 2.1 g jrujjrujdx % const n; kf k 1 ; jWj; GjAk j For technical simplicity we just consider the case a h g jhj

Ak

for k > k0 . The function f max u ÿ k; 0 is in the space WG1 W which follows from 1 1 W together with the identity W 11 W \ WG W WG1 W (see [4], f 2 W 11 W \ WG Theorem 2.1), hence f is admissible in (1.1), and we obtain 2.2 g jrujjrujdx f u ÿ kdx: Ak

Ak

296

M. FUCHS and L. GONGBAO

ARCH. MATH.

The right-hand side of (2.2) is estimated with the help of Youngs inequality nÿ1 n 1 n f u ÿ kdx % kf k 1 jAk jn ju ÿ kjnÿ1 dx Ak

Ak 1

% c nkf k 1 jAk jn

Ak

jruj dx

1 1 G 2c nkf k 1 jAk jn G jruj dx % 2 Ak 1 1 G 2c nkf k 1 jAk jn jAk j G jruj dx 2 Ak 1 1 1 jruj g jruj dx % G 2c nkf k 1 jAk jn jAk j 2 Ak 2 1 1 jruj g jrujdx: % G 2c nkf k 1 jAk jn jAk j 2 Ak This gives estimate (2.1). Next we show 2.3 jrujdx % const n; kf k 1 ; jWj; GjAk j

8k ^ k0 :

Ak

We have

Ak

jrujdx %

Ak \fjruj % 1g

% jAk j

jruj dx

Ak \fjruj>1g

jrujdx

1 g jrujjrujdx; g 1 Ak

and (2.3) follow from (2.1). In a final step we observe Ak

1

u ÿ kdx % jAk jn

Ak 1

% c njAk jn

nÿ1 n n ju ÿ kjnÿ1 dx Ak

jrujdx

and (2.3) implies 1 u ÿ k dx % const n; kf k 1 ; jWj; GjAk j1n Ak

8k ^ k0 :

From this estimate the statement of Theorem 1.1 follows via Lemma 2.1.

h

References [1] R. A. ADAMS, Sobolev Spaces. New York 1975. [2] J. FREHSE and G. SEREGIN, Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening. Preprint No. 421, SFB256, Universität Bonn. [3] M. FUCHS and G. LI, Variational Inequalities for Energy Functionals with Nonstandard Growth Conditions. Abstract Appl. Anal. 3, Nos. 1 ± 2, 41 ± 64 (1998).

Vol. 72, 1999

Equations on Orlicz-Sobolev spaces

297

[4] M. FUCHS and V. OSMOLOVSKI, Variational integrals on Orlicz-Sobolev spaces. Z. Anal. Anwendungen 17, No. 2, 393 ± 415 (1998). [5] M. FUCHS and G. SEREGIN, A regularity theory for variational integrals with L ln L-growth. Calc. of Variations 6, 171 ± 187 (1998). [6] M. FUCHS and G. SEREGIN, Variational methods for fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening. Preprint No. 476, SFB256, Universität Bonn, Math. Methods Appl. Sci., in press. [7] O. LADYZHENSKAYA and N. URALTSEVA, Linear and quasilinear elliptic equations. Academic Press, New York 1968. [8] D. GILBARG and N. S. TRUDINGER, Elliptic partial differential equations of second order. BerlinHeidelberg-New York 1983. Eingegangen am 9. 2. 1998 Anschriften der Autoren: Martin Fuchs Universität des Saarlandes Fachbereich 9 Mathematik Postfach 15 11 50 D-66041 Saarbrücken

Li Gongbao Wuhan Institute of Physics and Mathematics Young Scientist Lab. of Math. Physics P.O. Box 71010 Wuhan 430071 P.R. of China

Archiv der Mathematik

L 1 -bounds for elliptic equations on Orlicz-Sobolev spaces By MARTIN FUCHS and LI GONGBAO

Abstract. We apply a lemma of Stampacchia and Ladyzhenskaya-Uraltseva to weak solutions of elliptic boundary value problems whose principal part has nonstandard growth. Our main result is a maximum principle bounding the absolute values of the solution in terms of the supremum of the absolute values of the boundary data.

1. Introduction and results. Let W Rn denote a bounded Lipschitz domain and consider a weak solution u : W ! R of the equation 1.1

ÿdiv a ru f

on W

satisfying 1.2

uV

on @W;

with prescribed boundary datum V. The properties of u are mainly investigated under the assumption that the vectorfield a : Rn ! Rn satisfies a power growth condition which means that a h grows like jhjpÿ1 for some number p > 1 (see [8] and [7] for detailed statements). Instead of this we require a to satisfy 1.3

ja hj % Lg jhj;

a h h ^ ljhj g jhj

n

for all h 2 R , l; L > 0 denoting constants. Here g : 0; 1 ! R is a given function with the properties a) g 0 0; g t > 0 for t > 0; b) g is increasing; c) g is continuous.

lim g t 1 ;

t! 1

Following [1] we introduce the N-function t G t g s ds; 0

t ^ 0;

and its conjugate function G . From now on we assume that G satisfies a D2 -condition near infinity, i.e. G 2s % aG s Mathematics Subject Classification (1991): 35J.

294

M. FUCHS and L. GONGBAO

ARCH. MATH.

is true for all sufficiently large s and some constant a > 0. The choice G t tp ; p > 1, leads to the above mentioned power growth condition, nonstandard examples are G t t ln 1 t; t G t s1ÿa ar sinh sa ds; 0 % a % 1; 0

G t t ln 1 ln 1 t which occur in the mechanics of solids and fluids (see [2], [5], [6] and [4]). Let us briefly discuss the notion of a weak solution to equation (1.1). The defining 1 W (compare [1]) and the following function G generates the Orlicz-Sobolev space WG 1 W and calculations show that weak solutions u should belong to this class. Consider u 2 WG 1 f 2 C0 W. Then by Youngs inequality and (1.3) (G denoting the conjugate function of G) ja rujjrfj % Lg jrujjrfj % LfG g jruj G jrfjg which together with the identity G g jruj jruj g jruj ÿ G jruj implies finiteness of ja rujjrfjdx in case that W

W

jruj g jrujdx < 1 :

But this follows from convexity of G and the D2 -condition, since s g s % G 2s ÿ G s % G 2s % aG s for all large enough s. In conclusion we see that 1 W and f 2 C01 W. u 2 WG

W

a ru rfdx is well-defined for

1 D e f i n i t i o n 1 . 1. Suppose that we are given functions V 2 WG W and 1 1 f 2 L W: u 2 WG W is a weak solution to the boundary value problem (1.1), (1.2) iff

u ÿ V 2 WG1 W and a ru rfdx f fdx W

holds for all f 2

C01 W.

W

1 Here WG1 W is the closure of C01 W in WG W.

Now we can state our main result: 1 Theorem 1.1. Let u 2 WG W denote a weak solution of (1.1),(1.2) and assume that f is in L W. Then sup V < 1 implies that u is bounded from above, i.e. @W sup u % const sup V; kukL1 W ; n; jWj; G; kf kL 1 W ; l; L < 1 : 1

W

@W

R e m a r k 1 . 1. The method used in the proof of the above theorem also gives inf u > ÿ 1 W

provided that V is bounded from below. In particular, boundedness of V implies u 2 L 1 W:

Vol. 72, 1999

295

Equations on Orlicz-Sobolev spaces

R e m a r k 1 . 2. The minimisation problem 1.4 G jruj ÿ fudx V min in V WG1 W W

h : Another example was studied in jhj the paper [3]: suppose that Y : W ! R is sufficiently smooth satisfying Yj@W < 0: We then considered the obstacle problem 1.5 G jrujdx V min in fv 2 WG1 W : v ^ Ya:e:g immediately leads to equation (1.1) with a h g jhj

W

and proved for G of class C2 the existence of a function f 2 L 1 W such that any solution of (1.5) is also a solution of (1.1), (1.2) with V 0: 1 R e m a r k 1 . 3. Up to now the regularity theory for weak solutions u 2 WG W of (1.1) is very poor. Roughly speaking, in the papers [5], [6], [4] and [3] we demonstrated C1 -regularity in case n 2, whereas for n ^ 3 some partial C1 -results were established. But again G is required to be of class C2 together with some extra coercivity conditions. We hope to extend our Theorem 1.1 in the sense that under the stated hypothesis the solution satisfies an interior Hölder condition.

A c k n o w l e d g e m e n t . This note was finished during the second authors stay at the Universität des Saarlandes in June 1997. The second author was partially supported by NSFC. 2. Proof of Theorem 1.1. The following lemma due to Ladyzhenskaya and Uraltseva [7] will turn out useful. Lemma 2.1. Let k0 > 0; g > 0; " > 0 and a 2 0; 1 " denote constants and suppose that u 2 L1 W satisfies the estimate u ÿ kdx % gka jAk j1" Ak

for all k ^ k0 where Ak denotes the set of points x 2 W for which u x > k. Then sup u is W bounded by a finite constant depending on g; "; a; k0 and kukL1 Ak : 0

h . In order to verify the jhj hypothesis of Lemma 2.1 we let k0 sup V < 1 and claim (u denoting the weak solution @W of (1.1), (1.2)) 2.1 g jrujjrujdx % const n; kf k 1 ; jWj; GjAk j For technical simplicity we just consider the case a h g jhj

Ak

for k > k0 . The function f max u ÿ k; 0 is in the space WG1 W which follows from 1 1 W together with the identity W 11 W \ WG W WG1 W (see [4], f 2 W 11 W \ WG Theorem 2.1), hence f is admissible in (1.1), and we obtain 2.2 g jrujjrujdx f u ÿ kdx: Ak

Ak

296

M. FUCHS and L. GONGBAO

ARCH. MATH.

The right-hand side of (2.2) is estimated with the help of Youngs inequality nÿ1 n 1 n f u ÿ kdx % kf k 1 jAk jn ju ÿ kjnÿ1 dx Ak

Ak 1

% c nkf k 1 jAk jn

Ak

jruj dx

1 1 G 2c nkf k 1 jAk jn G jruj dx % 2 Ak 1 1 G 2c nkf k 1 jAk jn jAk j G jruj dx 2 Ak 1 1 1 jruj g jruj dx % G 2c nkf k 1 jAk jn jAk j 2 Ak 2 1 1 jruj g jrujdx: % G 2c nkf k 1 jAk jn jAk j 2 Ak This gives estimate (2.1). Next we show 2.3 jrujdx % const n; kf k 1 ; jWj; GjAk j

8k ^ k0 :

Ak

We have

Ak

jrujdx %

Ak \fjruj % 1g

% jAk j

jruj dx

Ak \fjruj>1g

jrujdx

1 g jrujjrujdx; g 1 Ak

and (2.3) follow from (2.1). In a final step we observe Ak

1

u ÿ kdx % jAk jn

Ak 1

% c njAk jn

nÿ1 n n ju ÿ kjnÿ1 dx Ak

jrujdx

and (2.3) implies 1 u ÿ k dx % const n; kf k 1 ; jWj; GjAk j1n Ak

8k ^ k0 :

From this estimate the statement of Theorem 1.1 follows via Lemma 2.1.

h

References [1] R. A. ADAMS, Sobolev Spaces. New York 1975. [2] J. FREHSE and G. SEREGIN, Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening. Preprint No. 421, SFB256, Universität Bonn. [3] M. FUCHS and G. LI, Variational Inequalities for Energy Functionals with Nonstandard Growth Conditions. Abstract Appl. Anal. 3, Nos. 1 ± 2, 41 ± 64 (1998).

Vol. 72, 1999

Equations on Orlicz-Sobolev spaces

297

[4] M. FUCHS and V. OSMOLOVSKI, Variational integrals on Orlicz-Sobolev spaces. Z. Anal. Anwendungen 17, No. 2, 393 ± 415 (1998). [5] M. FUCHS and G. SEREGIN, A regularity theory for variational integrals with L ln L-growth. Calc. of Variations 6, 171 ± 187 (1998). [6] M. FUCHS and G. SEREGIN, Variational methods for fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening. Preprint No. 476, SFB256, Universität Bonn, Math. Methods Appl. Sci., in press. [7] O. LADYZHENSKAYA and N. URALTSEVA, Linear and quasilinear elliptic equations. Academic Press, New York 1968. [8] D. GILBARG and N. S. TRUDINGER, Elliptic partial differential equations of second order. BerlinHeidelberg-New York 1983. Eingegangen am 9. 2. 1998 Anschriften der Autoren: Martin Fuchs Universität des Saarlandes Fachbereich 9 Mathematik Postfach 15 11 50 D-66041 Saarbrücken

Li Gongbao Wuhan Institute of Physics and Mathematics Young Scientist Lab. of Math. Physics P.O. Box 71010 Wuhan 430071 P.R. of China

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