Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum

M.Francesca Betta; Anna Mercaldo; François Murat; M.Michaela Porzio

doi:10.1016/S1631-073X(02)02338-5

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum
[Existence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures]

M.Francesca Betta ¹ ; Anna Mercaldo ² ; François Murat ³ ; M.Michaela Porzio ⁴

¹ Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
² Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, via Cintia, 80126 Napoli, Italy
³ Laboratoire Jacques-Louis Lions, Université Paris VI, boı̂te courrier 187, 75252 Paris cedex 05, France
⁴ Facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi del Sannio, via Port'Arsa 11, 82100 Benevento, Italy

Comptes Rendus. Mathématique, Volume 334 (2002) no. 9, pp. 757-762.

Résumé
Abstract

Dans cette Note nous considérons une classe de problèmes non linéaires et non coercifs dont le prototype est

- △_{p} u + b (x) {| \nabla u |}^{λ} = μ dans Ω, u = 0 sur \partial Ω,

où

Ω

est un ouvert borné de

ℝ^{N}

(N⩾2), △_p est le p-Laplacien (1<p<N) ou une variante de cet opérateur, μ est une mesure de Radon bornée ou une function de

L^{1} (Ω)

, λ⩾0 et b appartient à l'espace de Lorentz

L^{N, 1} (Ω)

ou à l'espace de Lebesgue

L^{\infty} (Ω)

. Nous démontrons l'existence et l'unicité de solutions renormalisées de ce problème.

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

- △_{p} u + b (x) {| \nabla u |}^{λ} = μ in Ω, u = 0 on \partial Ω,

where

Ω

is a bounded open subset of

ℝ^{N}

(N⩾2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on

Ω

or a function in

L^{1} (Ω)

, λ⩾0 and b belongs to the Lorentz space

L^{N, 1} (Ω)

or to the Lebesgue space

L^{\infty} (Ω)

. We prove existence and uniqueness of renormalized solutions.

Reçu le : 2001-11-27
Révisé le : 2002-02-28
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02338-5

Affiliations des auteurs :

M.Francesca Betta ¹ ; Anna Mercaldo ² ; François Murat ³ ; M.Michaela Porzio ⁴

¹ Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
² Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, via Cintia, 80126 Napoli, Italy
³ Laboratoire Jacques-Louis Lions, Université Paris VI, boı̂te courrier 187, 75252 Paris cedex 05, France
⁴ Facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi del Sannio, via Port'Arsa 11, 82100 Benevento, Italy

@article{CRMATH_2002__334_9_757_0,
     author = {M.Francesca Betta and Anna Mercaldo and Fran\c{c}ois Murat and M.Michaela Porzio},
     title = {Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {757--762},
     publisher = {Elsevier},
     volume = {334},
     number = {9},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02338-5},
     language = {en},
}

TY  - JOUR
AU  - M.Francesca Betta
AU  - Anna Mercaldo
AU  - François Murat
AU  - M.Michaela Porzio
TI  - Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 757
EP  - 762
VL  - 334
IS  - 9
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02338-5
LA  - en
ID  - CRMATH_2002__334_9_757_0
ER  -

%0 Journal Article
%A M.Francesca Betta
%A Anna Mercaldo
%A François Murat
%A M.Michaela Porzio
%T Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum
%J Comptes Rendus. Mathématique
%D 2002
%P 757-762
%V 334
%N 9
%I Elsevier
%R 10.1016/S1631-073X(02)02338-5
%G en
%F CRMATH_2002__334_9_757_0

M.Francesca Betta; Anna Mercaldo; François Murat; M.Michaela Porzio. Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum. Comptes Rendus. Mathématique, Volume 334 (2002) no. 9, pp. 757-762. doi : 10.1016/S1631-073X(02)02338-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02338-5/

Bibliographie
Cité par

[1] P. Bénilan; L. Boccardo; T. Gallouët; R. Gariepy; M. Pierre; J.L. Vazquez An L¹-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 22 (1995), pp. 241-273

[2] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side a measure, J. Math. Pures Appl., to appear

[3] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side $L^{1} (Ω)$ , ESAIM Control Optim. Calc. Var., Special issue dedicated to the memory of Jacques-Louis Lions (2002), to appear

[4] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Uniqueness results for nonlinear elliptic equations with a lower order term, in preparation

[5] L. Boccardo; T. Gallouët Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., Volume 87 (1989), pp. 149-169

[6] L. Boccardo; T. Gallouët; L. Orsina Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996), pp. 539-551

[7] G. Bottaro; M.E. Marina Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati, Boll. Un. Mat. Ital., Volume 8 (1973), pp. 46-56

[8] G. Dal Maso; F. Murat; L. Orsina; A. Prignet Renormalized solutions for elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 28 (1999), pp. 741-808

[9] T. Del Vecchio Nonlinear elliptic equations with measure data, Pot. Analysis, Volume 4 (1995), pp. 185-203

[10] P.-L. Lions, F. Murat, Solutions renormalisées d'équations elliptiques non linéaires, to appear

[11] F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023, Laboratoire d'Analyse Numérique de l'Université Paris VI, 1993

[12] F. Murat Équations elliptiques non linéaires avec second membre L¹ ou mesure, Actes du 26ème Congrès National d'Analyse Numérique, Les Karellis, France, 1994, p. A12-A24

[13] J. Serrin Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 18 (1964), pp. 385-387

[14] G. Stampacchia Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), Volume 15 (1965), pp. 189-258

Cité par Sources :

Commentaires - Politique