Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms

Grenon, Nathalie; Murat, François; Porretta, Alessio

doi:10.1016/j.crma.2005.09.027

Partial Differential Equations

Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms
[Existence et estimation a priori pour des problèmes elliptiques avec des termes sous quadratiques par rappport au gradient]

Présenté par : Pierre-Louis Lions

Grenon, Nathalie ¹ ; Murat, François ² ; Porretta, Alessio ³

¹ Centre universitaire de Bourges, rue Gaston Berger, BP 4043, 18028 Bourges cedex, France
² Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris cedex 05, France
³ Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Comptes Rendus. Mathématique, Tome 342 (2006) no. 1, pp. 23-28.

Résumé
Abstract

Dans cette Note nous démontrons une estimation a priori et l'existence d'une solution pour une classe de problèmes non linéaires dont le modèle est $- div A (x) D u + α_{0} u = γ {| D u |}^{q} + f (x)$ , où $1 < q < 2$ et où $f \in L^{m} (Ω)$ pour un m convenable. L'intérêt principal du résultat réside dans l'estimation a priori, dont la démonstration complète est donnée dans la Note.

In this Note we prove an a priori estimate and the existence of a solution for a class of nonlinear elliptic problems whose model is $- div A (x) D u + α_{0} u = γ {| D u |}^{q} + f (x)$ , when $1 < q < 2$ and $f \in L^{m} (Ω)$ for some suitable m. The main interest of the result lies in the a priori estimate, the complete proof of which is given in the Note.

Reçu le : 2005-08-31
Accepté le : 2005-09-14
Publié le : 2005-11-02

DOI : 10.1016/j.crma.2005.09.027

Affiliations des auteurs :

Grenon, Nathalie ¹ ; Murat, François ² ; Porretta, Alessio ³

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Grenon, Nathalie; Murat, François; Porretta, Alessio. Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms. Comptes Rendus. Mathématique, Tome 342 (2006) no. 1, pp. 23-28. doi : 10.1016/j.crma.2005.09.027. http://www.numdam.org/articles/10.1016/j.crma.2005.09.027/

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