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

Nathalie Grenon; François Murat; Alessio Porretta

doi:10.1016/j.crma.2005.09.027

Partial Differential Equations

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

Presented by: Pierre-Louis Lions

Nathalie Grenon ¹; François Murat ²; Alessio Porretta 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, Volume 342 (2006) no. 1, pp. 23-28.

Abstract
Résumé

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.

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.

Received: 2005-08-31
Accepted: 2005-09-14
Published online: 2005-11-02

DOI: 10.1016/j.crma.2005.09.027

Author's affiliations:

Nathalie Grenon ¹; François Murat ²; Alessio Porretta 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

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     author = {Nathalie Grenon and Fran\c{c}ois Murat and Alessio Porretta},
     title = {Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {23--28},
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     doi = {10.1016/j.crma.2005.09.027},
     language = {en},
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Nathalie Grenon; François Murat; Alessio Porretta. Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms. Comptes Rendus. Mathématique, Volume 342 (2006) no. 1, pp. 23-28. doi : 10.1016/j.crma.2005.09.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.027/

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