Partial Differential Equations
Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms
Comptes Rendus. Mathématique, Volume 342 (2006) no. 1, pp. 23-28.

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 divA(x)Du+α0u=γ|Du|q+f(x), when 1<q<2 and fLm(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 divA(x)Du+α0u=γ|Du|q+f(x), où 1<q<2 et où fLm(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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.09.027
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
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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, Volume 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/

[1] Alaa, N.; Pierre, M. Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., Volume 24 (1993), pp. 23-35

[2] G. Barles, A. Porretta, Uniqueness for unbounded solutions to stationary viscous Hamilton–Jacobi equations, Preprint

[3] Boccardo, L.; Murat, F. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., Volume 19 (1992), pp. 581-597

[4] Boccardo, L.; Murat, F.; Puel, J.-P. Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique (Brezis, H.; Lions, J.-L., eds.), Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. IV, Pitman Res. Notes in Math., vol. 84, Pitman, London, 1983, pp. 19-73

[5] Boccardo, L.; Murat, F.; Puel, J.-P. L Estimates for some nonlinear partial differential equations and an application to an existence result, SIAM J. Math. Anal., Volume 23 (1992), pp. 326-333

[6] L. Boccardo, M.M. Porzio, Quasilinear elliptic equations with subquadratic growth, Preprint

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

[8] Dall'Aglio, A.; Giachetti, D.; Puel, J.-P. Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl., Volume 181 (2002), pp. 407-426

[9] V. Ferone, B. Messano, Comparison results for nonlinear equations with general growth in the gradient, Preprint

[10] Ferone, V.; Murat, F. Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, Equations aux dérivées partielles et applications. Articles dédiés à Jacques-Louis Lions, Gauthier–Villars, Paris, 1998, pp. 497-515

[11] N. Grenon, F. Murat, A. Porretta, Elliptic equations with superlinear gradient dependent terms: existence and a priori estimates, in preparation

[12] Hansson, K.; Maz'ya, V.; Verbitsky, I. Criteria of solvability for multidimensional Riccati's equation, Ark. Mat., Volume 37 (1999), pp. 87-120

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