JournalsaihpcVol. 23, No. 1pp. 1–28

Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature

  • Francesca Da Lio

    Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
  • Annamaria Montanari

    Dipartimento di Matematica, Università di Bologna, piazza di Porta S. Donato 5, 40127 Bologna, Italy
Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature cover
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Abstract

In this paper we prove comparison principles between viscosity semicontinuous sub- and supersolutions of the generalized Dirichlet problem (in the sense of viscosity solutions) for the Levi Monge–Ampère equation. As a consequence of this result and of the Perron's method we get the existence of a continuous solution of the Dirichlet problem related to the prescribed Levi curvature equation under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by building Lipschitz continuous barriers and by applying a weak Bernstein method introduced by Barles in [Differential Integral Equations 4 (2) (1991) 241].

Résumé

Dans cet article, nous prouvons des principes de comparaison entre sous et sursolutions du problème de Dirichlet généralisé (dans le sens des solutions de viscosité) pour l'équation de Levi Monge–Ampère. Comme conséquence de ces résultats, nous obtenons l'existence d'une solution continue du problème de Dirichlet associé à l'équation de la courbure de Levi sous des hypothèses convenables sur les conditions au bord et sur l'ouvert. Nous prouvons que la solution est lipschitzienne par la méthode de Bernstein faible introduite par Barles dans [Differential Integral Equations 4 (2) (1991) 241].

Cite this article

Francesca Da Lio, Annamaria Montanari, Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 1, pp. 1–28

DOI 10.1016/J.ANIHPC.2004.10.006