Communications on Pure and Applied Analysis (CPAA)

Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators

Pages: 335 - 366, Volume 6, Issue 2, June 2007      doi:10.3934/cpaa.2007.6.335

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Isabeau Birindelli - Dipartimento di Matematica, Università di Roma, Italy (email)
Francoise Demengel - Laboratoire d'Analyse, Géométrie et Modelisation, Université de Cergy-Pontoise, Site de Saint-Martin, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France (email)

Abstract: The main scope of this article is to define the concept of principal eigenvalue for fully non linear second order operators in bounded domains that are elliptic, homogenous with lower order terms. In particular we prove maximum and comparison principle, Hölder and Lipschitz regularity. This leads to the existence of a first eigenvalue and eigenfunction and to the existence of solutions of Dirichlet problems within this class of operators.

Keywords:  Fully nonlinear equation, viscosity solution, eigenvalue.
Mathematics Subject Classification:  35J60, 35P30.

Received: February 2006;      Revised: November 2006;      Published: March 2007.