On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators

We study uniformly elliptic fully nonlinear equations of the type F(D(2)u, Du, u, x) = f(x). We - show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; - obtain existence and uniqueness results for non-proper operators whose principal eigenvalues (in some cases, only one of them) are positive; - obtain an existence result for non-proper Isaac's equations. To cite this article: A. Quaas, B. Sirakov, C R. Acad. Sci. Paris, Ser. 1342 (2006). (c) 2005 Academie des sciences. Published by Elsevier SAS. All rights reserved.