Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic 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; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations. (c) 2007 Elsevier Inc. All rights reserved.