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 , D u , 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.