Existence results for Bellman equations and maximum principles in unbounded domains

The purpose of this paper is to extend known existence results for Hamilton -Jacobi-Bellman equations. The classical results give existence, uniqueness and Holder regularity when all elliptic operators involved have nonpositiv e zero-order term. We want to handle here the more general case where they have principal eigenvalues bounded below by a positive constant. As a motiv ation for this work, we give an application to the study of the Maximum Pri nciple in infinite cylinders, following a work by Berestycki, Caffarelli an d Nirenberg [2]. This is used to extend the cylindrical symmetry result in [2] to a more general class of operators.