Directional ellipticity on special domains: weak Maximum and Phragm\`en-Lindel\"of principles

Abstract

We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains $\Omega \subset \mathbb R^n$ of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along the bounded directions of $\Omega$, with possible degeneracy along the unbounded directions.