Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

Abstract

Maximum Principles on unbounded domains play a crucial role in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators $\\mathcal{L}$ in $\\mathbb{R}^N$ and we establish some criteria for an unbounded open set to be a Maximum Principle set for $\\mathcal{L}$. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author).