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

TL;DR

This paper investigates maximum principles for a class of sub-elliptic operators on unbounded domains in ba3a3, extending classical results for Laplacians and sub-Laplacians on stratified Lie groups.

Contribution

It establishes new criteria for unbounded open sets to satisfy maximum principles for sub-elliptic operators, generalizing classical results.

Findings

Criteria for maximum principle sets in unbounded domains.

Extension of classical Laplacian results to sub-elliptic operators.

Application to sub-Laplacians on stratified Lie groups.

Abstract

Maximum Principles on unbounded domains play a crucial r\^ole 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).