Maximum principles in unbounded Riemannian domains

TL;DR

This paper investigates maximum principles for second order elliptic operators on unbounded Riemannian domains, establishing conditions under which these principles hold, with implications for the qualitative analysis of PDE solutions.

Contribution

It provides new criteria for maximum principles on unbounded Riemannian domains, considering both geometric and operator-related assumptions.

Findings

Maximum principles established under specific geometric conditions.

Criteria involving the differential operator and manifold geometry.

Applications to qualitative PDE analysis.

Abstract

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kind of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.