Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains

TL;DR

This paper explores various notions of the principal eigenvalue for elliptic operators in unbounded domains, establishing conditions for maximum principles and eigenfunction existence, and analyzing their properties and relations.

Contribution

It introduces and compares three notions of generalized principal eigenvalues, providing necessary and sufficient conditions for maximum principles and eigenfunction existence in unbounded domains.

Findings

Necessary and sufficient conditions for maximum principles

Existence criteria for positive eigenfunctions with Dirichlet boundary conditions

Relations and properties of different principal eigenvalues

Abstract

Using three different notions of generalized principal eigenvalue of linear second order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions satisfying Dirichlet boundary conditions. Relations between these principal eigenvalues, their simplicity and several other properties are further discussed.