Strict positivity for the principal eigenfunction of elliptic operators with various boundary conditions

TL;DR

This paper proves that the principal eigenfunction of elliptic operators with Robin, Dirichlet, or mixed boundary conditions remains strictly positive in the domain, even with minimal boundary regularity, using a new positivity approach.

Contribution

It introduces a novel method based on an abstract positivity improving condition to establish strict positivity of eigenfunctions under weak boundary regularity.

Findings

Principal eigenfunction is strictly positive under Lipschitz boundary conditions.

The new approach applies to Robin, Dirichlet, and mixed boundary conditions.

Results enable derivation of strong minimum and maximum principles.

Abstract

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge \delta > 0$ for all $x \in \overline{\Omega}$, even if the boundary $\partial \Omega$ is only Lipschitz continuous. Under such weak regularity assumptions the Hopf-Ole\u{\i}nik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map $L_p(\Omega)$ into $C(\overline{\Omega})$. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.