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

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.