  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.
