Monotonicity results for the principal eigenvalue of the generalized Robin problem

Abstract

We study domain monotonicity of the principal eigenvalue $\lambda_1^\Omega(\alpha)$ corresponding to $\Delta u=\lambda(\alpha) \, u \text{ in } \Omega, \frac{\partial u}{\partial \nu} =\alpha\, u \text{ on } \partial \Omega$, with $\Omega \subset {\mathcal R}^n$ a $C^{0,1}$ bounded domain, and $\alpha$ a fixed real. We show that contrary to intuition domain monotonicity might hold if one of the two domains is a ball.