Mean curvature bounds and eigenvalues of Robin Laplacians

Abstract

We consider the Laplacian with attractive Robin boundary conditions, $$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$ in a class of bounded smooth domains $$\Omega \in \mathbb {R}^\nu $$ ; here $$n$$ is the outward unit normal and $$\alpha >0$$ is a constant. We show that for each $$j\in \mathbb {N}$$ and $$\alpha \rightarrow +\infty $$ , the $$j$$ th eigenvalue $$E_j(Q^\Omega _\alpha )$$ has the asymptotics $$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$ where $$H_\mathrm {max}(\Omega )$$ is the maximum mean curvature at $$\partial \Omega $$ . The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of $$H_\mathrm {max}$$ . In particular, we show that the ball is the strict minimizer of $$H_\mathrm {max}$$ among the smooth star-shaped domains of a given volume, which leads to the following result: if $$B$$ is a ball and $$\Omega $$ is any other star-shaped smooth domain of the same volume, then for any fixed $$j\in \mathbb {N}$$ we have $$E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )$$ for large $$\alpha $$ . An open question concerning a larger class of domains is formulated.