Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions, $Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ,$ in a class of bounded smooth domains $Ω\in\mathbb{R}^ν$; here $n$ is the outward unit normal and $α>0$ is a constant. We show that for each $j\in\mathbb{N}$ and $α\to+\infty$, the $j$th eigenvalue $E_j(Q^Ω_α)$ has the asymptotics $E_j(Q^Ω_α)=-α^2 -(ν-1)H_\mathrm{max}(Ω)\,α+{\mathcal O}(α^{2/3}),$ where $H_\mathrm{max}(Ω)$ is the maximum mean curvature at $\partial Ω$. 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 $Ω$ is any other star-shaped smooth domain of the same volume, then for any fixed $j\in\mathbb{N}$ we have $E_j(Q^B_α)>E_j(Q^Ω_α)$ for large $α$. An open question concerning a larger class of domains is formulated.

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