Mean curvature bounds and eigenvalues of Robin Laplacians

Abstract

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 $\Omega \in {\mathbb{R}}^{\nu }$; here $n$ is the outward unit normal and $\alpha >0$ is a constant. We show that for each $j\in \mathbb{N}$ and $\alpha \to +\infty$, the $j$th eigenvalue $E_j(Q_{\alpha }^{\Omega })$ has the asymptotics \[ E_j(Q^Ω_α)=-α^2 -(ν-1)H_\mathrm{max}(Ω)\,α+{\mathcal O}(α^{2/3}), \] where $H_{\max }(\Omega )$ 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 $H_{\max }$. In particular, we show that the ball is the strict minimizer of $H_{\max }$ 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 $E_j(Q_{\alpha }^B)>E_j(Q_{\alpha }^{\Omega })$ for large $\alpha$. An open question concerning a larger class of domains is formulated.

15 pages

References in corpus (2)

• Asymptotic eigenvalue estimates for a Robin problem with a large parameter
• Tunneling between corners for Robin Laplacians

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