An eigenvalue estimate for a Robin $p$-Laplacian in $C^1$ domains

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded $C^1$ domain and $p>1$. For $\alpha >0$, define the quantity \[ Λ(α)=\inf_{u\in W^{1,p}(Ω),\, u\not\equiv 0} \Big(\int_Ω|\nabla u|^p\,\mathrm{d}x - α\int_{\partialΩ} |u|^p \,\mathrm{d} s\Big)\Big/ \int_Ω|u|^p\,\mathrm{d} x \] with $\mathrm{d}s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $\Omega$ with a non-linear $\alpha$-dependent Robin boundary condition. We show the asymptotics $\Lambda (\alpha )=(1-p){\alpha }^{p/(p-1)}+o({\alpha }^{p/(p-1)})$ as $\alpha$ tends to $+\infty$. The result was only known for the linear case $p=2$ or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for $C^{1,\lambda }$ domains.

6 pages. New results on the remainder for $C^{1,\lambda }$ domains are added