An eigenvalue estimate for a Robin 𝑝-Laplacian in 𝐶¹ domains

Pankrashkin, Konstantin

doi:10.1090/proc/15116

An eigenvalue estimate for a Robin $p$ -Laplacian in $C^1$ domains
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Proc. Amer. Math. Soc. 148 (2020), 4471-4477 Request permission

Abstract:

Let $\Omega \subset \mathbb {R}^n$ be a bounded $C^1$ domain and $p>1$. For $\alpha >0$, define the quantity \[ \Lambda (\alpha )=\inf _{u\in W^{1,p}(\Omega ), u\not \equiv 0} \Big (\int _\Omega |\nabla u|^p \mathrm {d}x - \alpha \int _{\partial \Omega } |u|^p \mathrm {d} s\Big )\Big / \int _\Omega |u|^p \mathrm {d} x \] 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.

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