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

Let $Ω\subset \mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $α>0$, 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 $Ω$ with a non-linear $α$-dependent Robin boundary condition. We show the asymptotics $Λ(α) =(1-p)α^{p/(p-1)}+o(α^{p/(p-1)})$ as $α$ 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 $C^{1,λ}$ domains.