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

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.