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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded $C^1$ domain and $p>1$. For $\alpha >0$, define the quantity $$\mathrm{\Lambda }(\alpha )=\underset{u\in W^{1,p}(\mathrm{\Omega }),u\equiv ̸0}{\inf }({\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p\mathrm{d}x-\alpha {\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^p\mathrm{d}s)/{\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^p\mathrm{d}x$$ with $\mathrm{d}s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $\mathrm{\Omega }$ with a non-linear $\alpha$-dependent Robin boundary condition. We show the asymptotics $\mathrm{\Lambda }(\alpha )=(1-p){\alpha }^{p\mathrm{/}(p-1)}+o({\alpha }^{p\mathrm{/}(p-1)})$ as $\alpha$ tends to $+\mathrm{\infty }$. 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.