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

Let $Omegasubset mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $alpha>0$, define the quantity [ Lambda(alpha)=inf_{uin W^{1,p}(Omega),, u otequiv 0} Big(int_Omega | abla u|^p,mathrm{d}x - alpha int_{partialOmega} |u|^p ,mathrm{d} sBig)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.