On the $p$-Laplacian with Robin boundary conditions and boundary trace theorems

Let $Omegasubsetmathbb{R}^ u$, $ uge 2$, be a $C^{1,1}$ domain whose boundary $partialOmega$ is either compact or behaves suitably at infinity. For $pin(1,infty)$ and $alpha>0$, define [ Lambda(Omega,p,alpha):=inf_{substack{uin W^{1,p}(Omega) u otequiv 0}}dfrac{displaystyle int_Omega | abla u|^p mathrm{d} x - alphadisplaystyleint_{partialOmega} |u|^pmathrm{d}sigma}{displaystyleint_Omega |u|^pmathrm{d} x}, ] where $mathrm{d}sigma$ is the surface measure on $partialOmega$. We show the asymptotics [ Lambda(Omega,p,alpha)=-(p-1)alpha^{frac{p}{p-1}} - ( u-1)H_mathrm{max}, alpha + o(alpha), quad alphato+infty, ] where $H_mathrm{max}$ is the maximum mean curvature of $partialOmega$. The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.