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

Let $\Omega\subset\mathbb{R}^\nu$, $\nu\ge 2$, be a $C^{1,1}$ domain whose boundary $\partial\Omega$ is either compact or behaves suitably at infinity. For $p\in(1,\infty)$ and $\alpha>0$, define $$\Lambda(\Omega,p,\alpha):=\inf_{\substack{u\in W^{1,p}(\Omega)\\ u\not\equiv 0}}\dfrac{\displaystyle \int_\Omega |\nabla u|^p \mathrm{d} x - \alpha\displaystyle\int_{\partial\Omega} |u|^p\mathrm{d}\sigma}{\displaystyle\int_\Omega |u|^p\mathrm{d} x},$$ where $\mathrm{d}\sigma$ is the surface measure on $\partial\Omega$. We show the asymptotics $$\Lambda(\Omega,p,\alpha)=-(p-1)\alpha^{\frac{p}{p-1}} - (\nu-1)H_\mathrm{max}\, \alpha + o(\alpha), \quad \alpha\to+\infty,$$ where $H_\mathrm{max}$ is the maximum mean curvature of $\partial\Omega$. 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.