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

Abstract

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 \[ Λ(Ω,p,α):=\inf_{\substack{u\in W^{1,p}(Ω)\\ u\not\equiv 0}}\dfrac{\displaystyle \int_Ω|\nabla u|^p \mathrm{d} x - α\displaystyle\int_{\partialΩ} |u|^p\mathrm{d}σ}{\displaystyle\int_Ω|u|^p\mathrm{d} x}, \] where $\mathrm{d}\sigma$ is the surface measure on $\partial \Omega$. We show the asymptotics \[ Λ(Ω,p,α)=-(p-1)α^{\frac{p}{p-1}} - (ν-1)H_\mathrm{max}\, α+ o(α), \quad α\to+\infty, \] where $H_{\max }$ 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.

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