Upper bound for the first $p$-Steklov eigenvalue in $\mathbb{R}^n$

Abstract

For $p\in(1,n]$ and any bounded convex domain $Ω\subset\mathbb{R}^n$, we prove the sharp inequality \[ Λ_p(Ω):=\frac{W_p(Ω)}{P(Ω)V(Ω)^{p/n}}\geqω_n^{-p/n}, \qquad W_p(Ω)=\int_{\partialΩ}|x|^p\ dS, \] with equality holding exactly at centered balls. Combining this with the isoperimetric inequality yields the explicit upper bound \[ σ_{1,p}(Ω)\leq \frac{A(n,p)}{r(Ω^*)^{p-1}}, \] where $Ω^*$ is a ball having the same perimeter as $Ω$, and $A(n,p)=1$ for $1<p\leq 2$, $A(n,p)=n^{p/2-1}$ for $2<p\leq n$. When $p=2$, the result recovers the higher-dimensional Weinstock inequality of Bucur et al. [J. Differential Geom. 2021]. We also obtain an explicit upper bound for the first Wentzell eigenvalue of the $p$-Laplacian on convex domains.