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

For $p\in(1,n]$ and any bounded convex domain $\Omega\subset\mathbb{R}^n$, we prove the sharp inequality \[ \Lambda_p(\Omega):=\frac{W_p(\Omega)}{P(\Omega)V(\Omega)^{p/n}}\geq\omega_n^{-p/n}, \qquad W_p(\Omega)=\int_{\partial\Omega}|x|^p\ dS, \] with equality holding exactly at centered balls.

Combining this with the isoperimetric inequality yields the explicit upper bound \[ \sigma_{1,p}(\Omega)\leq \frac{A(n,p)}{r(\Omega^*)^{p-1}}, \] where $\Omega^*$ is a ball having the same perimeter as $\Omega$, 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.