An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

We study the Stekloff eigenvalue problem for the so-called pseudo $p-$Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalue $\sigma_{2,p}$, providing various equivalent characterizations for it. We also prove an upper bound for $\sigma_{2,p}$ in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock-Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian. Such an estimate is obtained by exploiting a family of sharp weighted Wulff inequalities, which are here derived and appear to be interesting in themselves.