An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

Abstract

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 σ2,p, providing various equivalent characterizations for it. We also prove an upper bound for σ2,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.