An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities
Résumé
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 appears to be interesting in themselves.
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