Calculus of Variations and Geometric Measure Theory

L. Brasco - G. Franzina

An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

created by brasco on 08 Sep 2012
modified by franzina on 05 Dec 2013


Published Paper

Inserted: 8 sep 2012
Last Updated: 5 dec 2013

Journal: NoDEA
Volume: 20
Number: 6
Pages: 1795–1830
Year: 2012


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.

Keywords: Stekloff eigenvalue problem, pseudo $p-$Laplacian, Wulff inequality