Special Session 88: Diffusion problems with non-standard growth conditions

First Eigenvalue and Torsional Rigiditiy: Isoperimetric Inequalities for the Fractional Laplacian
Gianpaolo Piscitelli
Universita` degli studi di Napoli Parthenope
Italy
Co-Author(s):    B. Brandolini, I. De Bonis, V. Ferone, G. Piscitelli, B. Volzone
Abstract:
We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\Omega\subset \R^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the fractional Laplacian attains its minimum on balls. With the same arguments we also establish a reverse H\older inequality for an eigenfunction corresponding to $\lambda_1(\Omega)$.