Calculus of Variations and Geometric Measure Theory

L. Brasco - G. Franzina

On the Hong-Krahn-Szego inequality for the $p-$Laplace operator

created by brasco on 12 Sep 2011
modified by franzina on 12 Feb 2013


Published Paper

Inserted: 12 sep 2011
Last Updated: 12 feb 2013

Journal: Manuscripta Math.
Pages: 18
Year: 2012


Given an open set $\Omega$, we consider the problem of providing sharp lower bounds for $\lambda_2(\Omega)$, i.e. its second Dirichlet eigenvalue of the $p-$Laplace operator. After presenting the nonlinear analogue of the {\it Hong-Krahn-Szego inequality}, asserting that the disjoint unions of two equal balls minimize $\lambda_2$ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases $p=1$ and $p=\infty$ are considered as well.

Keywords: Nonlinear eigenvalue problems, Hong-Krahn-Szego inequality, Stability for eigenvalues