*Published Paper*

**Inserted:** 12 sep 2011

**Last Updated:** 12 feb 2013

**Journal:** Manuscripta Math.

**Pages:** 18

**Year:** 2012

**Abstract:**

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

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