*Published Paper*

**Inserted:** 31 mar 2009

**Last Updated:** 10 feb 2015

**Journal:** J. Math. Anal. Appl.

**Year:** 2010

**Abstract:**

In this paper we prove that if $O_1$ and $O_2$ are close enough for the complementary Hausdorff distance and their boundaries satisfy some geometrical and topological conditions then the difference $

\lambda_1-\mu_1

$, where $\lambda_1$ (resp. $\mu_1$) is the first Dirichlet eigenvalue of the Laplacian in $O_1$ (resp. $O_2$), is controlled by the Lebesgue measure of the symetric difference between $O_1$ and $O_2$.

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