An isoperimetric result for the fundamental frequency via domain derivative

Details An isoperimetric result for the fundamental frequency via domain derivative An isoperimetric result for the fundamental frequency via domain derivative

2012-01-25

arxiv.org/abs/1201.5328v2

Mathematics

Optimization and Control

12 pages, minor corrections, the proof of Lemma 2.3 was shortened and clarified

Scientific paper

The Faber-Krahn deficit $\delta\lambda$ of an open bounded set $\Omega$ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $\Omega$ and on the ball having same measure as $\Omega$. For any given family of open bounded sets of $\R^N$ ($N\ge 2$) smoothly converging to a ball, it is well known that both $\delta\lambda$ and the isoperimetric deficit $\delta P$ are vanishing quantities. It is known as well that, at least for convex sets, the ratio $\frac{\delta P}{\delta \lambda}$ is bounded by below by some positive constant (see \cite{BNT,PW}), and in this note, using the technique of the shape derivative, we provide the explicit optimal lower bound of such a ratio as $\delta P$ goes to zero.

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