An isoperimetric result for the fundamental frequency via domain derivative

Abstract

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 ${\mathbb{R}}^N$ ($N\ge 2$) 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.

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

References in corpus (1)

• A Selection Principle for the Sharp Quantitative Isoperimetric Inequality