An isoperimetric result for the fundamental frequency via domain derivative

TL;DR

This paper establishes an explicit optimal lower bound for the ratio of the isoperimetric deficit to the Faber-Krahn deficit for convex domains approaching a ball, using shape derivative techniques.

Contribution

It provides the first explicit optimal lower bound for the ratio of these deficits as the domain converges to a ball, advancing understanding of shape optimization.

Findings

Derived the explicit optimal lower bound for the ratio of deficits.

Showed the ratio remains bounded below by a positive constant.

Applied shape derivative techniques to analyze domain perturbations.

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 $\R^N$ ($N\ge 2$) 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.