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An isoperimetric result for the fundamental frequency via domain derivative

The Faber-Krahn deficit $δλ$ of an open bounded set $Ω$ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $Ω$ and on the ball having same measure as $Ω$. 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 $δλ$ and the isoperimetric deficit $δP$ are vanishing quantities. It is known as well that, at least for convex sets, the ratio $\frac{δP}{δλ}$ 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 $δP$ goes to zero.