An isoperimetric result for the fundamental frequency via domain derivative

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 (Brandolini et al., Arch Math (Basel) 94(4): 391–400, 2010; Payne and Weinberger, J Math Anal Appl 2:210–216, 1961), 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.