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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.

An isoperimetric result for the fundamental frequency via domain derivative