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 ≥ 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 (δP)/(δλ) is bounded by below by some positive constant (Brandolini et al., Arch Math (Basel) 94(4): 391–400, 2010; Payne andWeinberger, 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 δP goes to zero.
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机译：一个开放边界集Ω的Faber-Krahn赤字δλ是第一个Dirichlet Laplacian特征值在Ω和与Ω具有相同度量的球上获得的值之间的归一化间隙。对于任何给定的R〜N（N≥2）的开放边界集族，它们平滑地收敛到一个球上，众所周知δλ和等距赤字δP都消失了。众所周知，至少对于凸集，比率（δP）/（δλ）在下面受某个正常数的限制（Brandolini等人，Arch Math（Basel）94（4）：391-400， 2010; Payne and Weinberger，J Math Anal Appl 2：210-216，1961），在本说明中，使用形状导数技术，当δP趋于零时，我们提供了该比率的明确的最优下界。
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