利用区域导数求基频的一个等周结果
An isoperimetric result for the fundamental frequency via domain derivative
摘要：
开有有界集$Ω$的Faber-Krahn亏格$δλ$是第一狄利克雷拉普拉斯特征值在$Ω$上和在具有与$Ω$相同度量的球上实现的值之间的归一化间隙。对于任意给定的光滑收敛于球的开有界$\R^N$（$N\ge 2$）集族，$δλ$和等周亏量$δP$都是零量.众所周知，至少对于凸集，比率$\frac{δP}{δλ}$由某个正常数限定（见\cite{BNT,PW}），在本说明中，使用形状导数的技术，我们提供了当$δP$趋于零时这样的比率的显式最优下界。
英文摘要：
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.
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