论文标题
Kohler-Jobin遇到Ehrhard：高斯主频率的尖锐下限，而高斯扭转刚度是通过重排固定的
Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements
论文作者
论文摘要
在本说明中，我们提供了Kohler-Jobin重排技术对高斯空间设置的改编。结果，我们证明了Kohler-Jobin对Pólya-Szegö的猜想的分辨率的高斯类似物：当固定A（凸）域的高斯扭转刚度时，高斯主频率被固定在半空间中。这种重排技术的核心是考虑给定功能的“修改”扭转刚度，并以特定方式将其图层重新排列到半个空间；瑞利商随着此过程而降低。 我们强调，在这里不期望高斯案与勒贝格案例的类比，因为除了某些软对称思想外，该论点还依赖于某些特殊功能的特性。这个类比确实存在的事实在某种程度上是一个奇迹。
In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Pólya-Szegö: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a "modified" torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.
