奥恩斯坦-乌伦贝克算子第一特征值的布伦-闵科夫斯基不等式和相关特征函数的对数凹性

arXiv - MATH - Metric Geometry Pub Date : 2024-07-31 DOI:arxiv-2407.21354

Andrea Colesanti, Elisa Francini, Galyna Livshyts, Paolo Salani

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摘要

我们证明了奥恩斯坦-乌伦贝克算子 $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ 作为域的函数，其第一个（非微小）狄利克特特征值相对于明考斯基加法是凸的，并且我们描述了一些凸集类别中的相等情况。我们还证明，如果域是凸的，则相应的（正）特征函数是对数凹的。

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The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction

We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski addition, and we characterize the equality cases in some classes of convex sets. We also prove that the corresponding (positive) eigenfunction is log-concave if the domain is convex.

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arXiv - MATH - Metric Geometry

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