Eigenvalue problems in ^{8}: optimality conditions, duality, and relations with optimal transport
Eigenvalue problems in ^{8}: optimality conditions, duality, and relations with optimal transport
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^{8} 中的特征值问题：最优条件、对偶性以及与最优传输的关系
DOI：
10.1090/cams/11
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Bungert L
中科院分区：
文献类型：
--
作者：
Bungert L
在这篇文章中，我们刻画了与Rayleigh商$\left相关的特征值问题. {\|\nabla u\| _ {\mathrm {L}^\infty}}\middle/{\| u\| _\infty}\right.并将其与发散形式的偏微分方程联系起来，类似于已知的外特征值问题和拉普拉斯算子。相反，现有的方法，研究问题的限制问题，我们开发了一个新的框架，直接使用凸分析和几何测度理论分析的限制问题。为此，我们得到了一个新的精细刻画的Lipschitz常数泛函的次微分。我们证明了特征值问题的形式，其中和是非负测度，分别集中在其中最大，和是关于的切向梯度。最后，我们研究了一个对偶Rayleigh商，其极小化解决了与广义Kantorovich-Rubinstein范数相关的最优运输问题。我们的结果适用于Rayleigh商的所有驻点，包括无穷基态，无穷调和势，距离函数等，并推广了文献中的已知结果。引用
In this article we characterize theeigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\| _ {\mathrm {L}^\infty}}\middle/{\| u\| _\infty}\right. $ and relate it to a divergence-form PDE, similarly to what is known foreigenvalue problems and the-Laplacian for. Contrary to existing methods, which study-problems as limits of-problems for, we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a novel fine characterization of the subdifferential of the Lipschitz-constant-functional. We show that the eigenvalue problem takes the form, whereandare non-negative measures concentrated whererespectivelyare maximal, andis the tangential gradient ofwith respect to. Lastly, we investigate a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized Kantorovich–Rubinstein norm. Our results apply to all stationary points of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature. References