Geometric Variational Problems and Rearrangement Inequalities

这项建议提出了一个关于非局部几何变分问题的研究议程。在数学物理和几何中，许多地方都出现了非局部泛函。例如，库仑能量出现在静电学、天体力学和量子力学中，路径积分采用多重卷积的形式，粒子之间的相互作用在统计力学中用更复杂的碰撞核来描述。一个重要的结果是它们的极值，也就是所谓的“基态”，通常是旋转对称的。这些不等式可以看作是凸几何的Brunn-Minkowski不等式的泛函版本，进而推广到等周不等式。*一个关键问题是非局部泛函的几何稳定性-一个具有统计力学中连续统极限的基本含义的问题。几何稳定性的结果是，对于许多几何泛函，已经建立了“赤字”(泛函与其最佳值的偏差)来控制某种“不对称性”(到优化器流形的距离)的度量。涉及卷积的非局部泛函鲜为人知，但在过去的两年里有了显著的进展。*另一个基本问题涉及Riesz重排不等式在高维上的稳定性，这将意味着Brunn-Minkowski不等式在非凸情形下的新的稳定性结果。Riesz不等式对重积分的最终推广是BrasCamp-Lieb-Luttinger不等式。要稳定BLL不平等，需要首先对平等案件进行分类，这是一个长期存在的问题，最近也取得了很大进展。还将讨论Riesz重排不等式在球体上的扩展。*本提案中的其他主题是更简单对称化序列对对称递减重排的逼近，以及色散管理孤子的对称性和变分特征。*拟议的工作旨在解决上述问题，并开发可更广泛应用的分析工具。**

This proposal presents a research agenda on non-local geometric variational problems. Non-local functionals arise in many places in Mathematical Physics and Geometry. For example, the Coulomb energy appears in Electrostatics, Celestial Mechanics and Quantum Mechanics, path integrals take the form of multiple convolutions, and interactions of particles are described by more complicated collision kernels in Statistical Mechanics. ***Many of these functionals satisfy geometric inequalities. An important consequence is that their extremals, known as "ground states", are often rotationally symmetric. These inequalities can be viewed as functional versions of the Brunn-Minkowski inequality of convex geometry, and, by extension, the isoperimetric inequality. ***A key problem is the geometric stability of nonlocal functionals --- a question with fundamental implications for continuum limits in Statistical Mechanics. Geometric stability results where a "deficit" (the deviation of a functional from its optimal value) controls some measure of "asymmetry" (the distance from the manifold of optimizers) have been established for many geometric functionals. Little is known for non-local functionals that involve convolutions, but there has been notable progress in the last two years. ***Another fundamental problem concerns the stability of Riesz' rearrangement inequality in higher dimensions, which would imply new stability results for the Brunn-Minkowski inequality in the non-convex case. The ultimate generalization of Riesz' inequality to multiple integrals is the Brascamp-Lieb-Luttinger inequality. Stability for the BLL inequality will require to first classify the equality cases, a long-standing problem that has also seen very recent progress. Exensions of Riesz' rearrangement inequality to spheres will also be considered.***Other topics topics in this proposal are the approximation of the symmetric decreasing rearrangement by sequences of simpler symmetrization, and the symmetry and variational characterization of dispersion-managed solitons.***The proposed work seeks to address the questions described above, and to develop analytical tools that can be applied more broadly.**