Geometric Variational Problems and Rearrangement Inequalities

提出了非局部几何变分问题的研究方向。非局部泛函在数学、物理和几何中有广泛的应用。例如，库仑能出现在静电学、天体力学和量子力学中，路径积分采用多重卷积的形式，粒子的相互作用在统计力学中用更复杂的碰撞核来描述。这些泛函中有许多满足几何不等式。一个重要的结果是它们的极值，即所谓的“基态”，通常是旋转对称的。这些不等式可以看作是凸几何的布伦-闵可夫斯基不等式的泛函版本，并通过推广，可以看作是等周不等式。***一个关键问题是非局部泛函的几何稳定性——这是一个对统计力学中连续体极限具有基本含义的问题。在许多几何泛函中，“缺陷”（泛函与其最优值的偏差）控制某种“不对称性”（与优化器流形的距离）的几何稳定性结果已经建立起来。涉及卷积的非局部泛函鲜为人知，但在过去两年中已经取得了显著进展。***另一个基本问题涉及到Riesz重排不等式在高维中的稳定性，这将意味着Brunn-Minkowski不等式在非凸情况下的新的稳定性结果。Riesz不等式对多重积分的最终推广是Brascamp-Lieb-Luttinger不等式。为了稳定BLL不平等，首先需要对平等情况进行分类，这是一个长期存在的问题，最近也取得了进展。还将考虑将Riesz的重排不平等扩展到领域。***本提案的其他主题是通过更简单的对称化序列逼近对称递减重排，以及色散管理孤子的对称性和变分特性。***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.**