Geometric Variational Problems and Rearrangement Inequalities

几何变分问题和重排不等式

• 批准号: RGPIN-2015-05436
• 负责人: Burchard, Almut
• 金额: $ 1.46万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=646522
• 关键词: Geometric; Variational; Problems; Rearrangement; Inequalities

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.**

该提案提出了数学物理和几何中许多地方出现的非局部泛函问题的研究议程，例如，库仑能量出现在静电学、天体力学和量子力学中，路径积分采用以下形式。多重卷积和粒子的相互作用由统计力学中更复杂的碰撞核来描述***这些函数中的许多满足几何不等式。极值，称为“基态”，通常是旋转对称的。这些不等式可以被视为凸几何的 Brunn-Minkowski 不等式的函数版本，并且推而广之，等周不等式的一个关键问题是几何。非局部泛函的稳定性——一个对统计力学中的连续极限具有根本影响的问题，其中“赤字”（泛函与其最佳值的偏差）控制了某种程度的几何稳定性。对于许多几何泛函来说，“不对称”（与优化器流形的距离）已经被确定。对于涉及卷积的非局部泛函，人们知之甚少，但在过去两年中，另一个基本问题取得了显着的进展。涉及 Riesz 重排不等式在更高维度上的稳定性，这意味着非凸情况下 Brunn-Minkowski 不等式的新稳定性结果 Riesz 不等式对多重积分的最终推广是Brascamp-Lieb-Luttinger 不等式。BLL 不等式的稳定性需要首先对等式情况进行分类，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 重排不等式扩展到球体。***本提案中的其他主题是通过更简单的对称化序列进行对称递减重排的近似，以及色散管理孤子的对称性和变分表征。***拟议的工作寻求解决上述问题，并开发可以更广泛应用的分析工具。**