Stability, regularity and symmetry issues in geometric variational problems

几何变分问题中的稳定性、正则性和对称性问题

• 批准号: 1265910
• 负责人: Francesco Maggi
• 金额: $ 23.85万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265910&HistoricalAwards=false
• 关键词: Stability; regularity; symmetry; issues; geometric

This project is aimed at investigating stability, regularity, and symmetry issues in various geometric variational problems, and at exploiting the corresponding results in the effective description of equilibrium configurations of surface tension driven physical systems. For example, the stability theory for isoperimetric-type problems recently established by the PI and his collaborators, beyond its intrinsic mathematical interest, has revealed useful in studying minimizers of classical sharp interface energies, of the Gates-Lebowitz-Penrose energy, the Ohta-Kawasaki energy and variants, and in cavitation models in Non-linear Elasticity. This research program will achieve significant improvements in the stability theory for geometric inequalities, broadening the reach of the theory to include new and challenging situations, and opening new spaces for further applications to problems of applied interest. Specific stability issues considered in this project arise in the study of minimizing clusters, Plateau's problem, and isoperimetric problems in arbitrary codimension and in Gauss space. The project will also advance the mathematical theory of capillarity problems, by addressing regularity issues related to the validity of Young's law, and by providing a quantitative description of geometric properties of equilibrium configurations. Finally, the project aims to some conclusive developments in symmetrization theory, by characterizing, from a geometric viewpoint, those situations where equality cases in symmetrization inequalities imply symmetry of minimizers. This project aims to advance the mathematical understanding of geometric variational problems. Geometric variational problems play a fundamental role in the mathematical modeling of Nature, and in particular, in our quantitative and qualitative understanding of equilibrium states of physical systems. Despite their ubiquitous interest, and the very considerable amount of work that has been devoted to their study both from mathematicians, physicists, and engineers, several questions remain unanswered, or just partially understood, due to the mathematical challenges they arise. In turn, geometric variational problems play also a pivotal role in various area of Mathematics, including Analysis, Probability Theory, and Geometry. Several important contributions to the stability theory for geometric variational problems has been obtained in recent years by the PI and his collaborators, with applications to the effective description of equilibrium states of physical systems, and with the introduction of new mathematical ideas and techniques. An important part of this project will consist in the training of graduate students on these new mathematical developments.

该项目旨在研究各种几何变异问题中的稳定性，规律性和对称性问题，并在有效描述表面张力驱动物理系统的平衡构中利用相应的结果。例如，PI及其合作者最近在其内在的数学兴趣之外建立的等值型问题的稳定理论揭示了对gates-lebowitz-penrose Energy的经典尖锐界面能量的最小化，OHTA---ohta-ohta----川崎能量和变体，以及非线性弹药模型中的。该研究计划将在几何不平等的稳定理论上取得重大改进，扩大理论的覆盖范围，包括新的和具有挑战性的情况，并为应用于应用的问题的进一步应用开放新空间。该项目中考虑的特定稳定性问题是在最小化集群，高原问题以及任意编成和高斯空间中的等速度问题的研究中出现的。该项目还将通过解决与Young定律的有效性相关的规律性问题，并提供对平衡配置的几何特性的定量描述，从而推进毛细血管问题的数学理论。最后，该项目的目的是通过从几何学角度表征对称理论的一些结论性发展，这些情况是在对称不平等中的平等案例暗示着最小化的对称性的情况。该项目旨在提高对几何变异问题的数学理解。几何变异问题在自然的数学建模中起着基本作用，尤其是我们对物理系统平衡状态的定量和定性理解。尽管他们的兴趣无处不在，并且从数学家，物理学家和工程师那里致力于研究的大量工作，但由于他们产生的数学挑战，几个问题仍未得到解决或部分理解。反过来，几何变异问题在数学的各个领域（包括分析，概率理论和几何学）中也起着关键作用。近年来，PI及其合作者对稳定理论对几何变异问题的稳定理论做出了一些重要贡献，并应用了物理系统平衡状态的有效描述，并引入了新的数学思想和技术。该项目的重要部分将包括对这些新数学发展的研究生培训。