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分问题中的稳定性，规律性和对称性问题，并利用相应的结果有效描述表面张力驱动的物理系统的平衡构型。例如，PI和他的合作者最近建立的等周型问题的稳定性理论，超越了其固有的数学兴趣，在研究经典尖锐界面能，Gates-Lebowitz-Penrose能量，Ohta-Kawasaki能量和变体的极小化，以及非线性弹性中的空化模型中显示出有用。该研究计划将实现几何不等式的稳定性理论的显着改进，扩大理论的范围，包括新的和具有挑战性的情况，并为进一步应用于应用感兴趣的问题开辟新的空间。在这个项目中考虑的具体稳定性问题出现在研究最小化集群，高原的问题，在任意余维和高斯空间的等周问题。该项目还将通过解决与杨氏定律有效性相关的规律性问题，并通过提供平衡配置几何特性的定量描述，推进毛细作用问题的数学理论。最后，该项目旨在对称化理论的一些结论性的发展，通过表征，从几何的角度来看，这些情况下，在对称化不等式中的平等情况意味着极小的对称性。这个项目的目的是促进几何变分问题的数学理解。几何变分问题在自然界的数学建模中起着重要的作用，特别是在我们对物理系统平衡态的定量和定性理解中。尽管他们无处不在的兴趣，以及大量的工作，一直致力于他们的研究都从数学家，物理学家和工程师，一些问题仍然没有答案，或只是部分理解，由于数学的挑战，他们出现。反过来，几何变分问题在数学的各个领域，包括分析，概率论和几何中也起着举足轻重的作用。近年来，PI及其合作者对几何变分问题的稳定性理论做出了一些重要贡献，并应用于物理系统平衡态的有效描述，以及引入新的数学思想和技术。该项目的一个重要部分将包括对研究生进行这些新数学发展的培训。