Stability, regularity and symmetry issues in geometric variational problems

本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，并在表面张力驱动的物理系统的平衡构型的有效描述中利用相应的结果。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量的最小化，Gates-Lebowitz-Penrose能量，Ohta-Kawasaki能量及其变体，以及非线性弹性中的空化模型中显示出有用。本研究项目将在几何不等式稳定性理论方面取得重大进展，拓宽了几何不等式稳定性理论的研究范围，使其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了新的空间。在这个项目中考虑的具体稳定性问题出现在最小化簇的研究中，高原问题，以及在任意余维和高斯空间中的等周问题。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，并通过提供平衡构型几何特性的定量描述，推进毛细问题的数学理论。最后，该项目旨在通过从几何角度描述对称不等式中的相等情况意味着最小值对称的情况，从而在对称理论中取得一些结论性的发展。这个项目旨在促进对几何变分问题的数学理解。几何变分问题在自然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状态的定量和定性理解中。尽管对它们的兴趣无处不在，数学家、物理学家和工程师也为它们的研究投入了大量的工作，但由于它们在数学上的挑战，有几个问题仍然没有得到解答，或者只是部分理解。反过来，几何变分问题在数学的各个领域也起着关键作用，包括分析，概率论和几何。近年来，PI和他的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，应用于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。这个项目的一个重要部分是训练研究生掌握这些新的数学发展。

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.