Interacting Free Boundaries in the Calculus of Variations

该项目的目标是进一步开发研究非线性偏微分方程组、变分和自由边界问题中基本问题所需的数学工具。这些问题起源于弹性力学、流体力学或成本优化等应用科学，并与几何等纯数学领域相关。这些模型的理论方面对于更好地理解基本物理现象很重要，可能对更大的科学界有用。该项目为学生提供研究培训机会，其成果将广泛传播给不同的受众。首席调查员(PI)将研究与经典障碍问题有关的几个问题。其中之一涉及多层膜问题，它描述了N个弹性膜相互接触的相互作用。在数学上，它可以看作是具有N-1个相互作用的自由边界的障碍问题的耦合系统。另一个要解决的问题涉及薄障碍问题的某些爆破锥的唯一性，该问题也被称为西诺里尼问题。该项目的一个重要部分考虑了二次成本的最优运输问题，该成本的密度在支撑点的边界上为零。这个问题等价于一个高度退化的Monge-Ampère方程的第二边值问题。PI还旨在了解在二维最简单的多凸能量之一的变分中最小化地图的内在规律性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of the project is to develop further the mathematical tools needed in the study of fundamental questions in nonlinear partial differential equations, calculus of variations, and free boundary problems. These questions have their origins in the applied sciences like elasticity, fluid mechanics, or cost optimization, and are relevant in areas of pure mathematics like geometry. The theoretical aspects of these models are important towards a better understanding of basic physical phenomena and could be useful for the larger scientific community. The project provides research training opportunities for students and its outcomes will be broadly disseminated to diverse audiences. The principal investigator (PI) will study several questions related to the classical obstacle problem. One of them concerns the multiple membrane problem, which describes the interaction of N elastic membranes in contact with each other. Mathematically, it can be viewed as a coupled system of obstacle problems with N-1 interacting free boundaries. Another question to be addressed deals with the uniqueness of certain blow-up cones for the thin obstacle problem, also known as the Signorini problem. An important part of the project considers the optimal transportation problem for quadratic costs with densities that vanish on the boundary of their support. This problem is equivalent to the second boundary value problem for a highly degenerate Monge-Ampère equation. The PI also aims to understand the interior regularity of minimizing maps in the calculus of variations for one of the simplest polyconvex energies in two dimensions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.