Regularity Problems in Elliptic Equations

在物理科学以及经济学、金融学和计算机科学中，基本定律都是用偏微分方程(PDE)写成的。对于所研究的量，这些偏微分方程揭示了不同方向的变化率之间的关系，并决定了量的性质和演化。偏微分方程组研究的核心是解的一个特殊特征，称为正则性。从理论上讲，正则性对于证明模型是必要的，并且是解决方案的其他属性(存在性、唯一性和长期行为)的垫脚石。在实际应用中，模型的应用涉及到偏微分方程组的数值解，为了得到合理的结果，需要了解的稳定性，这往往是正则性的结果。广义地说，看起来粗糙的物体如果满足椭圆方程或约束条件，往往会变成规则的。研究人员将在这一一般哲学下研究三个具体现象：自由界面问题、最优传输图和自由边界问题。该项目的成果将在计算机视觉、数据挖掘、机器学习、材料科学和生物学等其他领域得到广泛的应用。第一个项目将研究电场下液滴的界面。目标是表明这些界面在低空间维度上是平滑的。这与物理学文献中的一个重要猜想有关。第二个项目是关于欧氏空间中凸域之间的最优传输映射。这个计划是为了表明，在没有任何关于域的任何额外假设的情况下，这些地图可以非常规则地“一直到边界”。第三个项目是关于自由边界问题，特别是涉及多个相互作用的自由边界的问题。在这三个项目中，都需要新的技术来整合几何信息。这些技术将有助于解决目前技术遥不可及的广泛问题。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。

In the physical sciences, as well as economics, finance, and computer science, the fundamental laws are written in terms of partial differential equations (PDEs). For the quantity under study, these PDEs reveal the relation between the rates of change in different directions, and dictate the properties and evolution of the quantity. At the heart of the study of PDEs is a particular characteristic of solutions known as regularity. Theoretically, regularity is necessary to justify the models, and is the stepping stone for other properties of the solution (existence, uniqueness, and long term behavior). In practice, applications of the models involve numerical solutions of the PDEs and to obtain reasonable results requires stability of the solutions, which is often a consequence of regularity. Broadly speaking, objects that might seem rough often turn out to be regular if they satisfy an elliptic equation or constraint. The investigator will investigate three specific phenomena under this general philosophy: free interface problems, optimal transport maps, and free boundary problems. The results of this project will have numerous applications in other fields such as computer vision, data mining, machine learning, material sciences and biology.The first project will investigate interfaces arising in the study of liquid drops exposed to electric fields. The goal is to show that these interfaces are smooth in low spatial dimensions. This is related to an important conjecture in the physics literature. The second project concerns the optimal transport map between convex domains in Euclidean spaces. The plan is to show that without any extra assumptions on the domains, these maps can be very regular "up to the boundary". The third project is about free boundary problems, in particular problems involving multiple interacting free boundaries. In all three projects, new techniques are needed to incorporate geometric information. These techniques will be useful in addressing a wide range of problems that have been so far out of reach using current techniques.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.