Unique continuation and the regularity of elliptic PDEs and generalized minimal submanifolds

该奖项支持椭圆型偏微分方程解的正则性和广义极小子流形的正则性的研究。椭圆微分方程支配着各种物理现象的平衡态，例如，那些由自然能量泛函的最小化问题引起的平衡态。例子包括自由悬挂的桥梁的形状，肥皂泡的形状和鼓声。椭圆微分方程也被用来量化物理对象弯曲或扭曲的程度，在几何和拓扑学中具有深远的影响和应用。拟议的研究重点是这些方程的解的规律性。要解决的问题包括以下几点：非光滑点（奇点）存在吗？奇点的集合能有多大？解在奇点附近的行为是什么？有没有可能通过扰动底层环境来消除奇点？该项目还将通过个人指导和组织关于几何分析和几何测量理论的指导性学习研讨会，为研究生的专业发展提供机会。首先，首席研究员将研究椭圆型偏微分方程解的唯一延拓，重点是这些解的奇异集的大小和结构的定量估计。第二个要考虑的主题是正则性理论的广义极小子流形（一个广义的概念，光滑子流形出现的临界点的面积功能下的局部变形）。特别是，首席研究员将研究分支奇异点的内部以及在边界的广义极小子流形，根据面积最小化或稳定性假设。后一个主题的研究可以被视为椭圆方程的定量唯一延拓的非线性模拟，需要整合几何测度理论、偏微分方程和几何分析的思想。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports research on the regularity of solutions to elliptic partial differential equations and regularity of generalized minimal submanifolds. Elliptic differential equations govern the equilibrium configurations of various physical phenomena, for instance, those arising from minimization problems for natural energy functionals. Examples include the shape of free-hanging bridges, the shape of soap bubbles, and the sound of drums. Elliptic differential equations are also used to quantify the degree to which physical objects are bent or distorted, with far-reaching implications and applications in geometry and topology. The proposed research focuses on the regularity of solutions to such equations. Questions to be addressed include the following: Do non-smooth points (singularities) exist? How large can the set of singularities be? What is the behavior of the solution near a singularity? Is it possible to perturb the underlying environment in order to eliminate the singularity? The project will also provide opportunities for the professional development of graduate students, both via individual mentoring and via the organization of a directed learning seminar on geometric analysis and geometric measure theory.The mathematical objectives of the project are twofold. First, the principal investigator will study unique continuation for solutions to elliptic partial differential equations, with a focus on quantitative estimates on the size and structure of the singular set of these solutions. A second topic for consideration is the regularity theory for generalized minimal submanifolds (a generalized notion of smooth submanifolds which arise as critical points for the area functional under local deformations). In particular, the principal investigator will study branch singular points in the interior as well as at the boundary of a generalized minimal submanifold, under an area-minimizing or stability assumption. Research on the latter topic, which can be viewed as a non-linear analogue of quantitative unique continuation for elliptic equations, requires the integration of ideas from geometric measure theory, partial differential equations and geometric analysis.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.