Shape Optimization, Free Boundary Problems, and Geometric Measure Theory

优化形状，使其具有最佳的物理性能或最有效地利用材料，是应用数学中的一种基本问题，出现在电子元件、绝缘、空气动力学、成像、声学、制造以及液滴和晶体形成等物理过程中。在数学上，这种最优形状被解释为自由边界问题的解，这是一种广义微分方程，其中形状的边是要求解的未知数之一。自由边界问题是数学分析中的一个经典而又困难的话题，这个项目的目标是开发更健壮的工具来理解更广泛的这类方程的局部和全局特征。通过严格的近似格式、扰动下的稳定性分析和解的刚性定性性质，更好的数学理解可能会导致对应用问题的更智能和更安全的方法。这个项目为本科生、研究生和博士后研究人员提供了在具有重要工业应用的数学领域的培训机会。要考虑的具体主题包括多相或矢量伯努利问题、各种类型的两相抛物型自由边界问题、缺乏尺度不变性的间断半线性问题、非局部算子的自由边界以及传输问题。一种方法将重点放在定量单调公式方法上，并结合几何测量理论来证明具有较小刚性结构的问题的估计。另一种方法是为目前在已知方法范围之外的情况开发线性化论点，在这些情况下，切线对象对问题的局部逼近相对较差。该项目的一个主要焦点是新的和改进的技术，这些技术可能在各种情况下有用，而不是在个别问题上有用。这一奖项反映了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。

Optimizing a shape to have the best physical properties or make the most efficient use of a material is a basic type of problem in applied mathematics, appearing in the design of electronic components, insulation, aerodynamics, imaging, acoustics, manufacturing, as well as in physical processes like the formation of liquid drops and crystals. Mathematically, such optimal shapes are interpreted as solutions to free boundary problems, a kind of generalized differential equation where the edge of the shape is one of the unknowns being solved for. Free boundary problems are a classical but difficult topic in mathematical analysis, and the goal of this project is to develop more robust tools for understanding the local and global characteristics of wider classes of such equations. Better mathematical understanding may lead to smarter and safer approaches to the applied problems through rigorous approximation schemes, analysis of stability under perturbations, and rigid qualitative properties of solutions. This project offers training opportunities for undergraduate students, graduate students, and postdoctoral researchers, in a mathematical area with important industrial applications.The specific topics to be considered include multi-phase or vectorial Bernoulli problems, two-phase parabolic free boundary problems of various types, discontinuous semilinear problems lacking scale invariance, free boundaries for nonlocal operators, and transmission problems. One approach will focus on quantitative monotonicity formula methods combined with geometric measure theory to prove estimates for problems with little rigid structure. Another will be to develop linearization arguments for situations currently outside the scope of known approaches, where the tangent objects are relatively poor approximations for the problem locally. A major focus of the project is on novel and improved techniques which may be useful in a variety of contexts rather than on individual problems.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.