Shape Optimization, Free Boundary Problems, and Geometric Measure Theory

优化形状以具有最佳的物理特性或最有效地利用材料是应用数学中的一个基本类型的问题，出现在电子元件，绝缘，空气动力学，成像，声学，制造的设计以及物理过程中，如液滴和晶体的形成。在数学上，这种最佳形状被解释为自由边界问题的解，自由边界问题是一种广义微分方程，其中形状的边缘是被求解的未知数之一。自由边界问题是数学分析中一个经典但困难的课题，本项目的目标是开发更强大的工具来理解更广泛类别的此类方程的局部和全局特征。更好的数学理解可能会导致更聪明和更安全的方法来解决应用问题，通过严格的近似方案，扰动下的稳定性分析，以及解决方案的严格定性性质。本项目为本科生、研究生和博士后研究人员提供具有重要工业应用的数学领域的培训机会。具体研究课题包括多相或向量伯努利问题、各种类型的两相抛物自由边界问题、缺乏标度不变性的不连续半线性问题、非局部算子的自由边界和传输问题。一种方法将集中在定量单调公式方法结合几何测度理论，以证明估计的问题很少刚性结构。另一个将是开发线性化参数的情况下，目前已知的方法的范围之外，切线对象是相对较差的近似的问题局部。该项目的一个主要重点是新颖的和改进的技术，这可能是在各种情况下，而不是在个别问题上有用。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.