Shape Optimization, Free Boundary Problems, and Geometric Measure Theory

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.