Time and Place: MWF 12:30pm–1:20pm REC 123
Instructor: Arshak Petrosyan
Office Hours: MWF 11:30am-12:30pm, or by appointment, in MATH 610
Course Description: Free boundary problems are boundary value problems for partial differential equations (PDEs) which are defined in a domain with an apriori unknown part of its boundary; this part is accordingly named a free boundary. A further quantitative condition must be then provided at the free boundary to exclude indeterminacy. Typical examples include interfaces between different phases or different types of media, moving boundaries, shocks and discontinuities, etc.
This course will serve as an introduction to the theory of regularity of free boundaries on the example of so-called obstacle-type problems, including the classical obstacle problem, a problem from potential theory, and the Signorini problem.
We are going to discuss classical as well as more recent methods in such problems, including the optimal regularity of solutions, several types of monotonicity formulas (Alt-Caffarelli-Friedman, Almgren, Weiss, Monneau), classification of global solutions, criteria for the regularity of the free boundary, structure of the singular set.
[PSU] A. Petrosyan, H. Shahgholian, N. Uraltseva, Regularity of free boundaries in obstacle-type problems, Graduate Studies in Mathematics 136, American Mathematical Society, Providence, RI, 2012