Textbook

The lectures we will follow the book

A. Petrosyan, H. Shahgholian, N. Ural'tseva, Regularity of free boundaries in obstacle type problems

[Front Matter]
[Chap 1]
[Chap 2] (Updated Sep 1)
[Chap 3] (Updated Sep 15)
[Chap 4] (Updated Oct 7)
[Chap 5]
[Chap 6]
[Chap 7]


More chapters will be available due course.

(Note: the PDFs are in a secured area. Login information will be provided in class)

Course Information

Schedule: TTh 10:30-11:45 in MATH 211
Instructor: Arshak Petrosyan
Office Hours: TTh 9:30 -10:30am, or by appointment, in MATH 610

Course Description: Free boundaries are apriori unknown sets, coming up in solutions of partial differential equations and variational problems. Typical examples are the interfaces and moving boundaries in problems on phase
transitions and fluid mechanics. Main questions of interest are the regularity (smoothness) of free boundaries and their structure.

A well-known (and well-studied) example is the obstacle problem of minimizing the energy of the membrane subject to remaining above a given obstacle: the free boundary is the boundary of the contact set. The objective in this course is to give an introduction to the theory of the regularity of the free boundaries in problems of the obstacle type, pioneered in the works of Luis Caffarelli, et al.

We are going to discuss classical and more recent methods in such problems, including the optimal regularity of solutions, monotonicity formulas, classification of global solutions, geometric and energy criterions for the regularity of the free boundary, singular points.

Prerequisite: MA64200 or the consent of the instructor (absolute minimum is MA52300).

Textbook:
A. Petrosyan, H. Shahgholian, N. Ural'tseva, Regularity of free boundaries in obstacle type problems, unpublished.