Lecture notes

Lecture 1 Catalog of Problems
Lecture 2 Mathematical Formulation
Lecture 3 Existence
Lecture 4 Optimal Regularity
Lecture 5 ACF-type Monotonicity Formulas
Lecture 6 Optimal Regularity (Continued)
Lecture 7 Optimal Regularity (Continued)
Lecture 8 Nondegeneracy and (n-1)-Hausdorff measure
Lecture 9 Normalized solutions, Rescalings and Blowups
Lecture 10 Weiss's Monotonicity Formulas
Lecture 11 Classification of Free Boundary Points
Lecture 12 First Results on the Regularity of the Free Boundary: A
Lecture 13 First Results on the Regularity of the Free Boundary: B
Lecture 14 First Results on the Regularity of the Free Boundary: C
Lecture 15 Global Solutions
Lecture 16 Approximation by Global Solutions
Lecture 17 Lipschitz Regularity of the Free Boundary
Lecture 18 C ^1,\alpha Regularity of the Free Boundary: Problems A, B updated
Lecture 19 C ¹ Regularity of the Free Boundary: Problems C
Lecture 20 Higher Regularity: Problems A and B draft
Lecture 21 The Singular Set draft

References

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Course Information

Schedule: TTh 12:00-1:15pm 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: MA 642 or the consent of the instructor (absolute minimum is MA523).

Textbook:
Lecture Notes