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Regularity of Free Boundaries in Obstacle Type Problems - 4

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Free Boundary Problems and Related Topics

The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).

Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions.

Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas.

Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points.

Lecture 4. In this lecture, we will show how to prove the regularity of the “regular set” and obtain a structural theorem on the singular set.

Suggested reading:

[1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN : 978-0-8218-8794-3

[2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383—402.

[3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23—50.

[4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269—292.

[5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827—869.

[6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415461.

[7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213

This talk is part of the Isaac Newton Institute Seminar Series series.

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