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SUMMARY:Regularity of Free Boundaries in Obstacle Type Problems - 2 - Shah
 gholian\, H (KTH - Royal Institute of Technology)
DTSTART:20140109T100000Z
DTEND:20140109T110000Z
UID:TALK49669@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:The aim of these lectures is to give an introduction to the re
 gularity theory of free \nboundaries related to the obstacle problem. Besi
 des the classical obstacle problem\, we will \nconsider the problem on har
 monic continuation of Newtonian potentials\, the thin obstacle problem\, a
 nd their parabolic counterparts (as much as the time permits).\n\nLecture 
 1. In this lecture\, we will introduce the problems we will be working on 
 and discuss \ninitial regularity results for the solutions.\n\nLecture 2. 
 In this lecture\, we will discuss the optimal regularity of solutions and 
 give proofs by using monotonicity formulas.\n\nLecture 3. In this lecture\
 , we will consider the blowups of the solutions at free boundary points. W
 e will then classify the blowups and thereby classify the free boundary po
 ints.\n\nLecture 4. In this lecture\, we will show how to prove the regula
 rity of the "regular set" and \nobtain a structural theorem on the singula
 r set.\n\nSuggested reading:\n\n[1] Petrosyan\, Arshak \;  Shahgholian\, H
 enrik\;  Uraltseva\, Nina . Regularity of free \nboundaries in obstacle-ty
 pe problems. Graduate Studies in Mathematics\, 136. American \nMathematica
 l Society\, Providence\, RI\,  2012. x+221 pp. ISBN: 978-0-8218-8794-3\n\n
 [2] Caffarelli\, L. A.  The obstacle problem revisited. J. Fourier Anal. A
 ppl.  4  (1998)\,  no. 4-5\, 383--402.\n\n[3] Weiss\, Georg S.  A homogene
 ity improvement approach to the obstacle problem. Invent. Math.  138  (199
 9)\,  no. 1\, 23--50.\n\n[4] Caffarelli\, Luis A. \;  Karp\, Lavi \;  Shah
 gholian\, Henrik . Regularity of a free \nboundary with application to the
  Pompeiu   problem. Ann. of Math. (2)  151  (2000)\,  no. 1\, 269--292.\n\
 n[5] Caffarelli\, Luis \;  Petrosyan\, Arshak \;  Shahgholian\, Henrik . R
 egularity of a free \nboundary in parabolic potential theory. J. Amer. Mat
 h. Soc.  17  (2004)\,  no. 4\, 827--869.\n\n[6] Garofalo\, Nicola \;  Petr
 osyan\, Arshak . Some new monotonicity formulas and the singular set in th
 e lower dimensional obstacle problem. Invent. Math.  177  (2009)\,  no. 2\
 , 415461.\n\n[7] Danielli\, Donatella \; Garofalo\, Nicola \; Petrosyan\, 
 Arshak \; To\, Tung . Optimal regularity and the free boundary in the para
 bolic Signorini problem. arXiv:1306.5213\n
LOCATION:Seminar Room 1\, Newton Institute
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