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SUMMARY:Obstacle type problems : An overview and some recent results - Hen
 rik Shagolian (KTH\, Stockholm)
DTSTART:20100211T150000Z
DTEND:20100211T160000Z
UID:TALK22711@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:In this talk I will present recent developments of the obstacl
 e type problems\, with various applications ranging from Industry to Finan
 ce\, local to nonlocal operators\, and one to multi-phases.\nThe  theory h
 as evolved from a single equation\n$$\n\\Delta u = \\chi_{u>0}\,  \\qquad 
 u \\geq 0\n$$\nto embrace a more general (two-phase) form\n$$\n\\Delta u =
  \\lambda_+ \\chi_{u>0} -  \\lambda_- \\chi_{u<0}\n$$\nwith $\\lambda_\\pm
 $ reasonably smooth functions (down to Dini continuous).\n\nAstonishing re
 sults of Yuval Peres and his collaborators has shown remarkable relationsh
 ips between obstacle problem and various forms of random walks\, including
  Smash sum of Diaconis-Fulton (Lattice sets)\, and there is more to come.\
 n\nThe two-phase form (and its multi-phase form) has been under investigat
 ion in the last 10 years\, and interesting discoveries has been made about
  the behavior of the free boundaries in such problems. Existing methods ha
 s so far only allowed us to consider $\\lambda_\\pm >0$.\n\nThe above prob
 lem changes drastically if one allows $\\lambda_\\pm$ to have the incorrec
 t sign (that appears in composite membrane problem)!\n In part of my talk 
  I will focus on the simple _unstable_ case\n$$\n\\Delta u = - \\chi_{u>0}
 \n$$\nand present very recent results (Andersson\, Sh.\, Weiss) that class
 ifies the set of singular points ($\\{u=\\nabla u =0\\}$) for the above pr
 oblem. The techniques developed recently by our team also shows an unortho
 dox approach to such problems\, as the classical technique fails.\n\n At t
 he end of my talk I will explain the technique in a heuristic way.\n
LOCATION:MR14\, CMS
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