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SUMMARY:Obstacle type problems : An overview and some recent results - Hen
 rik Shahgholian (KTH\, Stockholm)
DTSTART:20100211T150000Z
DTEND:20100211T160000Z
UID:TALK22712@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\n\\Delta u = \\chi_{u>0}\,  \\qquad 
 u \\geq 0\n\nto embrace a more general (two-phase) form\n\n\\Delta u = \\l
 ambda_+ \\chi_{u>0} -  \\lambda_- \\chi_{u<0}\n\nwith $\\lambda_\\pm$ reas
 onably smooth functions (down to Dini continuous).\n\nAstonishing results 
 of Yuval Peres and his collaborators has shown remarkable relationships be
 tween 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 investigation in
  the last 10 years\, and interesting discoveries has been made about the b
 ehavior of the free boundaries in such problems. Existing methods has so f
 ar only allowed us to consider $\\lambda_\\pm >0$.\n\nThe above problem ch
 anges drastically if one allows $\\lambda_\\pm$ to have the incorrect sign
  (that appears in composite membrane problem)!\n In part of my talk  I wil
 l focus on the simple _unstable_ case\n\n\\Delta u = - \\chi_{u>0}\n\nand 
 present very recent results (Andersson\, Sh.\, Weiss) that classifies the 
 set of singular points ($\\{u=\\nabla u =0\\}$) for the above problem. The
  techniques developed recently by our team also shows an unorthodox approa
 ch to such problems\, as the classical technique fails.\n\nAt the end of m
 y talk I will explain the technique in a heuristic way.\n
LOCATION:MR14\, CMS
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