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CATEGORIES:Applied and Computational Analysis
SUMMARY:Obstacle type problems : An overview and some rece
 nt results - Henrik Shagolian (KTH\, Stockholm)
DTSTART;TZID=Europe/London:20100211T150000
DTEND;TZID=Europe/London:20100211T160000
UID:TALK22711AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/22711
DESCRIPTION:In this talk I will present recent developments of
  the obstacle type problems\, with various applica
 tions ranging from Industry to Finance\, local to 
 nonlocal operators\, and one to multi-phases.\nThe
   theory has evolved from a single equation\n$$\n\
 \Delta u = \\chi_{u>0}\,  \\qquad u \\geq 0\n$$\nt
 o embrace a more general (two-phase) form\n$$\n\\D
 elta u = \\lambda_+ \\chi_{u>0} -  \\lambda_- \\ch
 i_{u<0}\n$$\nwith $\\lambda_\\pm$ reasonably smoot
 h functions (down to Dini continuous).\n\nAstonish
 ing results of Yuval Peres and his collaborators h
 as shown remarkable relationships between obstacle
  problem and various forms of random walks\, inclu
 ding 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 investig
 ation in the last 10 years\, and interesting disco
 veries has been made about the behavior of the fre
 e boundaries in such problems. Existing methods ha
 s so far only allowed us to consider $\\lambda_\\p
 m >0$.\n\nThe above problem changes drastically if
  one allows $\\lambda_\\pm$ to have the incorrect 
 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 classifies the set of singular poin
 ts ($\\{u=\\nabla u =0\\}$) for the above problem.
  The techniques developed recently by our team als
 o shows an unorthodox approach to such problems\, 
 as the classical technique fails.\n\n At the end o
 f my talk I will explain the technique in a heuris
 tic way.\n
LOCATION:MR14\, CMS
CONTACT:
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