BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Applied and Computational Analysis
SUMMARY:Obstacle type problems : An overview and some rece
nt results - Henrik Shahgholian (KTH\, Stockholm)
DTSTART;TZID=Europe/London:20100211T150000
DTEND;TZID=Europe/London:20100211T160000
UID:TALK22712AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/22712
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\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 func
tions (down to Dini continuous).\n\nAstonishing re
sults of Yuval Peres and his collaborators has sho
wn remarkable relationships between obstacle probl
em and various forms of random walks\, including S
mash sum of Diaconis-Fulton (Lattice sets)\, and t
here is more to come.\n\nThe two-phase form (and i
ts multi-phase form) has been under investigation
in the last 10 years\, and interesting discoveries
has been made about the behavior of the free boun
daries in such problems. Existing methods has so f
ar only allowed us to consider $\\lambda_\\pm >0$.
\n\nThe above problem changes drastically if one a
llows $\\lambda_\\pm$ to have the incorrect sign (
that appears in composite membrane problem)!\n In
part of my talk I will focus on the simple _unsta
ble_ case\n\n\\Delta u = - \\chi_{u>0}\n\nand pres
ent very recent results (Andersson\, Sh.\, Weiss)
that classifies the set of singular points ($\\{u=
\\nabla u =0\\}$) for the above problem. The techn
iques developed recently by our team also shows an
unorthodox approach to such problems\, as the cla
ssical technique fails.\n\nAt the end of my talk I
will explain the technique in a heuristic way.\n
LOCATION:MR14\, CMS
CONTACT:
END:VEVENT
END:VCALENDAR