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SUMMARY:The Signorini problem\, fractional Laplacians and the language of 
 semigroups - Stinga\, PR (University of Texas at Austin)
DTSTART:20140624T103000Z
DTEND:20140624T110000Z
UID:TALK53126@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:The Signorini problem can be equivalently formulated as a thin
  obstacle problem for an elastic membrane. The resulting free boundary pro
 blem turns out to be equivalent to the obstacle problem for the fractional
  Laplacian on the whole space. We will show how to understand this problem
  under the light of the language of semigroups that I developed in my PhD 
 thesis (2010). In particular\, we are able to consider different kinds of 
 Signorini problems that are equivalent to obstacle problems for fractional
  powers of operators different than the Laplacian on the whole space. Boun
 dary conditions of different kinds (Dirichlet\, Neumann\, periodic) and ra
 dial solutions can also be treated with this unified language. Another adv
 antage is that this language avoids the use of the Fourier transform. The 
 basic regularity results (Harnack inequalities\, Schauder estimates) for t
 hese fractional nonlocal operators can be studied by means of the generali
 zation of the Caffarelli--Silvestre extensio n problem that I proved in my
  PhD thesis. It turns out that the solution for the extension problem can 
 be written in terms of the heat semigroup.\n\n\n
LOCATION:Seminar Room 2\, Newton Institute Gatehouse
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