University of Toronto
                  PDE/Applied Math/Analysis Seminar
                  Friday 23. November, 1:10-2:00pm
                          6183 Bahen Center      



SPEAKER:  Macieij Zworski

TITLE:    Local smothing in the presence of lots of trapping.

ABSTRACT: The Schr"odinger propagator for the Laplacian in the 
          Euclidean space is unitary on any Sobolev space so 
          regularity is not improved in propagation. Remarkably, 
          and as has been known for about 20 years, the 
          regularity improves when we integrate in time and 
          cut-off in space and this much exploited effect is known 
          as _local smoothing_. Roughly, half a derivative is 
          gained in this way.

          The same is true on many other noncompact manifolds 
          under _nontrapping_ assumptions. In fact, any trapping 
          (e.g. presence of closed geodesics) will destroy local 
          smoothing.  Using recent results obtained by Stephane 
          Nonnenmacher and the speaker we show that local smoothing 
          with a minimal loss of regularity (1/2 - logarithmic loss) 
          can be obtained under the assumptions on the dimension 
          of the trapped set, or more generally on the topological 
          pressure of the classical flow.


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University of Toronto PDE/Applied Math/Analysis Seminar 
http://www.math.toronto.edu/appmath/

2007-2008 organizers:
          Walid Abou-Salem      walid@ math.toronto.edu
          Almut Burchard        almut@ math.toronto.edu
          Robert McCann		mccann@ math.toronto.edu
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