[CvGmt News] Seminari di Calcolo delle Variazioni e Analisi Geometrica

magnani at dm.unipi.it magnani at dm.unipi.it
Fri Mar 25 09:33:51 CET 2011

Slow time behavior of different approximations of the Perona-Malik equation
Speaker: Maria Colombo (Scuola Normale Superiore)
Seminari di Calcolo delle Variazioni e Analisi Geometrica
30 Mar 2011

   Dipartimento di Matematica - Sala Riunioni - ore 18:00

   ABSTRACT: According to the general idea that ``the limit of the
   gradient-flows is the gradient-flow of the limit functional'', we prove
   an abstract result for passing to the limit in the theory of maximal
   slope curves in metric spaces, and then we apply this result to the
   study of the Perona-Malik equation. In a recent paper, P. Guidotti
   introduced a mild regularization of this problem. We prove that
   solutions of the regularized problem converge, in a slow time scale, to
   solutions of the total variation flow. The convergence result is
   global-in-time, and holds true in any space dimension. Then we consider
   the long time behavior of the semidiscrete scheme for the Perona-Malik
   equation in dimension one. We prove that the rescaled approximated
   solutions converge to solutions of a limit problem. This limit problem
   evolves piecewise constant functions by moving their plateaus in the
   vertical direction according to a system of ordinary differential
   equations. In this case, the main difficulty is the renormalization of
   the functionals after each collision in order to have a nontrivial
   Gamma-limit for all times. (Joint work with Massimo GOBBINO)

You find this news in the cvgmt preprint server: http://cvgmt.sns.it/news/20110330a/

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