[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
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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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