# Slow time behavior of different approximations of the Perona-Malik equation

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Maria Colombo
(University of Zurich / ETH)

created by magnani on 22 Mar 2011

30 mar 2011

**Abstract.**

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 works with Massimo GOBBINO)