The $\Gamma$-limit and the related gradient flow for singular perturbation functionals of Perona-Malik type

Abstract:

We consider a class of nonconvex functionals of the gradient in one dimension, which we regularize with a second order derivative term. After a proper rescaling, suggested by the associated dynamical problems, we show that the sequence $\{F_\nu \}$ of regularized functionals $\Gamma$-converges, as $\nu \to 0^+$, to a particular class of free-discontinuity functionals $\mathcal {F}$, concentrated on $SBV$ functions with finite energy and having only the jump part in the derivative. We study the singular dynamic associated with $\mathcal {F}$, using the minimizing movements method. We show that the minimizing movement starting from an initial datum with a finite number of discontinuities has jump positions fixed in space and whose number is nonincreasing with time. Moreover, there are a finite number of singular times at which there is a dropping of the number of discontinuities. In the interval between two subsequent singular times, the vector of the survived jumps is determined by the system of ODEs which expresses the $L^2$-gradient of the $\Gamma$-limit. Furthermore the minimizing movement turns out to be continuous with respect to the initial datum. Some properties of a minimizing movement starting from a function with an infinite number of discontinuities are also derived.