The Γ-limit and the Related Gradient Flow for Singular Perturbation Functionals of Perona-malik Type

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 {Fν} of regularized functionals Γ-converges, as ν → 0+, to a particular class of free-discontinuity functionals F , concentrated on SBV functions with finite energy and having only the jump part in the derivative. We study the singular dynamic associated with 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 L2-gradient of the Γ-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.