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

Authors: G. Bellettini and G. Fusco
Journal: Trans. Amer. Math. Soc. 360 (2008), 4929-4987
MSC (2000): Primary 49J45, 35B25, 74H40, 74G65
Published electronically: April 25, 2008
MathSciNet review: 2403710
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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.

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Additional Information

G. Bellettini
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 00133 Roma, Italy – and – INFN Laboratori Nazionali di Frascati, Italy

G. Fusco
Affiliation: Dipartimento di Matematica, Università di L’Aquila, via Vetoio, loc. Coppito, 67100 l’Aquila, Italy

Keywords: Singular perturbations, $\Gamma $-convergence, gradient flows, minimizing movements
Received by editor(s): December 1, 2004
Received by editor(s) in revised form: September 22, 2006
Published electronically: April 25, 2008
Additional Notes: The authors gratefully acknowledge the hospitality and the support of the Centro De Giorgi of the Scuola Normale Superiore di Pisa, where this paper was completed. The first author gratefully acknowledges also the hospitality and the support of the Max Planck Institute for Gravitational Physics in Golm. The authors are also grateful to the referee for some useful comments.
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