Publication 26-CNA-003
Asymptotic analysis of higher-order perturbations of the Perona–Malik functional
Andrea Braides
University of Rome Tor Vergata
Rome, Italy
braides@mat.unroma2.it
Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu
Abstract: The $\Gamma$-limit of higher-order singular perturbations of the Perona-Malik functional is analyzed. The energies considered combine the critically scaled logarithmic term with a
k-th order regularization designed to balance bulk and interfacial effects. A compactness result is obtained, and the $\Gamma$-limit is identified as a free-discontinuity functional on SBV, given by the sum of the Dirichlet energy and a surface term proportional to the jump amplitude to the power 1/
k. The surface density is characterized through a one-dimensional optimal-profile problem with homogeneous boundary conditions on derivatives up to order
k — 1. As a consequence, the limit of the same energies at a different scaling is determined. That scaling had been previously studied in the second-order case to address the so-called staircasing phenomenon.
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