Asymptotic analysis of higher-order perturbations of the Perona–Malik functional

The Γ-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 Γ-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 {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 {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.