Accepted Paper

Inserted: 2 aug 2026
Last Updated: 2 aug 2026

Journal: Adv. Calc. Var.
Year: 2026

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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