2022/05/05 by Massimo Gobbino, Nicola Picenni, Gobbino, Massimo +1 · 1 citation
Computer Science · Engineering · Mathematics · #35B25 #49J45 #49Q20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2205.02467
openalex publication_date 2022/05/05 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28
We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient. We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale. Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models.