2024/02/16 by Margherita Solci, Solci, Margherita · 1 citation
Engineering · Mathematics · #26B30 #35B25 #49J10 #49J45 #74A45 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Material Science and Thermodynamics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2402.10656
openalex publication_date 2024/02/16 · openalex created_date 2024/02/20 · openalex updated_date 2026/07/28
We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order k≥ 2. When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order k-1 in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for k-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant mk times the k-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford-Shah functional can be obtained by letting k tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.