On De Giorgi's Conjecture of Nonlocal approximations for free-discontinuity problems: The symmetric gradient case

TL;DR

This paper extends De Giorgi's conjecture to nonlocal free-discontinuity problems involving the symmetric gradient, establishing Gamma-convergence and compactness results for suitable finite-difference approximations.

Contribution

It introduces a new class of finite-difference approximants and proves their Gamma-convergence in the context of symmetric gradient-based free-discontinuity problems.

Findings

Proved Gamma-convergence of nonlocal functionals involving symmetric gradients.

Established compactness of deformations with bounded energies.

Characterized the limiting space of admissible deformations in GSBD.

Abstract

We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis is a crucial hurdle, which we overcome by generalizing a Fr\'echet-Kolmogorov approach previously introduced by two of the authors. A second essential difficulty is the identification of the limiting space of admissible deformations, since a control on the directional variations is, a priori, only available in average. A limiting representation in GSBD is eventually established via a novel characterization of this space.