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

Stefano Almi, Elisa Davoli, Anna Kubin, Emanuele Tasso

TL;DR

This work extends De Giorgi's nonlocal approximation framework for free-discontinuity problems to the symmetric-gradient setting, using continuous finite-difference approximants. It develops the GBV$^{\mathcal{E}}$ and GSBD spaces, proves a Γ-convergence result to Griffith-type energies, and identifies the limiting deformation space via Fréchet–Kolmogorov and integral-geometric slicing techniques. The results provide a rigorous bridge from nonlocal, directionally integrated energies to local Griffith energies with explicit bulk density $\varphi_p$ and surface density $\beta_p$, and they establish convergence of quasi-minimisers under Dirichlet data. The approach opens avenues for broader nonlocal approximations in elasticity-type free-discontinuity problems and potential extensions to other differential operators and discrete settings.

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 Fréchet-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.

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

TL;DR

This work extends De Giorgi's nonlocal approximation framework for free-discontinuity problems to the symmetric-gradient setting, using continuous finite-difference approximants. It develops the GBV and GSBD spaces, proves a Γ-convergence result to Griffith-type energies, and identifies the limiting deformation space via Fréchet–Kolmogorov and integral-geometric slicing techniques. The results provide a rigorous bridge from nonlocal, directionally integrated energies to local Griffith energies with explicit bulk density and surface density , and they establish convergence of quasi-minimisers under Dirichlet data. The approach opens avenues for broader nonlocal approximations in elasticity-type free-discontinuity problems and potential extensions to other differential operators and discrete settings.

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 Fréchet-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.

Paper Structure

This paper contains 13 sections, 24 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Structure of the space $\textnormal{GBV}^{\mathcal{E}}$
  4. Preliminaries
  5. Proof of Theorem \ref{['t:main-emanuele']}
  6. Proof of Theorem \ref{['t:slicejs']}
  7. Proof of Theorem \ref{['t:symmdiff']}
  8. Proof of Theorem \ref{['compattezza']}
  9. Preliminary results
  10. Proof of Theorem \ref{['compattezza']}
  11. $\Gamma$-convergence and convergence of quasi-minimisers
  12. $\Gamma$-convergence
  13. Convergence of quasi-minimisers

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^n$ be open, let $p\ge1$, and let $\{u_{\varepsilon}\}_{\varepsilon >0} \subset L^0(\Omega;\mathbb{R}^n)$ be such that Then, there exists a subsequence $\varepsilon_k \to 0$ as $k \to \infty$ such that the set has finite perimeter, and $u_{\varepsilon_k}\to u$ pointwise almost everywhere in $\Omega \setminus A$ for some measurable function $u \colon \Omega \setminus

Theorems & Definitions (55)

  • Theorem 1.1: Closure and compactness
  • Theorem 1.2: $\Gamma$-convergence
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 45 more