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$Γ$-convergence of convolution-type functionals for free discontinuity problems

Giuseppe Cosma Brusca, Davide Donati, Sergio Scalabrino, Chiara Trifone, Edoardo Voglino

Abstract

We prove compactness with respect to $Γ$-convergence for a general class of non-local energies modelled after the ones considered in [Gobbino, CPAM (1998)]. We give an integral representation result for the limits, which are free discontinuity functionals defined on the space of generalised special functions of bounded variation. We then characterise the bulk and surface energy densities of the obtained limits by means of minimisation problems on small cubes for the approximating energies.

$Γ$-convergence of convolution-type functionals for free discontinuity problems

Abstract

We prove compactness with respect to -convergence for a general class of non-local energies modelled after the ones considered in [Gobbino, CPAM (1998)]. We give an integral representation result for the limits, which are free discontinuity functionals defined on the space of generalised special functions of bounded variation. We then characterise the bulk and surface energy densities of the obtained limits by means of minimisation problems on small cubes for the approximating energies.
Paper Structure (7 sections, 24 theorems, 278 equations)

This paper contains 7 sections, 24 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Setting of the Problem and Main results
  4. Reduction to finite-range interactions
  5. Compactness
  6. Integral Representation
  7. Convergence of minima and Γ-convergence

Key Result

Lemma 2.1

${\rm GSBV}^2(\Omega;\mathbb R^m)$ is a vector space. Moreover, if $u\in {\rm GSBV^2}(\Omega;\mathbb R^m)$ then $u_i\in {\rm GSBV^2}(\Omega)$ for all $i\in\{1,...,m\}$. In particular, for all $M>0$, if $u^M$ denotes the function whose $i$-th component is given by $u^M_i:=(u_i\lor -M)\land M$, then $

Theorems & Definitions (56)

  • Lemma 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • ...and 46 more