Γ-convergence of free-discontinuity problems

Filippo Cagnetti, Gianni Dal Maso, Lucia Scardia*, Caterina Ida Zeppieri

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)
18 Downloads (Pure)

Abstract

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

Original languageEnglish
Pages (from-to)1035–1079
Number of pages45
JournalAnnales de l'Institut Henri Poincaré (C) Analyse Non Linéaire
Volume36
Issue number4
Early online date15 Nov 2018
DOIs
Publication statusPublished - Jul 2019

Keywords

  • Free-discontinuity problems
  • Homogenisation
  • Γ-convergence

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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