Homogenization of high-contrast Mumford-Shah energies

Xavier Pellet, Lucia Scardia, Caterina Ida Zeppieri

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6 Citations (Scopus)
44 Downloads (Pure)


We prove a homogenization result for Mumford-Shah-type energies associated to a brittle composite material with weak inclusions distributed periodically at a scale ϵ > 0. The matrix and the inclusions in the material have the same elastic moduli but very different toughness moduli, with the ratio of the toughness modulus in the matrix and in the inclusions being 1/β ϵ, with β ϵ > 0 small. We show that the high-contrast behavior of the composite leads to the emergence of interesting effects in the limit: The volume and surface energy densities interact by Γ -convergence, and the limit volume energy is not a quadratic form in the critical scaling β ϵ = ϵ, unlike the ϵ-energies, and unlike the extremal limit cases.

Original languageEnglish
Pages (from-to)1696–1729
Number of pages34
JournalSIAM Journal on Mathematical Analysis
Issue number3
Publication statusPublished - 2 May 2019


  • brittle fracture
  • free-discontinuity problems
  • high-contrast materials
  • homogenization
  • Γ -convergence

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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