A general integral representation result for continuum limits of discrete energies with superlinear growth

Roberto Alicandro, Marco Cicalese

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91 Scopus citations

Abstract

We study the asymptotic behavior, as the mesh size ε tends to zero, of a general class of discrete energies defined on functions u: α ∈ ε ℤN ∩ Ω →u(α) ∈ ℝd of the form Fε(u) = Σ gε(α,β, u(α) - u(β)) α,β ∈ ε ℤN [α,β] ⊂ Ω and satisfying superlinear growth conditions. We show that all the possible variational limits are defined on W 1,p(Ωℝd) of the local type ∫ f(x,▽u)dx. We show that, in general, f may be a quasi-convex nonconvex function even if very simple interactions are considered. We also treat the case of homogenization, giving a general asymptotic formula that can be simplified in many situations (e.g., in the case of nearest neighbor interactions or under convexity hypotheses).

Original languageEnglish
Pages (from-to)1-37
Number of pages37
JournalSIAM Journal on Mathematical Analysis
Volume36
Issue number1
DOIs
StatePublished - 2005
Externally publishedYes

Keywords

  • Discrete systems
  • Homogenization
  • Γ-convergence

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