A class of homogenization problems in the calculus of variations

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Abstract

We study a class of integral functionals for which the integrand fe(x, u, ∇u) is an oscillatory function of both x and u. Our method is based on the concept of Γ‐convergenee. Technical difficulties arise because fe(x, u, ∇u) is not convex or equi‐continuous in u with respect to e. Two somewhat different approaches, based respectively on abstract convergence theorems and the study of affine functions, are exploited together to overcome these technical difficulties. As an application, we give another proof of a homogenization result of P. L. Lions, G. Papanicolaou, and S. R. S. Varadhan for Hamilton‐Jacobi equations.

Original languageEnglish (US)
Pages (from-to)733-759
Number of pages27
JournalCommunications on Pure and Applied Mathematics
Volume44
Issue number7
DOIs
StatePublished - Jan 1 1991
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Mathematics(all)

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