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A class of homogenization problems in the calculus of variations. (English) Zbl 0773.49007
We study a class of integral functionals for which the integrand \(f^ \varepsilon(x,u,\nabla u)\) is an oscillatory function of both \(x\) and \(u\). Our method is based on the concept of \(\Gamma\)-convergence. Technical difficulties arise because \(f^ \varepsilon(x,u,\nabla u)\) is not convex or equi-continuous in \(u\) with respect to \(\varepsilon\). 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.
Reviewer: W.E (Princeton)

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
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