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A class of homogenization problems in the calculus of variations
Author(s) -
Weinan E.
Publication year - 1991
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160440702
Subject(s) - homogenization (climate) , mathematics , affine transformation , regular polygon , hamilton–jacobi equation , pure mathematics , convex function , class (philosophy) , calculus (dental) , geometry , computer science , medicine , biodiversity , ecology , dentistry , artificial intelligence , biology
We study a class of integral functionals for which the integrand f e ( 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 f e ( 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.