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Compactness methods in the theory of homogenization II: Equations in non‐divergence form
Author(s) -
Avellaneda Marco,
Lin FangHua
Publication year - 1989
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.3160420203
Subject(s) - mathematics , homogenization (climate) , a priori and a posteriori , divergence (linguistics) , compact space , nonlinear system , mathematical analysis , boundary value problem , constant coefficients , periodic boundary conditions , section (typography) , elliptic operator , class (philosophy) , pure mathematics , biodiversity , ecology , philosophy , linguistics , physics , epistemology , quantum mechanics , artificial intelligence , advertising , computer science , business , biology
We prove C 0, α , C 1, α and C 1, 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form Σ i n , j=1 a i j (x/ϵ)δ 2 /δx i δx j . The constants in these estimates are independent of the small parameter ϵ, and hence our results imply strengthened versions of the classical averaging theorems. These results generalize to a wide class of linear operators in non‐divergence form, including equations with lower‐order terms and nonuniformly oscillating coefficients, as well as to certain nonlinear problems, which we discuss in the last section.