Compactness methods in the theory of homogenization II: Equations in non‐divergence form

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Abstract

We prove C0, α, C1, α and C1, 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form Σ in,j=1ai j(x/ϵ)δ2/δxiδxj. 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.

Original languageEnglish (US)
Pages (from-to)139-172
Number of pages34
JournalCommunications on Pure and Applied Mathematics
Volume42
Issue number2
DOIs
StatePublished - 1989

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Homogenization
Boundary value problems
Compactness
Mathematical operators
Oscillating Coefficients
Periodic Coefficients
A Priori Estimates
Elliptic Operator
Small Parameter
Linear Operator
Averaging
Nonlinear Problem
Boundary Value Problem
Imply
Generalise
Term
Theorem
Estimate
Form
Class

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "We prove C0, α, C1, α and C1, 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form Σ in,j=1ai j(x/ϵ)δ2/δxiδxj. 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.",
author = "Marco Avellaneda and Lin, {Fang‐Hua ‐H}",
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AU - Avellaneda, Marco

AU - Lin, Fang‐Hua ‐H

PY - 1989

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AB - We prove C0, α, C1, α and C1, 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form Σ in,j=1ai j(x/ϵ)δ2/δxiδxj. 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.

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