Operator error estimates for homogenization of the elliptic dirichlet problem in a bounded domain

2012/01/10 by Pakhnin, M. A., Suslina, T. A.
#35B27 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1201.2140

Abstract

Let O ⊂ ℝ^d be a bounded domain of class C². In the Hilbert space L₂(O;ℂⁿ), we consider a matrix elliptic second order differential operator A_D,ε with the Dirichlet boundary condition. Here ε>0 is the small parameter. The coefficients of the operator are periodic and depend on x/ε. We find approximation of the operator A_D,ε^-1 in the norm of operators acting from L₂(O;ℂⁿ) to the Sobolev space H¹(O;ℂⁿ) with an error term O(√(ε)). This approximation is given by the sum of the operator (A⁰_D)^-1 and the first order corrector, where A⁰_D is the effective operator with constant coefficients and with the Dirichlet boundary condition.

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