Asymptotic analysis of second-order boundary layer correctors

Abstract

In this article we extend the ideas presented in Onofrei and Vernescu [Asymptotic Anal. 54 (2007), pp. 103–123] and introduce suitable second-order boundary layer correctors, to study the H ¹-norm error estimate for the classical problem of homogenization, i.e.

Previous second-order boundary layer results assume either smooth enough coefficients (which is equivalent to assuming smooth enough correctors χ _j , χ _ij  ∈ W ^1,∞), or smooth homogenized solution u ₀, to obtain an estimate of order . For this we use some ideas related to the periodic unfolding method proposed by Cioranescu et al. [C. R. Acad. Sci. Paris, Ser. I 335 (2002), pp. 99–104]. We prove that in two dimensions, for non-smooth coefficients and general data, one obtains an estimate of order . In three dimensions the same estimate is obtained assuming χ _j , χ _ij  ∈ W ^1,p , with p > 3.

Keywords:

• homogenization
• error estimates
• nonsmooth coefficients

AMS Subject Classifications::

• 35J15
• 35B27