manuscripta math. 154, 225–256 (2017) © Springer-Verlag Berlin Heidelberg 2016
Regularity of boundary data in periodic
homogenization of elliptic systems in layered
Received: 15 September 2015 / Accepted: 1 December 2016
Published online: 21 December 2016
Abstract. In this note we study periodic homogenization of Dirichlet problem for diver-
gence type elliptic systems when both the coefﬁcients and the boundary data are oscillating.
One of the key difﬁculties here is the determination of the ﬁxed boundary data correspond-
ing to the limiting (homogenized) problem. This issue has been addressed in recent papers
by Gérard-Varet and Masmoudi (Acta Math. 209:133–178, 2012), and by Prange (SIAM J.
Math. Anal. 45(1):345–387, 2012), however, not much is known about the regularity of this
ﬁxed data. The main objective of this note is to initiate a study of this problem, and to prove
several regularity results in this connection.
For a bounded domain D ⊂ R
(d ≥ 2) consider the following problem
(x) = 0, x ∈ D, (1.1)
with oscillating Dirichlet data
u(x) = g
, x ∈ ∂ D. (1.2)
Here ε>0 is a small parameter, A(x) = (A
(x)) is R
deﬁned on R
, where 1 ≤ α, β ≤ d,1≤ i, j ≤ N, and the boundary data g(x, y)
-valued function deﬁned on ∂ D × R
. The action of the operator in (1.1)on
a vector-function u = (u
) is deﬁned as
where 1 ≤ i ≤ N. Here and throughout the text, if not stated otherwise, we use the
summation convention for repeated indices.
H. Aleksanyan: School of Mathematics, The University of Edinburgh, JCMB The King’s
Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK.
Present address: H. Aleksanyan (
) Department of Mathematics, KTH Royal Institute of
Technology, 100 44 Stockholm, Sweden
Mathematics Subject Classiﬁcation: Primary 35B27; Secondary 35B40 · 35J08 · 35J57 ·