Homogenization for Locally Periodic Elliptic Problems on a Domain
Nikita N. Senik · SIAM Journal on Mathematical Analysis · 2023
Abstract. Let [Formula: see text] be a Lipschitz domain in [Formula: see text], and let [Formula: see text] be a strongly elliptic operator on [Formula: see text]. We suppose that [Formula: see text] is small and the function [Formula: see text] is Hölder continuous of order [Formula: see text] in the first variable and periodic in the second, so the coefficients of [Formula: see text] are locally periodic and rapidly oscillate. Given [Formula: see text] in the resolvent set, we are interested in finding the rates of approximations, as [Formula: see text], for [Formula: see text] and [Formula: see text] in the operator topology on [Formula: see text] for suitable [Formula: see text]. It is well-known that the rates depend on regularity of the effective operator [Formula: see text]. We prove that if [Formula: see text] and its adjoint are bounded from [Formula: see text] to the Lipschitz–Besov space [Formula: see text] with the same [Formula: see text] as for [Formula: see text], then the rates are, respectively, [Formula: see text] and [Formula: see text]. The results are applied to the Dirichlet, Neumann, and mixed Dirichlet–Neumann problems for strongly elliptic operators with uniformly bounded and vanishing mean oscillation coefficients.