Slow convergence in periodic homogenization problems for divergence type elliptic operators

Abstract

We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show that solutions to boundary layer problems for divergence type elliptic equations set in halfspaces and with infinitely smooth data, may converge to their corresponding boundary layer tails as slow as one wish depending on the position of the hyperplane. Second, we construct a Dirichlet problem for divergence type elliptic operators set in a bounded domain, and with all data being $C^{\infty }$-smooth, for which the boundary value homogenization holds with arbitrarily slow speed.

Referee remarks implemented. 35 pages, 1 figure

References in corpus (4)

• The Green function estimates for strongly elliptic systems of second order
• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• Applications of Fourier analysis in homogenization of Dirichlet problem III: Polygonal Domains
• Homogenization of the boundary value for the Dirichlet Problem

Cited by in corpus (3)

• Regularity of boundary data in periodic homogenization of elliptic systems in layered media
• Homogenization and boundary layers in domains of finite type
• Regularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems