Slow convergence in periodic homogenization problems for divergence type elliptic operators

TL;DR

This paper introduces a new method to demonstrate that convergence rates in periodic homogenization of divergence type elliptic operators can be arbitrarily slow, even with smooth data and boundary conditions.

Contribution

It provides a constructive approach to establish lower bounds on convergence rates, showing that slow convergence can occur in both half-space and bounded domain settings.

Findings

Solutions can converge arbitrarily slowly depending on hyperplane position.

Homogenization can occur with arbitrarily slow speed despite smooth data.

Constructs examples of boundary value problems with slow convergence.

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.