Continuity Properties for Divergence Form Boundary Data Homogenization Problems

TL;DR

This paper investigates how the effective boundary condition behaves in periodic homogenization of oscillating Dirichlet data, showing continuity for linear systems and discontinuity for nonlinear equations.

Contribution

It provides a rigorous analysis of boundary condition continuity in homogenization, highlighting differences between linear and nonlinear systems.

Findings

Continuity of boundary conditions for linear systems.

Discontinuity examples for nonlinear equations.

Differentiated behavior based on system linearity.

Abstract

We study the continuity/discontinuity of the effective boundary condition for periodic homogenization of oscillating Dirichlet data for nonlinear divergence form equations and linear systems. For linear systems we show continuity, for nonlinear equations we give an example of discontinuity.