Vol. 12, No. 8, 2019

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Continuity properties for divergence form boundary data homogenization problems

William M. Feldman and Yuming Paul Zhang

Vol. 12 (2019), No. 8, 1963–2002
Abstract

We study the asymptotic behavior at rational directions of the effective boundary condition in periodic homogenization of oscillating Dirichlet data. We establish a characterization for the directional limits at a rational direction in terms of a relatively simple two-dimensional boundary layer problem for the homogenized operator. Using this characterization we show continuity of the effective boundary condition for divergence form linear systems, and for divergence form nonlinear equations we give an example of discontinuity.

Keywords
homogenization, boundary layers, oscillating boundary data, nonlinear elliptic equations, elliptic systems
Mathematical Subject Classification 2010
Primary: 35B27, 35J57, 35J60
Milestones
Received: 7 February 2018
Revised: 28 September 2018
Accepted: 30 November 2018
Published: 28 October 2019
Authors
William M. Feldman
Department of Mathematics
University of Chicago
Chicago, IL
United States
Yuming Paul Zhang
Department of Mathematics
University of California
Los Angeles, CA
United States