A cell problem for the boundary density and approximation of boundary correctors in homogenization theory
Abstract
We study the boundary corrector θε in the two-scale expansion for periodic elliptic problems with Dirichlet conditions. By results of Gérard-Varet--Masmoudi and Prange, on smooth uniformly convex domains θε converges to the solution θ* of a homogenized problem with boundary data given by the boundary layer tail d(n), which is determined by a half-space problem for each normal n. In the lifted form of Gérard-Varet and Masmoudi, d(n) is given explicitly by the cell correctors and the Dirichlet-to-Neumann operator Nn of this half-space problem. Our main result is that Nn satisfies a nonlinear equation on the period cell alone, so that d(n) and the boundary density that represents it can be computed on the cell. This equation is a Riccati equation in which the derivative in depth is replaced by a commutator with the derivative along n. This cell problem can therefore be solved by a Riccati doubling algorithm, and we can compute d(n) at irrational normals without rational approximation. Dropping the commutator term in the cell equation gives an operator square root approximation for Nn that captures the leading term of d(n) for small contrast, for the truncated problem. Finally, we approximate θε at finite ε by the sum of two solutions of constant coefficient problems, one with oscillatory boundary data and the other θ*, and test it on the disk.
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BibTeX
@misc{moskow2026cell,
title = {{A cell problem for the boundary density and approximation of boundary correctors in homogenization theory}},
author = {Shari Moskow},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17026},
eprint = {2610.17026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17026},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A cell problem for the boundary density and approximation of boundary correctors in homogenization theory
AU - Shari Moskow
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17026
UR - https://arxiv.org/abs/2610.17026
AB - We study the boundary corrector θ_ε in the two-scale expansion for periodic elliptic problems with Dirichlet conditions. By results of Gérard-Varet--Masmoudi and Prange, on smooth uniformly convex domains θ_ε converges to the solution θ^* of a homogenized problem with boundary data given by the boundary layer tail d(n), which is determined by a half-space problem for each normal n. In the lifted form of Gérard-Varet and Masmoudi, d(n) is given explicitly by the cell correctors and the Dirichlet-to-Neumann operator N_n of this half-space problem. Our main result is that N_n satisfies a nonlinear equation on the period cell alone, so that d(n) and the boundary density that represents it can be computed on the cell. This equation is a Riccati equation in which the derivative in depth is replaced by a commutator with the derivative along n. This cell problem can therefore be solved by a Riccati doubling algorithm, and we can compute d(n) at irrational normals without rational approximation. Dropping the commutator term in the cell equation gives an operator square root approximation for N_n that captures the leading term of d(n) for small contrast, for the truncated problem. Finally, we approximate θ_ε at finite ε by the sum of two solutions of constant coefficient problems, one with oscillatory boundary data and the other θ^*, and test it on the disk.
ER -