Boundary asymptotics for two-dimensional vectorial Allen-Cahn systems
Abstract
We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.
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BibTeX
@misc{dai2026boundary,
title = {{Boundary asymptotics for two-dimensional vectorial Allen-Cahn systems}},
author = {Zhiyuan Dai and Haotong Fu and Huaijie Wang and Wei Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00739},
eprint = {2609.00739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00739},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Boundary asymptotics for two-dimensional vectorial Allen-Cahn systems
AU - Zhiyuan Dai
AU - Haotong Fu
AU - Huaijie Wang
AU - Wei Wang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00739
UR - https://arxiv.org/abs/2609.00739
AB - We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.
ER -