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Entire monotone solutions of the anisotropic Allen-Cahn equation in dimension 5
Abstract
In this paper, we consider the anisotropic Allen-Cahn equation −div a(Du)+W'(u)=0 in ℝN, where a(p):=DH(p) with H(p)=1/2F(p)2 and F a uniformly elliptic integrand, and W(u)=1/4(1−u2)2. Based on the Mooney-Yang anisotropic minimal graph, we prove that the anisotropic Allen-Cahn equation admits a stable solution for N≥4 in the weak sense, whose level sets are not hyperplanes. As a byproduct, we also construct a smooth solution of the above anisotropic Allen-Cahn equation for N≥5 that is monotone in one direction but is not one-dimensional.
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BibTeX
@misc{lu2026entire,
title = {{Entire monotone solutions of the anisotropic Allen-Cahn equation in dimension 5}},
author = {Haowen Lu and Song Wang and Juncheng Wei and Yuanze Wu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30904},
eprint = {2608.30904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30904},
}
RIS
TY - UNPB
TI - Entire monotone solutions of the anisotropic Allen-Cahn equation in dimension 5
AU - Haowen Lu
AU - Song Wang
AU - Juncheng Wei
AU - Yuanze Wu
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30904
UR - https://arxiv.org/abs/2608.30904
AB - In this paper, we consider the anisotropic Allen-Cahn equation −div a(Du)+W'(u)=0 in ℝ^N, where a(p):=DH(p) with H(p)=1/2F(p)^2 and F a uniformly elliptic integrand, and W(u)=1/4(1−u^2)^2. Based on the Mooney-Yang anisotropic minimal graph, we prove that the anisotropic Allen-Cahn equation admits a stable solution for N≥4 in the weak sense, whose level sets are not hyperplanes. As a byproduct, we also construct a smooth solution of the above anisotropic Allen-Cahn equation for N≥5 that is monotone in one direction but is not one-dimensional.
ER -