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arXiv · 2608.30904

Entire monotone solutions of the anisotropic Allen-Cahn equation in dimension 5

Haowen Lu·Song Wang·Juncheng Wei·Yuanze Wu
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Abstract

In this paper, we consider the anisotropic Allen-Cahn equation $-\operatorname{div} a(Du)+W'(u)=0$ in $\mathbb{R}^N$, where $a(p):=DH(p)$ with $H(p)=\frac{1}{2}F(p)^2$ and $F$ a uniformly elliptic integrand, and $W(u)=\frac{1}{4}(1-u^2)^2$. Based on the Mooney-Yang anisotropic minimal graph, we prove that the anisotropic Allen-Cahn equation admits a stable solution for $N\geq4$ 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 $N\geq5$ that is monotone in one direction but is not one-dimensional.

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Haowen Lu, Song Wang, Juncheng Wei, Yuanze Wu. 2026-08-31. Entire monotone solutions of the anisotropic Allen-Cahn equation in dimension 5. https://arxiv.org/abs/2608.30904

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