发表机构
Chinese University of Hong Kong; Yunnan Normal University（香港中文大学; 云南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究各向异性Allen-Cahn方程，基于Mooney-Yang各向异性极小图证明了$N\neq4$时存在水平集非超平面的弱稳定解，并构造出$N\neq5$时沿单方向单调的非一维光滑解。
AI 中文摘要
本文研究$\u211d^N$中的各向异性Allen-Cahn方程$-\u00f6peratorname{div} a(Du)+W'(u)=0$，其中$a(p):=DH(p)$，$H(p)=\frac{1}{2}F(p)^2$，$F$为一致椭圆被积函数，$W(u)=\frac{1}{4}(1-u^2)^2$。基于Mooney-Yang各向异性极小图，我们证明当$N\neq4$时，该方程在弱意义下存在稳定解，其水平集不是超平面。作为副产品，我们还构造了$N\neq5$时上述各向异性Allen-Cahn方程的光滑解，该解沿一个方向单调但非一维。
英文摘要
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.
Comments46 pages
