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

Abstract

In this paper, we consider the anisotropic Allen-Cahn equation $-\mathrm{div}a(Du)+W^{\prime }(u)=0$ 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 $N\ge 4$ 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 $N\ge 5$ that is monotone in one direction but is not one-dimensional.

46 pages