Qualitative properties of two-end solutions to the Allen–Cahn equation in R3 ${\mathbb{R}}^{3}$

Abstract

A solution of the Allen–Cahn equation in R3 ${\mathbb{R}}^{3}$ is called a two-end solution if its nodal set is asymptotic to (x′,z)∈R3:z=ki⁡ln|x′|+ci,1≤i≤2 $\left\{\left({x}^{\prime },z\right)\in {\mathbb{R}}^{3}:z={k}_{i}\mathrm{ln}\vert {x}^{\prime }\vert +{c}_{i},1\le i\le 2\right\}$ at infinity. In this paper, we show that two-end solutions are axially symmetric and monotonic if k 1, k 2 satisfy k1−k2>22 ${k}_{1}-{k}_{2}{ >}2\sqrt{2}$ . We also establish the nonexistence of two-end solution with k 1, k 2 satisfying −22<k2<k1<22 $-\frac{\sqrt{2}}{2}{< }{k}_{2}{< }{k}_{1}{< }\frac{\sqrt{2}}{2}$ or k1=−k2=22 ${k}_{1}=-{k}_{2}=\frac{\sqrt{2}}{2}$ .

Keywords

• two-end solutions
• symmetry
• nonexistence
• curvature estimate
• 35a01
• 35b06
• 35j15
• 35j91