A two end family of solutions for the Inhomogeneous Allen-Cahn equation in R^2

TL;DR

This paper constructs a family of bounded solutions to the inhomogeneous Allen-Cahn equation in R^2, with nodal sets near two nonparallel lines, extending previous work on related equations in Riemannian manifolds.

Contribution

It introduces a new family of solutions for the inhomogeneous Allen-Cahn equation in the plane, with explicit description of their nodal sets near two asymptotically straight lines.

Findings

Solutions have nodal sets close to nondegenerate curves

Applicable to specific curves and functions a(x)

Extends previous compact case results

Abstract

In this work we construct a family of entire bounded solution for the singulary perturbed Inhomogeneous Allen-Cahn Equation $\ep^2\div\left(a(x)\nabla u\right)-a(x)F'(u)=0$ in $\R^2$, where $\ep\to 0$. The nodal set of these solutions is close to a "nondegenerate" curve which is asymptotically two non paralell straight lines. Here $F'$ is a double-well potential and $a$ is a smooth positive function. We also provide example of curves and functions $a$ where our result applies. This work is in connection with the results found by Z.Du and B.Lai, Z.Du and C.Gui, and F. Pacard and M. Ritore, in "Transition layers for an inhomogeneus Allen-Cahn equation in Riemannian Manifolds", "Interior layers for an inhomogeneous Allen-Cahn equation", "From the constant mean curvature hypersurfaces to the gradient theory of phase transitions" respectively, where they handle the compact case.