Calculus of Variations and Geometric Measure Theory

A. Zuniga - O. Agudelo

A two-end family of solutions for the inhomogeneous Allen-Cahn equation in $\mathbb{R}^2$

created by zuniga on 31 Jan 2019


Published Paper

Inserted: 31 jan 2019
Last Updated: 31 jan 2019

Journal: J. Differential Equations
Volume: 256
Number: 1
Pages: 157--205
Year: 2014
Doi: 10.1016/j.jde.2013.08.018

ArXiv: 1305.5573 PDF
Links: Journal link


In this work we construct a family of entire bounded solutions for the singularly perturbed inhomogeneous Allen-Cahn equation $\varepsilon^2\textrm{div}(a(x)\nabla u) -a(x)F'(u)=0$ in $\mathbb{R}^2$, where $\varepsilon\to 0$. The nodal set of these solutions is close to a ``nondegenerate" curve which is asymptotically two non-parallel straight lines. Here $F'$ is a double-well potential and $a$ is a smooth positive function. We also provide examples of curves and functions $a$ where our result applies. This work is in connection with the results found in [Z.Du and B.Lai, Transition layers for an inhomogeneus Allen-Cahn equation in Riemannian manifolds, Preprint], [Z.Du and C.Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Differential Equations, 249 (2010), pp 215-239] and [F. Pacard and M. Ritoré, From the constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom. 64 (2003), pp. 359-423], handling the compact case.

Keywords: phase transitions, entire solution, inhomogeneity, noncompact nodal set, Allen-Cahn equation