On the De Giorgi type conjecture for an elliptic system modeling phase separation

2012/07/23 by Kelei Wang, Wang, Kelei
Computer Science · Mathematics · #35B06 #35B08 #35B25 #35J91 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1207.5285

openalex publication_date 2012/07/23 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

In this paper we study the one dimensional symmetry problem of entire solutions to the problem Δu=uv²,Δv=vu²,u,vgt;0 in ℝⁿ, for all n≥ 2. We prove that, if a solution (u,v) is a local minimizer and has linear growth at infinity, then it is one dimensional, i.e. depending only on one variable. In the proof we also obtain the global Lipschitz continuity of solutions only under the linear growth assumption.

Citations

• On Phase-Separation Models: Asymptotics and Qualitative Properties
• Variational problems with two phases and their free boundaries

Related

• Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D
• Harmonic approximation and improvement of flatness in a singular perturbation problem
• De Giorgi type results for elliptic systems
• Some symmetry results for entire solutions of an elliptic system arising in phase separation
• On Entire Solutions of an Elliptic System Modeling Phase Separations