On an elliptic system with symmetric potential possessing two global minima

2008/10/28 by Nicholas D. Alikakos, Alikakos, Nicholas D., Giorgio Fusco +1
Mathematics · #35J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35J50

paper · pdf · doi:10.48550/arxiv.0810.5009

19 pages, 5 figures; major revision

arxiv created 2010/10/28 · arxiv updated 2010/10/29

Abstract

We consider the system Δu - W_u (u) = 0, for u: R² -> R², W: R² -> R, where W_u (u) is a smooth potential, symmetric with respect to the u₁, u₂ axes, possessing two global minima a^± := (\pma,0) and two connections e^±(x₁) connecting the minima. We prove that there exists an equivariant solution u(x₁, x₂) satisfying u(x₁, x₂) -> a^±, as x₁ -> \pminfiniti, and u(x₁, x₂) -> e^±(x₁), as x₂ -> \pminfiniti. The problem above was first studied by Alama, Bronsard, and Gui under related hypotheses to the ones introduced in the present paper. At the expense of one extra symmetry assumption, we avoid their considerations with the normalized energy and strengthen their result. We also provide examples for W.

Related

• A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system Δu - W_u (u) = 0
• Entire solutions to equivariant elliptic systems with variational structure
• On the connection problem for the p-Laplacian system for potentials with several global minima
• Asymptotic behavior and rigidity results for symmetric solutions of the elliptic system Δu = W_u(u)
• Existence of periodic orbits near heteroclinic connections