2011/02/15 by Xavier Cabré, Xavier Cabre, Cabre, Xavier · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics #math.AP
paper · pdf · doi:10.48550/arxiv.1102.3111
arxiv created 2011/02/15 · openalex publication_date 2011/02/15 · arxiv updated 2011/02/16 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We establish the uniqueness of a saddle-shaped solution to the diffusion equation -Δu = f(u) in all of ℝ2m, where f is of bistable type, in every even dimension 2m ≥ 2. In addition, we prove its stability whenever 2m ≥ 14. Saddle-shaped solutions are odd with respect to the Simons cone \mathcal C = \(x1,x2) ∈ ℝm × ℝm : |x1|=|x2| \ and exist in all even dimensions. Their uniqueness was only known when 2m=2. On the other hand, they are known to be unstable in dimensions 2, 4, and 6. Their stability in dimensions 8, 10, and 12 remains an open question. In addition, since the Simons cone minimizes area when 2m ≥ 8, saddle-shaped solutions are expected to be global minimizers when 2m ≥ 8, or at least in higher dimensions. This is a property stronger than stability which is not yet established in any dimension.