2019/01/23 by François Hamel, Yong Liu, Hamel, Francois +7
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1901.07671
openalex publication_date 2019/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we obtain rigidity results for a bounded non-constant entire solution u of the Allen-Cahn equation in ℝn, whose level set \u=0\ is contained in a half-space. If n≤ 3 we prove that the solution must be one-dimensional. In dimension n≥ 4, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.