2023/01/19 by Jared Marx-Kuo, Marx-Kuo, Jared
Materials Science · Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2301.08286
openalex publication_date 2023/01/19 · openalex created_date 2023/01/24 · openalex updated_date 2026/07/28
We study the asymptotic behavior of Dirichlet minimizers to the Allen--Cahn equation on manifolds with boundary, and we relate the Neumann data to the geometry of the boundary. We show that Dirichlet minimizers are asymptotically local in orders of ε and compute expansions of the solution to high order. A key tool is showing that the linearized allen-cahn operator is invertible at the heteroclinic solution, on functions with 0 boundary condition. We apply our results to separating hypersurfaces in closed Riemannian manifolds. This gives a projection theorem about Allen--Cahn solutions near minimal surfaces, as constructed by Pacard--Ritore.