A Dirichlet-to-Neumann Map for the Allen-Cahn Equation on Manifolds with Boundary

TL;DR

This paper investigates the asymptotic behavior of solutions to the Allen-Cahn equation on manifolds with boundary, relating boundary data to geometry, and provides high-order expansions and a projection theorem near minimal surfaces.

Contribution

It establishes the invertibility of the linearized Allen-Cahn operator at heteroclinic solutions and relates Dirichlet minimizers to boundary geometry, extending understanding of phase transitions on manifolds.

Findings

Dirichlet minimizers are asymptotically local in psilon.

High-order expansions of solutions are computed.

A projection theorem near minimal surfaces is proved.

Abstract

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 $\epsilon$ 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.