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

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.