A Dirichlet-to-Neumann map for the Allen–Cahn equation on manifolds with boundary

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Analysis reveals the asymptotic behavior of Dirichlet minimizers in manifolds, highlighting Neumann data implications.

Dirichlet minimizers display asymptotic local behavior in orders of ε, connecting them to boundary geometry.
The linearized allen-cahn operator's invertibility at the heteroclinic solution proves crucial for understanding solutions.
Expansions of the allen-cahn solution to high orders were computed, specifically for manifolds with boundary.
Implications for separating hypersurfaces in closed Riemannian manifolds are discussed, showcasing geometric applications.

Jared Marx-Kuo (2025) studied this question.

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