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arXiv · 2609.32829

Second order estimates on transition layers for the Allen-Cahn equation with homogeneous Neumann boundary conditions: Part I - Boundary-orthogonal transition layers

Akashdeep Dey·Wenkui Du·Davide Parise·Lorenzo Sarnataro
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Abstract

We study stable solutions of the Allen-Cahn equation with homogeneous Neumann boundary condition on Riemannian manifolds with boundary in the regime where the transition layers meet the boundary orthogonally. Under an a priori bound for the enhanced second fundamental form in the transition region, we establish uniform second-order Hölder estimates and quantitative mean curvature decay for the nodal hypersurfaces up to the boundary in ambient dimensions at most 10. The proof adapts the technique pioneered by Wang-Wei, adding a further correction determined by the geometry of the ambient boundary. Its leading contribution to mean curvature cancels on the nodal set but survives on non-zero level sets in the transition region, whose mean curvature need not decay uniformly. This phenomenon has no interior analogue and is exhibited by stable solutions constructed by Kowalczyk.

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Akashdeep Dey, Wenkui Du, Davide Parise, Lorenzo Sarnataro. 2026-09-26. Second order estimates on transition layers for the Allen-Cahn equation with homogeneous Neumann boundary conditions: Part I - Boundary-orthogonal transition layers. https://arxiv.org/abs/2609.32829

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