Second order estimates on transition layers for the Allen-Cahn equation with homogeneous Neumann boundary conditions: Part I - Boundary-orthogonal transition layers
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.
Related
- Dimension Monotonicity in Laguerre Ensembles II: Average Singular Values and the Rectangularity Transition in the Orthogonal Case
- A dual approach to Lp estimates for orthogonal martingales satisfying the differential subordination
- C1,1 Regularity Global Estimates for Homogeneous Complex Hessian Equations on Punctured Domains
- Almost-orthogonal Strichartz estimates and radial improvements
- Sharp global L2 estimates for oscillatory integral operators with quasi-homogeneous phases
BibTeX
@misc{dey2026second,
title = {{Second order estimates on transition layers for the Allen-Cahn equation with homogeneous Neumann boundary conditions: Part I - Boundary-orthogonal transition layers}},
author = {Akashdeep Dey and Wenkui Du and Davide Parise and Lorenzo Sarnataro},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32829},
eprint = {2609.32829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Second order estimates on transition layers for the Allen-Cahn equation with homogeneous Neumann boundary conditions: Part I - Boundary-orthogonal transition layers
AU - Akashdeep Dey
AU - Wenkui Du
AU - Davide Parise
AU - Lorenzo Sarnataro
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32829
UR - https://arxiv.org/abs/2609.32829
AB - 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.
ER -