发表机构
Chennai Mathematical Institute; School of Mathematics, Hunan University; Department of Mathematics, University of Toronto（金奈数学研究所; 湖南大学数学学院; 多伦多大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究带齐次Neumann边界条件的Allen-Cahn方程稳定解，在边界正交过渡层情形下，建立了节点超曲面到边界的二阶Hölder估计与平均曲率衰减，并揭示了边界几何修正项在非零水平集上的非均匀效应。
AI 中文摘要
我们研究了在带边界的黎曼流形上，具有齐次Neumann边界条件的Allen-Cahn方程的稳定解，其过渡层与边界正交相交。在过渡区域内增强第二基本形式的先验界假设下，我们建立了环境维度至多为10时，节点超曲面直到边界的均匀二阶Hölder估计以及定量平均曲率衰减。证明方法改编了Wang-Wei开创的技术，并添加了由环境边界几何决定的进一步修正项。该修正项对平均曲率的主导贡献在节点集上相互抵消，但在过渡区域的非零水平集上仍然存在，这些水平集的平均曲率不必均匀衰减。这一现象没有内部对应物，并且由Kowalczyk构造的稳定解所展现。
英文摘要
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.
