| 摘要: |
| 本文针对黎曼流形上的Allen-Cahn方程的椭圆型和抛物型正经典解，建立了Hessian矩阵的上界。基于先前涉及曲率假设的工作，我们将分析推广到截面曲率存在下界的情况。利用最大值原理、辅助函数方法和梯度估计等技术，在仅确定截面曲率下界的条件下，建立了Hessian矩阵的全局和局部估计。 |
| 关键词: Hessian估计 Allen-Cahn方程 黎曼流形 截面曲率 最大值原理 |
| DOI： |
| 分类号:O175.1 |
| 基金项目: |
|
| HESSIAN ESTIMATES FOR THE ALLEN-CAHN EQUATION ON RIEMANNIAN MANIFOLDS |
|
daiiyifei, zhuanqiang
|
| Abstract: |
| In this paper, we establish upper bounds for the Hessian of positive classical solutions to both the elliptic and parabolic forms of the Allen-Cahn equation on Riemannian manifolds. Building on previous work involving curvature assumptions, we extend the analysis to cases where the sectional curvature has a lower bound. Using techniques such as the maximum principle, auxiliary function methods, and gradient estimation, we establish global and local Hessian estimates under the condition that only the lower bound of the sectional curvature is determined. |
| Key words: Hessian estimate, Allen-Cahn equation, Riemannian manifold, sectional curvature, maximum principle |
