用于Dirichlet热方程的边界校正Li--Yau估计的热核期望方法
A Heat Kernel Expectation Approach to Boundary-Corrected Li--Yau Estimates for the Dirichlet Heat Equation
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中文总结 AI 辅助
本文提出一种热核期望方法，针对欧氏半空间Dirichlet热方程正解建立显式边界校正Li--Yau型梯度估计，为边界效应提供核解释并暗示可扩展至更一般区域。
中文摘要 AI 辅助
本文针对欧氏半空间上Dirichlet热方程的正解，建立了显式的边界校正Li--Yau型梯度估计。主要思路是利用Dirichlet热核的反射结构，引入归一化的核诱导概率测度；在该表示下，热核的对数导数成为显式核量的期望。反射高斯分量生成了涉及双曲余切函数\\( \coth\left(\frac{x_n y_n}{2t}\right) \\)的双曲校正项，这在整个欧氏热方程中无对应项。利用Jensen不等式和尖锐估计\\( 0<z\\,\csch z<1 \\)（\\( z>0 \\)），本文证明每个正解满足\\( \Delta\log w(x,t) \geq -\frac{n}{2t} -\frac{1}{x_n^2} \\)：第一项代表经典欧氏Li--Yau扩散标度，第二项是由到Dirichlet边界的距离决定的显式逆平方校正。本文方法为边界效应提供了直接的核解释，并暗示了向更一般区域扩展的可能性。
英文摘要
In this paper, we establish an explicit boundary-corrected Li--Yau type gradient estimate for positive solutions of the Dirichlet heat equation on the Euclidean half-space. The main idea is to exploit the reflection structure of the Dirichlet heat kernel and introduce a normalized kernel-induced probability measure. Under this representation, logarithmic derivatives of the heat kernel become expectations of explicit kernel quantities. The reflected Gaussian component generates a hyperbolic correction term involving \[ \coth\left(\frac{x_n y_n}{2t}\right), \] which has no analogue in the whole Euclidean heat equation. Using Jensen's inequality and the sharp estimate \[ 0<z\,\csch z<1 , \qquad z>0, \] we prove that every positive solution satisfies \[ Δ\log w(x,t) \geq -\frac n{2t} -\frac1{x_n^2}. \] The first term represents the classical Euclidean Li--Yau diffusion scaling, while the second term is an explicit inverse-square correction determined by the distance to the Dirichlet boundary. Our approach provides a direct kernel interpretation of the boundary effect and suggests possible extensions to more general domains.
