迁移微分算子的相容性矩阵的递归表示
A Recursive Representation of Compatibility Matrices for Transported Differential Operators
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中文总结 AI 辅助
本文针对热方程移动边界问题的相容性矩阵，构建递归代数框架，将其分解为通用参考矩阵与迁移扰动，为相关算子的后续研究奠定代数基础。
中文摘要 AI 辅助
我们针对热方程的移动边界问题中出现的相容性矩阵，构建了一种递归代数框架。这类矩阵自然出现在边界迹算子的分析中，包括Dirichlet-to-Neumann映射和布朗运动的首通问题。该构造基于两个递归层级：内部的迁移微分算子，以及通过边界条件逐次求导生成的边界算子。二者的相互作用产生了一个递归的相容性矩阵族，其大小随迁移算子的阶数线性增长。本文的主要贡献在于，将每个相容性矩阵分解为一个通用参考矩阵和一系列迁移扰动。该表示将边界几何与递归迁移修正分离，表明在每个递归层级仅迁移行发生变化。因此，相容性矩阵的行列式可解释为参考矩阵的行列式与逐次低秩扰动的组合。通过显式示例说明了递归构造，这些示例暗示参考行列式中存在额外的代数结构。所得的递归表示为相容性矩阵提供了系统的组织方式，并为后续研究移动边界问题的递归行列式公式、边界迹算子及Dirichlet-to-Neumann映射奠定了代数基础。
英文摘要
We develop a recursive algebraic framework for the compatibility matrices arising in moving-boundary problems for the heat equation. Such matrices naturally appear in the analysis of boundary trace operators, including Dirichlet-to-Neumann maps and first-passage problems for Brownian motion. The construction is based on two recursive hierarchies: transported differential operators in the interior and boundary operators generated by successive differentiation of the boundary condition. Their interaction produces a recursive family of compatibility matrices whose size grows linearly with the order of the transported operator. The main contribution of the paper is a decomposition of every compatibility matrix into a universal reference matrix and a sequence of transport perturbations. This representation isolates the boundary geometry from the recursive transport corrections and shows that only the transport rows change at each recursive level. Consequently, the determinant of the compatibility matrix may be interpreted as the determinant of a reference matrix together with successive low-rank perturbations. The recursive construction is illustrated through explicit examples, which suggest additional algebraic structure in the reference determinants. The resulting recursive representation provides a systematic organization of compatibility matrices and establishes an algebraic foundation for future investigations of recursive determinant formulas, boundary trace operators, and Dirichlet-to-Neumann maps for moving-boundary problems.
