横向弱Stieltjes导数：Stieltjes–Sobolev空间的全局刻画

中文摘要 AI 辅助

Stieltjes微分方程的弱形式必须保留驱动测度原子所携带的方向信息。我们推导出一个横向分部积分恒等式，其中测试函数的后迹由原子乘积规则强制确定。由此得到的对偶形式给出了指数范围从1到无穷的完整区间内积分Stieltjes–Sobolev空间的精确全局刻画，并从其带迹弱状态重构出唯一的强代表元。我们证明了相关的带迹弱算子是闭的，建立了Poincaré–Stieltjes估计，并通过一个典范导数测度刻画了有界变差函数内部的Stieltjes绝对连续性，该导数测度在开区间上的限制即为经典分布导数。该框架随后被应用于有限维非线性测度驱动系统，得到了弱、积分和测度三种表述的等价性，在可积Lipschitz假设下的存在唯一性，基于紧性的存在性，以及内在的复位律。三次和逻辑斯蒂模型说明了连续演化与瞬时干预的同时编码。

英文摘要

Weak formulations for Stieltjes differential equations must retain the directional information carried by atoms of the driving measure. We derive a lateral integration-by-parts identity in which the posterior trace of the test function is forced by the atomic product rule. The resulting dual formulation gives an exact global characterization of integral Stieltjes--Sobolev spaces for the full exponent range from one to infinity and reconstructs the unique strong representative from its traced weak state. We prove that the associated traced weak operator is closed, establish Poincaré--Stieltjes estimates, and \rev{characterize Stieltjes absolute continuity inside bounded variation through a canonical derivative measure whose restriction to the open interval is the classical distributional derivative}. The framework is then applied to finite-dimensional nonlinear measure-driven systems, yielding equivalence of weak, integral, and measure formulations, existence and uniqueness under integrable Lipschitz assumptions, compactness-based existence, and intrinsic reset laws. Cubic and logistic models illustrate the simultaneous encoding of continuous evolution and instantaneous interventions.

发表机构

• Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学）

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