AI 中文总结
研究黎曼流形上有界变差函数与有限周长集的结构理论，通过构建局部化框架融合相关技术，恢复欧几里得理论关键结果，为边值问题提供背景，证明有限周长集逼近结果及应用于毛细问题能量泛函Gamma收敛。
AI 中文摘要
我们发展了任意黎曼流形上有界变差函数的理论以及有限周长集的结构理论，不依赖于流形的全局曲率界或完备性。为此，构建了一个局部化框架，融合欧几里得几何测度理论和度量测度空间分析技术，保留如极向量场、法向量场、约化边界和近似切空间等黎曼特征。由此恢复了欧几里得理论的关键结果，如向量值测度的黎曼推广的微分定理以及De Giorgi和Federer的结构定理。还为流形区域上的边值问题提供理论背景，并证明了有限周长集的逼近结果及应用于毛细问题能量泛函的Gamma收敛。
英文摘要
We develop the theory of functions of bounded variation and the structure theory of finite-perimeter sets on arbitrary Riemannian manifolds without relying on global curvature bounds or completeness of the manifold. To this end, we build a localization framework that permits a synthesis of techniques from Euclidean geometric measure theory and analysis on metric measure spaces while preserving genuinely Riemannian features, such as polar and normal vector fields, reduced boundaries, and approximate tangent spaces. As a consequence, we recover key results of the Euclidean theory, such as a differentiation theorem for a Riemannian generalization of vector-valued measures and the structure theorems by De Giorgi and Federer, formulated intrinsically on Riemannian manifolds. This makes a large portion of the classical Euclidean $BV$ theory available to the Riemannian setting. We demonstrate this by providing the theoretical background for boundary value problems on domains in manifolds, including trace and Gauss-Green theorems. Finally, we prove an approximation result for finite-perimeter sets in a strict sense that respects a prescribed Dirichlet boundary portion of a given ambient domain and apply this to prove Gamma-convergence of a family of energy functionals for capillarity problems with mixed boundary conditions.
Comments39 pages, 1 figure
