可求长解析割空间上的异质各向异性Kac极限

中文摘要 AI 辅助

环境收敛并不能确定割体上的非局部能量：割具有零体积，但微观键保留其路径所穿越的标记岸。我们给出了这一信息的直接测度论解析，并证明了相应的尖锐界面Gamma-极限。割可以是具有有限总表面质量的Borel、可数可求长超曲面的有限或可数族，具有可测的近似法向量和可测的二元岸数据。不假设闭性、正触及半径、有限图册、曲率界、角度条件或分支集的正则性。对于由一致生成的可能非交换流族生成的异质Kac相互作用，极限为解析各向异性变差加上外边界接触以及每岸一个系数-一个接触。Gamma-泛函在裂纹接近边界处不包含强制低维项；常数恢复在该处不产生集中。证明结合了流穿越的精确时空面积公式、一维分裂-扩展下界和有序流紧性。我们还去除了紧径向支撑。有限一阶矩产生相同的局部Griffith泛函；在幂律阈值处，极限分为局部、对数重正化局部和真正分数阶区域。光滑割缝和结探测器被恢复为直接穿越规则的均匀等价实现。

英文摘要

Ambient convergence does not determine a nonlocal energy on a cut body: a cut has zero volume, but a microscopic bond retains which labelled bank its path crosses. We give a direct measure-theoretic resolution of this information and prove the corresponding sharp-interface Gamma-limit. The cut may be a finite or countable family of Borel, countably rectifiable hypersurfaces with finite total surface mass, measurable approximate normals, and measurable binary bank data. No closedness, positive reach, finite atlas, curvature bound, angle condition, or regularity of the branch set is assumed. For heterogeneous Kac interactions generated by a uniformly spanning family of possibly noncommuting flows, the limit is the resolved anisotropic variation plus the outer boundary contact and one coefficient-one contact on each bank. The Gamma-functional contains no compulsory lower-dimensional term where a crack approaches the boundary; constant recovery creates no concentration there. The proof combines an exact spacetime area formula for flow crossings, a one-dimensional split-extension lower bound, and ordered-flow compactness. We also remove compact radial support. A finite first moment yields the same local Griffith functional; at the power-law threshold the limits separate into local, logarithmically renormalized local, and genuinely fractional regimes. Smooth slit and junction detectors are recovered as uniformly equivalent realizations of the direct crossing rule.

发表机构

• School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）

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