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A note on the structure of prescribed gradient–like domains of non–integrable vector fields
Mathematical Methods in the Applied Sciences ( IF 2.9 ) Pub Date : 2021-05-07 , DOI: 10.1002/mma.7488
Răzvan M. Tudoran 1
Affiliation  

Given a geometric structure on n with n = 2 m (e.g., Euclidean, symplectic, Minkowski, and pseudo-Euclidean), we analyze the set of points inside the domain of definition of an arbitrary given C 1 vector field, where the value of the vector field equals the value of the left/right gradient–like vector field of some fixed C 2 potential function, although a natural non-integrability condition holds at each such a point. More precisely, we prove that if not empty, this is a Borel set which admits a cover consisting of at most 2 m m regularly embedded C 1 m−dimensional submanifolds of 2 m , and consequently, its Hausdorff dimension is less or equal to m. Particular examples of gradient–like vector fields include the class of gradient vector fields with respect to Euclidean or pseudo-Euclidean inner products, and the class of Hamiltonian vector fields associated to symplectic structures on n . The main result of this article provides a geometric version of the similar result for the classical gradient vector field.

中文翻译:

关于不可积向量场的规定梯度样域结构的注记

给定几何结构 n n = 2 (例如,欧几里得、辛、闵可夫斯基和伪欧几里得),我们分析任意给定定义域内的点集 C 1 向量场,其中向量场的值等于某些固定的左/右梯度样向量场的值 C 2 潜在函数,尽管在每个这样的点上都有一个自然的不可积条件。更准确地说,我们证明如果不是空的,这是一个 Borel 集,它允许覆盖最多由 2 定期嵌入 C 1 的 m维子流形 2 ,因此,其 Hausdorff 维数小于或等于m。类梯度向量场的具体例子包括关于欧几里得或伪欧几里得内积的梯度向量场类,以及与辛结构相关的哈密顿向量场类 n . 本文的主要结果为经典梯度向量场提供了类似结果的几何版本。
更新日期:2021-05-07
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