论文标题
无差异措施的两个近似结果
Two Approximation Results for Divergence Free Measures
论文作者
论文摘要
在本文中，我们证明了无差异措施的两个近似结果。第一种是主张J. Bourgain和H. Brezis关于严格拓扑中螺立电荷近似的近似值的一种形式：给定$ f \ in m_b（\ Mathbb {r}^d; \ sathbb {r}^d）$ $γ_{i，l} $带有关联的度量$μ_{γ_{i，l}} $，这样，\ [f = \ lim_ {l \ to \ to \ infty} \ frac {\ | f \ | f \ | _____ {m_b {m_b {\ nm_b { l} \ sum_ {i = 1}^{n_l}μ_{γ_{γ_{i，l}} \]在测量意义上弱 - \ [\ lim_ {\ lim_ {l \ to \ to \ to \ infty} \ frac \ frac {1} {1} \ | |μ_{γ_{i，l}} \ | _ {m_b（\ Mathbb {r}^d; \ Mathbb {r}^d）} = 1。\ \ \ \ \ \ \ \]，几乎是第一个的直接结果，是第一个平滑的合并功能，就是\左IN \左\ [ M_b（\ Mathbb {r}^d; \ Mathbb {r}^d）：\ peripatorName*{div} f = 0 \ right \} \]相对于严格的拓扑。
In this paper we prove two approximation results for divergence free measures. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given $F \in M_b(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{div} F=0$ in the sense of distributions, there exist oriented $C^1$ loops $Γ_{i,l}$ with associated measures $μ_{Γ_{i,l}}$ such that \[ F= \lim_{l \to \infty} \frac{\|F\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)}}{n_l \cdot l} \sum_{i=1}^{n_l} μ_{Γ_{i,l}} \] weakly-star in the sense of measures and \[ \lim_{l \to \infty} \frac{1}{n_l \cdot l} \sum_{i=1}^{n_l} \|μ_{Γ_{i,l}}\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)} = 1. \] The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \[ \left\{ F \in M_b(\mathbb{R}^d;\mathbb{R}^d): \operatorname*{div}F=0 \right\} \] with respect to the strict topology.
