Two Approximation Results for Divergence Free Measures

2020/10/27 by Jesse Goodman, Goodman, Jesse, Felipe Hernández +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2010.14079

openalex publication_date 2020/10/27 · openalex created_date 2024/02/23 · openalex updated_date 2026/07/28

Abstract

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 ∈ M_b(ℝ^d;ℝ^d) such that \operatorname*div F=0 in the sense of distributions, there exist oriented C¹ loops Γ_i,l with associated measures μ_Γ_i,l such that F= lim_{l → ∞} \frac‖F‖_{Mb(ℝ^d;ℝ^d)}n_l ⋅ l ∑_i=1^n_l μ_Γ_i,l weakly-star in the sense of measures and lim_{l → ∞} (1)/(n_l ⋅ l) ∑_i=1^n_l ‖μ_Γ_i,l‖_{Mb(ℝ^d;ℝ^d)} = 1. The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \ F ∈ M_b(ℝ^d;ℝ^d): \operatorname*divF=0 \ with respect to the strict topology.

Related

• Beyond BV: new pairings and Gauss-Green formulas for measure fields with divergence measure
• Divergence-free measures in the plane and inverse potential problems in divergence form
• On the structure of divergence-free measures on \mathbb R²
• Extensions of divergence-free fields in L¹-based function spaces
• Measures in the dual of BV: perimeter bounds and relations with divergence-measure fields