Two approximation results for divergence free measures

Jesse Goodman; Felipe Hernandez; Daniel Spector

doi:10.4171/pm/2126

JournalspmVol. 81, No. 3/4pp. 247–264

Two approximation results for divergence free measures

Jesse Goodman
University of Auckland, Auckland, New Zealand
Felipe Hernandez
Stanford University, Stanford, USA; MIT, Cambridge, USA
Daniel Spector
National Taiwan Normal University, Taipei City, Taiwan, R.O.C.

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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 $F \in M_{b} (R^{d}; R^{d})$ such that $div F = 0$ in the sense of distributions, there exist oriented $C^{1}$ loops $Γ_{i, l}$ with associated measures $μ_{Γ_{i, l}}$ such that

F = l \to \infty lim \frac{∥ F ∥ _{M_{b} (R^{d}; R^{d})}}{n _{l} \cdot l} i = 1 \sum n_{l} μ_{Γ_{i, l}}

weakly-star in the sense of measures and

l \to \infty lim \frac{1}{n _{l} \cdot l} i = 1 \sum n_{l} ∥ μ_{Γ_{i, l}} ∥_{M_{b} (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 ${F \in M_{b} (R^{d}; R^{d}) : div F = 0}$ with respect to the strict topology.

Cite this article

Jesse Goodman, Felipe Hernandez, Daniel Spector, Two approximation results for divergence free measures. Port. Math. 81 (2024), no. 3/4, pp. 247–264

DOI 10.4171/PM/2126