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On the distributional divergence of vector fields vanishing at infinity

Published online by Cambridge University Press:  11 February 2011

Thierry De Pauw  and
Monica Torres
Thierry De Pauw
Affiliation:
Institut de Recherches en Mathématiques et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium (thierry.depauw@uclouvain.be)
Monica Torres
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA (torres@math.purdue.edu)
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Abstract

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The equation div υ = F has a solution υ in the space of continuous vector fields vanishing at infinity if and only if F acts linearly on BVm/(m−1)(ℝm) (the space of functions in Lm/(m−1)(ℝm) whose distributional gradient is a vector-valued measure) and satisfies the following continuity condition: F(uj) converges to zero for each sequence {uj} such that the measure norms of ∇j are uniformly bounded and uj ⇀ 0 weakly in Lm/(m−1)(ℝm).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 1 , February 2011 , pp. 65 - 76
Copyright
Copyright © Royal Society of Edinburgh 2011