The divergence of continuously differentiable vector fields

Abstract

For $m\geq 2$, we give a complete characterisation of the continuous functions on $\mathbb{R}^m$ that are divergences of continuously differentiable vector fields. The characterisation is formulated in terms of distributions and, equivalently, in terms of the continuous dual of a suitably topologised subspace of the Arens--Eells space. The two implications yield contrasting applications. First, there exists a compactly supported continuous function which is the divergence of a bounded Lipschitz vector field but not of any continuously differentiable vector field. Second, every continuous function whose distributional partial derivatives are locally $m$-integrable (the critical exponent for Sobolev embedding) is the divergence of a continuously differentiable vector field.