Extension and Representation of Divergence-free Vector Fields on Bounded Domains

Abstract

(1.1) gj(x) := x− yj |x− yj |n , x ∈ Rn {yj}. Let Hs,p denote the usual scale of Lp-Sobolev spaces, and denote by div and Div the divergence of 1-tensors (i.e., vector fields) and 2-tensors, respectively. Given a divergence-free vector field u on Ω, possessing a certain regularity, e.g., u ∈ Hs,p(Ω), we want to investigate two closely related problems. One is to extend u to a divergence-free vector field ũ defined on a neighborhood of Ω (in fact, defined on Rn \{yj}), such that ũ is as smooth as u. The second is to produce an anti-symmetric 2-tensor field v such that (1.2) u = Div v + ∑