Approximation of divergence-free vector fields vanishing on rough planar sets

TL;DR

This paper investigates the approximation of divergence-free vector fields by smooth divergence-free fields in various Sobolev and Hölder spaces within bounded planar domains, with conditions depending on boundary regularity and domain complement structure.

Contribution

It establishes new approximation results for divergence-free fields in Sobolev and Hölder spaces under specific geometric and measure-theoretic conditions on the domain boundary.

Findings

Approximation holds for p>2 if boundary has zero Lebesgue measure.

Approximation holds for p≤2 if the domain complement decomposes into finitely many well-behaved sets.

In Hölder spaces, the approximation property holds in all bounded domains.

Abstract

Given any divergence-free vector field of Sobolev class $W^{m,p}_0(\Omega)$ in a bounded open subset $\Omega \subset \mathbb{R}^2$, we are interested in approximating it in the $W^{m,p}$ norm with divergence-free smooth vector fields compactly supported in $\Omega$. We show that this approximation property holds in the following cases: For $p>2$, this holds given that $\partial \Omega$ has zero Lebesgue measure (a weaker but more technical condition is sufficient); For $p \leq 2$, this holds if $\Omega^c$ can be decomposed into finitely many disjoint closed sets, each of which is connected or $d$-Ahlfors regular for some $d\in[0,2)$. This has links to the uniqueness of weak solutions to the Stokes equation in $\Omega$. For H\"older spaces, we prove this approximation property in general bounded domains.