The Helmholtz–Weyl decomposition of $L^r$ vector fields for two dimensional exterior domains

Abstract

Let $\Omega$ be a two-dimensional exterior domain with smooth boundary $\partial \Omega$ and $1< r < \infty$. Then $L^r(\Omega )^2$ allows a Helmholtz–Weyl decomposition, i.e., for every $\mathbf{u}\in L^r(\Omega )^2$ there exist $\mathbf{h} \in X^r_{\tiny {\text{ har }}}(\Omega )$, $w \in {\dot{H}}^{1,r}(\Omega )$ and $p \in {\dot{H}}^{1,r}(\Omega )$ such that

$$\begin{aligned} \mathbf{u} = \mathbf{h} + \mathrm{rot}\, w + \nabla p. \end{aligned}$$

The function $\mathbf{h}$ can be chosen alternatively also from $V^r_{\tiny {\text{ har }}}(\Omega )$, another space of harmonic vector fields subject to different boundary conditions. These spaces $X^r_{\tiny {\text{ har }}}(\Omega )$ and $V^r_{\tiny {\text{ har }}}(\Omega )$ of harmonic vector fields are known to be finite dimensional. The above decomposition is unique if and only if $1< r \leqq 2$, while in the case $2< r < \infty$, uniqueness holds only modulo a one dimensional subspace of $L^r(\Omega )^2$. The corresponding result for the three dimensional setting was proved in our previous paper, where in contrast to the two dimensional case, there are two threshold exponents, namely $r=3/2$ and $r =3$. In our two dimensional situation, $r=2$ is the only critical exponent, which determines the validity of a unique Helmholtz–Weyl decomposition.