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

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.

令\（\ Omega \）是具有光滑边界\（\ partial \ Omega \）和\（1 <r <\ infty \）的二维外部域。然后\（L ^ r（\ Omega）^ 2 \）可以进行亥姆霍兹-韦尔分解，即，对于L ^ r（\ Omega ^^ 2 \）中的每个\（\ mathbf {u} \都存在\（\ mathbf {h} \ in X ^ r _ {\ tiny {\ text {har}}}（\ Omega} \），\（w \ in {\ dot {H}} ^ {1，r}（\ Omega} \ ）和\（p \在{\点{H}} ^ {1，R}（\欧米茄）\） ，使得

也可以从\（V ^ r _ {\ tiny {\ text {har}}}（（Omega）\）中选择函数\（\ mathbf {h} \ ），这是另一个受不同边界限制的谐波矢量场空间条件。谐波矢量的这些空间\（X ^ r _ {\ tiny {\ text {har}}}（\ Omega）\）和\（V ^ r _ {\ tiny {\ text {har}}}（（Omega）\）已知场是有限维的。当且仅当\（1 <r \ leqq 2 \）时，上述分解才是唯一的，而在\（2 <r <\ infty \）的情况下，唯一性仅对\（L ^ r（ \ Omega）^ 2 \）。在先前的论文中证明了三维设置的相应结果，与二维情况相反，存在两个阈值指数，即\（r = 3/2 \）和\（r = 3 \）。在我们的二维情况下，\（r = 2 \）是唯一的临界指数，它决定了唯一的亥姆霍兹-韦尔分解的有效性。