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

Let Ω be a two-dimensional exterior domain with smooth boundary ∂Ω and 1 < r< ∞. Then L^r(Ω) ² allows a Helmholtz–Weyl decomposition, i.e., for every u∈ L^r(Ω) ² there exist h∈Xharr(Ω), w∈ H˙ ^1,r(Ω) and p∈ H˙ ^1,r(Ω) such that u=h+rotw+∇p.The function h can be chosen alternatively also from Vharr(Ω), another space of harmonic vector fields subject to different boundary conditions. These spaces Xharr(Ω) and Vharr(Ω) of harmonic vector fields are known to be finite dimensional. The above decomposition is unique if and only if 1 < r≦ 2 , while in the case 2 < r< ∞, uniqueness holds only modulo a one dimensional subspace of L^r(Ω) 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.