A Characterization of Harmonic L^r -Vector Fields in Two-Dimensional Exterior Domains

Consider the space of harmonic vector fields h in L^r(Ω) for 1 < r< ∞ in the two-dimensional exterior domain Ω with the smooth boundary ∂Ω subject to the boundary conditions h· ν= 0 or h∧ ν= 0 , where ν denotes the unit outward normal to ∂Ω. Denoting these spaces by Xharr(Ω) and Vharr(Ω), respectively, it is shown that, in spite of the lack of compactness of Ω , both of these spaces are finite dimensional and that their dimension of both spaces coincides with L for 2 < r< ∞ and L- 1 for 1 < r≤ 2. Here L is the number of disjoint simple closed curves consisting of the boundary ∂Ω.