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

Consider the space of harmonic vector fields u in L^r(Ω ) for 1 < r< ∞ for three dimensional exterior domains Ω with smooth boundaries ∂Ω subject to the boundary conditions u· ν= 0 or u× ν= 0 , where ν denotes the unit outward normal on ∂Ω. Denoting these spaces by Xharr(Ω) and Vharr(Ω), it is shown that, in spite of the lack of compactness of Ω , both of these spaces are finite dimensional and that dimVharr(Ω) equals L for 3 / 2 < r< ∞ and L- 1 for 1 < r≤ 3 / 2. Here L is a number representing topologically invariant quantities of ∂Ω and which in the case of bounded domains coincides with the first Betti number. In contrast to the situation of bounded domains, the dimension of Vharr(Ω) in exterior domains is depending on the Lebesgue exponent r. The critical value of this exponent for exterior domains is determined to be 3/2.