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

Abstract

Consider the space of harmonic vector fields h in $$L^r(\Omega )$$ for $$1<r<\infty $$ in the two-dimensional exterior domain $$\Omega $$ with the smooth boundary $$\partial \Omega $$ subject to the boundary conditions $$h\cdot \nu =0$$ or $$h\wedge \nu =0$$ , where $$\nu $$ denotes the unit outward normal to $$\partial \Omega $$ . Denoting these spaces by $$X^r_{\tiny {\text{ har }}}(\Omega )$$ and $$V^r_{\tiny {\text{ har }}}(\Omega )$$ , respectively, it is shown that, in spite of the lack of compactness of $$\Omega $$ , both of these spaces are finite dimensional and that their dimension of both spaces coincides with L for $$2< r<\infty $$ and $$L-1$$ for $$1<r\le 2$$ . . Here L is the number of disjoint simple closed curves consisting of the boundary $$\partial \Omega $$ .