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$L^r$ harmonic vector fields in 3D exterior domains

We characterize the space of harmonic vector fields in $L^r$ on the 3D exterior domain with smooth boundary. There are two kinds of boundary conditions. One is such a condition as the vector fields are tangential to the boundary, and another is such one as those are perpendicular to the boundary. In bounded domains both harmonic vector spaces are of finite dimensions and characterized in terms of topologically invariant quantities which we call the first and the second Betti numbers. These properties are closely related to characterization the null spaces of solutions to the elliptic boundary value problems associated with the operators div and rot. We shall show that, in spite of lack of compactness, spaces of harmonic vector fields in $L^r$ on the 3D exterior domain are of finite dimensions and characterized similarly to those in bounded domains. It will be also clarified a significant difference between interior and exterior domains in accordance with the integral exponent $1 < r < \infty$. This is based on the joint work with Profs. Matthias Hieber, Aoton Seyferd, Senjo Shimizu and Taku Yanagisawa.