The Poisson equation in homogeneous Sobolev spaces

We consider Poisson’s equation in an n-dimensional exterior domain G( n≥ 2) with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certain L q (G)-spaces there exists a solution in the homogeneous Sobolev space S 2,q (G), containing functions being local in L q (G) and having second-order derivatives in L q (G). Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimension n + 1, independent of q.

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