The Poisson equation in homogeneous Sobolev spaces (Q1777654)

Summary: We consider Poisson's equation in an \(n\)-dimensional exterior domain \(G\) \((n \geq 2)\) 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 \(q\).