Duality in spaces of polyharmonic functions (Q1397544)

The author gives a short summary of results which were submitted with full proofs to the same journal in 1998. Let \(D\) be a bounded domain in \(\mathbb{R}^n\) whose boundary is connected and real-analytic. Let \(\Delta\) denote the Laplace operator in \(\mathbb{R}^n\) and, for some integer \(m\), let \(\text{sol} (\Delta^m,D)\) denote the space of solutions of \(\Delta^m u=0\) in \(D\), with the topology of uniform convergence on compact sets of \(u\) and all of its derivatives. His aim is to give a representation of the dual space of \(\text{sol} (\Delta^m,D)\) with the topology of uniform convergence on bounded subsets of \(\text{sol} (\Delta^m,D)\). His main result says that this is topologically isomorphic to the space \(\text{sol} (\Delta^m, \overline{D})\) which is the inductive limit of spaces of solutions of \(\Delta^mu=0\) in a certain decreasing sequence of neighborhoods of the closure \(\overline{D}\) of \(D\).