The use of Rellich identities on certain nongraph boundaries (Q2767869)

In recent years, the method of layer potentials for solving boundary value problems in general Lipschitz domains has had spectacular successes; cf. \textit{C. E. Kenig} [Harmonic analysis techniques for second order elliptic boundary value problems, Regional Conference Series in Mathematics 83 (1994; Zbl 0812.35001)] for a survey. On the other hand, the boundaries of polyhedral domains are not given as graphs of Lipschitz functions in general. The author presents an extension of the method based on Rellich identities to general polyhedra in three dimensions. As an application the invertibility of the classical layer potentials and the well-posedness of the Dirichlet and Neumann problems for the Laplacian are then proven. Among further possible developments, the extension of the results to cones with Lipschitz facets is finally discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00020].