Vladimir Maz'ya (Liverpool and Linkoeping)

Higher order elliptic problems in non-smooth domains

Abstract: We discuss sharp continuity and regularity results for solutions of the polyharmonic equation in an arbitrary open set. The absence of information about geometry of the domain puts the question of regularity properties beyond the scope of applicability of the methods devised previously, which typically rely on specific geometric assumptions. Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron. The techniques developed recently allow to establish the boundedness of derivatives of solutions to the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain and to show that the order of the derivatives is maximal. An appropriate notion of polyharmonic capacity is introduced which allows one to describe the precise correlation between the smoothness of solutions and the geometry of the domain.