Higher Order Elliptic Problems in Arbitrary Domains
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.
We also study the 3D Lam\'e system and establish its weighted
positive definiteness for a certain range of elastic constants. By modifying
the general theory developed by Maz'ya (Duke, 2002), we then show, under the
assumption of weighted positive definiteness, that the divergence
of the classical Wiener integral for a boundary point guarantees the
continuity of solutions to the Lam\'e system at this point.
The talk is based on my joint work with
S.Mayboroda (Minnesota) and Guo Luo (Caltech)