Higher Order Elliptic Problems in Non-Smooth Domains


Lunedì 15 Marzo alle ore 14:45 in Aula 1C/150 il prof. Vladimir Maz'ya delle Universita' di Liverpool e Linkoeping terra' un seminario dal titolo "Higher Order Elliptic Problems in Non-Smooth 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 Lame' 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 Lame' system at this point. The lectures are based on my work and recent joint papers with S.Mayboroda (Purdue) and Guo Luo (Caltech).

Rif. int. P.D. Lamberti