Lp Boundary Value Problems on Lipschitz Domains.

摘要

In this dissertation we study the Lp Dirichlet and the regularity of the Lp Dirichlet problem for the Stokes system of linearized hydrostatics and higher, as well as second, order elliptic systems and equations on Lipschitz domains. We are able to establish two sufficient conditions for the solvability of the Lp Dirichlet problem for the Stokes system on Lipschitz domains, which leads to the solvability of the Lp Dirichlet problem for 2 - epsilon < p < 2d-1d-3 + epsilon. In the case of higher order elliptic systems we are able to establish a necessary and sufficient condition for the solvability of the Lp regularity problem when p > 2. This allows us to show that the solvability of the Lp regularity problem implies the solvability of the Lp Dirichlet problem for general higher order elliptic systems when 2 < q < q0 + epsilon where 1q0=1p -1d-1 . Then, we show that in the case of second order elliptic systems the solvability of the Lp regularity problem is equivalent to the solvability of the Lp' Dirichlet problem when 1p+1p' = 1. Finally, we show that the Lp regularity problem for the biharmonic equation implies the solvability of the Lp' Dirichlet problem. This leads to the solvability of the Lp regularity problem in a new range of p when d ≥ 4.;KEYWORDS: Lipschitz domains, regularity problem, Dirichlet problem, elliptic systems, biharmonic equation.