The Neumann problem in rough domains
Derek Sparrius · 2024
The purpose of this thesis is to study the solvability of the Neumann problem for divergence form elliptic operators on chord arc domains. We begin, in the setting of a 1 sided chord arc domain, by constructing the Neumann function associated with the differential operator and prove that the Neumann function is Holder continuous up to the boundary. Then, in the setting of a 2 sided chord arc domain, we establish the solvability of the Neumann problem with data in L^q, 1 [less than] q [less than] p as well as in the Hardy space H^1 under the assumptions we have solvability of the Neumann problem and Regularity problem with data in L^p and the Dirichlet problem for the transpose operator with data in L^p.