The Neumann problem in graph Lipschitz domain in the plane

Abstract

We study new aspects of the solvability of the classical Neumann boundary value problem in a graph Lipschitz domain in the plane. When the domain is the upper half-plane, the boundary data is assumed to belong to weighted Lebesgue or weighted Lorentz spaces; we show that the solvability of the Neumann problem in these settings may be characterized in terms of Muckenhoupt weights and related weights, respectively. For a general graph Lipschitz domain , as proved in an unpublished work by E. Fabes and C. Kenig, there exists such that the Neumann problem is solvable with data in for we review the proof of this result and show that the Neumann problem is solvable at the endpoint with data in the Lorentz space We present examples of our results in Schwarz–Christoffel Lipschitz domains and related domains.