The Neumann function and the L p Neumann problem in chord-arc domains

Abstract

We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in which L p solvability of the Neumann problem is shown to yield solvability in L q for 1 < q < p, and in the Hardy space H 1, in 2-sided chord-arc domains, under suitable background hypotheses.

Keywords

• neumann function
• neumann problem
• second order elliptic equations
• chord-arc domains
• 35j25
• 42b37
• 42b30