The mixed problem for the Laplacian in Lipschitz domains

TL;DR

This paper investigates the existence and uniqueness of solutions to the mixed boundary value problem for the Laplacian in Lipschitz domains, establishing results for data in L^p and Hardy spaces.

Contribution

It extends the theory of mixed boundary problems for the Laplacian to Lipschitz domains with boundary data in L^p and Hardy spaces, including cases near p=1.

Findings

Unique solutions exist with non-tangential maximal function in L^p

Results hold for boundary data in Hardy spaces at p=1

Boundary between Dirichlet and Neumann parts is Lipschitz

Abstract

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.