The mixed problem for the Laplacian in Lipschitz domains

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.

Version 5 includes a correction to one step of the main proof. Since the paper appeared long ago, this submission includes the complete paper, followed by a short section that gives the correction to one step in the proof

References in corpus (7)

• The mixed problem in L^p for some two-dimensional Lipschitz domains
• The mixed problem for the Laplacian in Lipschitz domains
• The mixed problem in Lipschitz domains with general decompositions of the boundary
• Lp estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains
• The L^p Boundary Value Problems on Lipschitz Domains
• Estimates for the $L^q$-mixed problem in $C^{1,1}$-domains
• The $L^p$ Regularity Problem on Lipschitz Domains

Cited by in corpus (10)

• The Green function for elliptic systems in two dimensions
• The mixed problem for the Laplacian in Lipschitz domains
• The mixed problem for the Lamé system in two dimensions
• Estimates for the $L^q$-mixed problem in $C^{1,1}$-domains
• Explicit estimates for solutions of mixed elliptic problems
• Higher order Poisson Kernels and $L^p$ polyharmonic boundary value problems in Lipschitz domains
• Weighted Global Regularity Estimates for Elliptic Problems with Robin Boundary Conditions in Lipschitz Domains
• BIE and BEM approach for the mixed Dirichlet-Robin boundary value problem for the nonlinear Darcy-Forchheimer-Brinkman system
• The Dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions
• Mixed Boundary Value Problems on Cylindrical Domains