The Zaremba problem in two-dimensional Lipschitz graph domains

Abstract

We study the Zaremba problem, or mixed problem associated to the Laplace operator, in two-dimensional Lipschitz graph domains with mixed Dirichlet and Neumann boundary data in Lebesgue and Lorentz spaces. We obtain an explicit value such that the Zaremba problem is solvable in for Lp and in the Lorentz space L(p,1). Applications include those where the domain is a cone with vertex at the origin and, more generally, a Schwarz–Christoffel domain. The techniques employed are based on results of the Zaremba problem in the upper half-plane, the use of conformal maps and the theory of solutions to the Neumann problem. For the case when the domain is the upper half-plane, we work in the weighted setting, establishing conditions on the weights for the existence of solutions and estimates for the non-tangential maximal function of the gradient of the solution. In particular, in the unweighted case, where known examples show that the gradient of the solution may fail to be squared-integrable, we prove restricted weak-type estimates.