The Zaremba problem in two-dimensional Lipschitz graph domains
Marı́a J. Carro, Teresa Luque, Virginia Naibo · Transactions of the American Mathematical Society · 2025
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 r r such that the Zaremba problem is solvable in L p L^p for 1 > p > r 1>p>r and in the Lorentz space L r , 1 L^{r,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 L 2 L^2 -unweighted case, where known examples show that the gradient of the solution may fail to be squared-integrable, we prove restricted weak-type estimates.