RT Journal Article
T1 The Zaremba problem in two-dimensional Lipschitz graph domains
A1 Carro Rossell, María Jesús
A1 Luque Martínez, Teresa Elvira
A1 Naibo, V.
AB 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.
PB American Mathematical Society
YR 2025
FD 2025
LK https://hdl.handle.net/20.500.14352/130468
UL https://hdl.handle.net/20.500.14352/130468
LA eng
NO Carro, M., Luque, T., & Naibo, V. The Zaremba problem in two-dimensional Lipschitz graph domains. Transactions of the American Mathematical Society. 2025; 378(10): 6885-6911.
NO Ministerio de Ciencia, Innovación y Universidades
NO Agencia Estatal de Investigación
NO National Sanitation Foundation
DS Docta Complutense
RD 19 mar 2026