## Abstract

We consider the mixed boundary value problem, or Zaremba's problem, for the Laplacian in a bounded Lipschitz domain Ω in R^{n}, n ≥ 2. We decompose the boundary ∂Ω= D ∪ N with D and N disjoint. The boundary between D and N is assumed to be a Lipschitz surface in ∂Ω. We find an exponent q_{0} > 1 so that for p between 1 and q_{0} we may solve the mixed problem for L^{p}. Thus, if we specify Dirichlet data on D in the Sobolev space W^{1,p}(D) and Neumann data on N in L^{p} (N), the mixed problem with data f_{D} and f_{N} has a unique solution and the non-tangential maximal function of the gradient lies in L^{p}(∂Ω). We also obtain results for p = 1 when the data comes from Hardy spaces.

Original language | English |
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Pages (from-to) | 1333-1364 |

Number of pages | 32 |

Journal | Potential Analysis |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - May 2013 |

### Bibliographical note

Funding Information:Research supported, in part, by the National Science Foundation.

Funding Information:

This work was partially supported by a grant from the Simons Foundation (#195075 to Russell Brown).

## Keywords

- Laplacian
- Mixed boundary value problem
- Non-smooth domain

## ASJC Scopus subject areas

- Analysis