### Abstract

We consider the mixed problem, {Δ u = 0 in Ω ∂u = f _{N} on N u = f_{D} on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f _{D} , has one derivative in L ^{p} (D) of the boundary and the Neumann data, f _{N} , is in L ^{p} (N). We find a p _{0} > 1 so that for p in an interval (1, p _{0}), we may find a unique solution to the mixed problem and the gradient of the solution lies in L ^{p} .

Original language | English (US) |
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Pages (from-to) | 91-124 |

Number of pages | 34 |

Journal | Mathematische Annalen |

Volume | 342 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1 2008 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Lanzani, L., Capogna, L., & Brown, R. M. (2008). The mixed problem in L

^{p}for some two-dimensional Lipschitz domains.*Mathematische Annalen*,*342*(1), 91-124. https://doi.org/10.1007/s00208-008-0223-6