### 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) |
---|---|

Pages (from-to) | 91-124 |

Number of pages | 34 |

Journal | Mathematische Annalen |

Volume | 342 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2008 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Annalen*,

*342*(1), 91-124. https://doi.org/10.1007/s00208-008-0223-6

**The mixed problem in L p for some two-dimensional Lipschitz domains.** / Lanzani, Loredana; Capogna, Luca; Brown, Russell M.

Research output: Contribution to journal › Article

*Mathematische Annalen*, vol. 342, no. 1, pp. 91-124. https://doi.org/10.1007/s00208-008-0223-6

}

TY - JOUR

T1 - The mixed problem in L p for some two-dimensional Lipschitz domains

AU - Lanzani, Loredana

AU - Capogna, Luca

AU - Brown, Russell M.

PY - 2008/9

Y1 - 2008/9

N2 - We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD 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 .

AB - We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD 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 .

UR - http://www.scopus.com/inward/record.url?scp=46649096818&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46649096818&partnerID=8YFLogxK

U2 - 10.1007/s00208-008-0223-6

DO - 10.1007/s00208-008-0223-6

M3 - Article

AN - SCOPUS:46649096818

VL - 342

SP - 91

EP - 124

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1

ER -