## Abstract

We consider the mixed problem for L the Lamé system of elasticity in a bounded Lipschitz domain Ω⊂R^{2}. We suppose that the boundary is written as the union of two disjoint sets, ∂Ω=D∪N. We take traction data from the space L^{p}(N) and Dirichlet data from a Sobolev space W^{1,p}(D) and look for a solution u of Lu=0 with the given boundary conditions. In our main result, we find a scale-invariant condition on D and an exponent p_{0}>1 so that for 1<p<p_{0}, we can find a unique solution of this boundary value problem with the non-tangential maximal function of the gradient in L^{p}(∂Ω). We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with p in (p_{1}, 1] for some p_{1}<1.

Original language | English |
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Pages (from-to) | 4373-4400 |

Number of pages | 28 |

Journal | Journal of Differential Equations |

Volume | 254 |

Issue number | 12 |

DOIs | |

State | Published - Jun 15 2013 |

### Bibliographical note

Funding Information:E-mail address: russell.brown@uky.edu (R.M. Brown). 1 Katharine Ott is partially supported by a grant from the US National Science Foundation, DMS 1201104. 2 Russell Brown is partially supported by a grant from the Simons Foundation (#195075).

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics