### Abstract

We prove the H_{p,q} ^{1} solvability of second order systems in divergence form with leading coefficients A^{αβ} only measurable in (t, x^{1}) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A^{11} is measurable in t and has a small BMO semi-norm in the other variables; (ii) A^{11} is measurable in x^{1} and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition.

Original language | English |
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Pages (from-to) | 357-389 |

Number of pages | 33 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Jan 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics