### 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 |
---|---|

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

### Cite this

**L _{p} solvability of divergence type parabolic and elliptic systems with partially BMO coefficients.** / Dong, Hongjie; Kim, Doyoon.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Lp solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

AU - Dong, Hongjie

AU - Kim, Doyoon

PY - 2011/1/1

Y1 - 2011/1/1

N2 - We prove the Hp,q 1 solvability of second order systems in divergence form with leading coefficients Aαβ only measurable in (t, x1) 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) A11 is measurable in t and has a small BMO semi-norm in the other variables; (ii) A11 is measurable in x1 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.

AB - We prove the Hp,q 1 solvability of second order systems in divergence form with leading coefficients Aαβ only measurable in (t, x1) 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) A11 is measurable in t and has a small BMO semi-norm in the other variables; (ii) A11 is measurable in x1 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.

UR - http://www.scopus.com/inward/record.url?scp=78751574391&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78751574391&partnerID=8YFLogxK

U2 - 10.1007/s00526-010-0344-0

DO - 10.1007/s00526-010-0344-0

M3 - Article

VL - 40

SP - 357

EP - 389

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 3

ER -