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L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

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Abstract

We prove the \({{\mathcal{H}}^{1}_{p,q}}\) 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.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Hongjie Dong

  2. Department of Applied Mathematics, Kyung Hee University, 1 Seochun-dong, Gihung-gu, Yongin-si, Gyeonggi-do, 446-701, Korea

    Doyoon Kim

Authors
  1. Hongjie Dong
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  2. Doyoon Kim
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Correspondence to Doyoon Kim.

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Communicated by N. Trudinger.

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Dong, H., Kim, D. L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients. Calc. Var. 40, 357–389 (2011). https://doi.org/10.1007/s00526-010-0344-0

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  • DOI: https://doi.org/10.1007/s00526-010-0344-0

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