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Elliptic Equations in Divergence Form with Partially BMO Coefficients

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Abstract

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients a ij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that a ij have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients.

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Authors and Affiliations

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

    Hongjie Dong

  2. Department of Mathematics, University of Southern California, 3620 South Vermont Avenue, KAP 108, Los Angeles, CA, 90089-2532, USA

    Doyoon Kim

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

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Communicated by V. Sverak

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Dong, H., Kim, D. Elliptic Equations in Divergence Form with Partially BMO Coefficients. Arch Rational Mech Anal 196, 25–70 (2010). https://doi.org/10.1007/s00205-009-0228-7

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  • DOI: https://doi.org/10.1007/s00205-009-0228-7

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