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Optimal Estimates for the Conductivity Problem by Green’s Function Method

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Abstract

We study a class of second-order elliptic equations of divergence form, with discontinuous coefficients and data, which models the conductivity problem in composite materials. We establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions. These extend the known results in the literature and answer open problem (b) proposed by Li and Vogelius (2000) for the isotropic conductivity problem. We also obtain more interesting higher-order derivative estimates, which answers open problem (c) of Li and Vogelius (2000). It is worth pointing out that the equations under consideration in this paper are non-homogeneous.

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Funding

Hongjie Dong was partially supported by the NSF under agreements DMS-1056737 and DMS-1600593. Haigang Li was partially supported by NSFC (11571042) (11631002), Fok Ying Tung Education Foundation (151003).

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Authors and Affiliations

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

    Hongjie Dong

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    Haigang Li

Authors
  1. Hongjie Dong
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  2. Haigang Li
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Correspondence to Hongjie Dong.

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The authors declare that they have no conflict of interest.

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Communicated by F. Otto

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Dong, H., Li, H. Optimal Estimates for the Conductivity Problem by Green’s Function Method. Arch Rational Mech Anal 231, 1427–1453 (2019). https://doi.org/10.1007/s00205-018-1301-x

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  • DOI: https://doi.org/10.1007/s00205-018-1301-x

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