Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1427–1453 | Cite as

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

  • Hongjie DongEmail author
  • Haigang Li


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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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).

Conflict of Interest

The authors declare that they have no conflict of interest.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijingChina

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