Inclusions of General Shapes Having Constant Field Inside the Core and NonElliptical Neutral Coated Inclusions With Anisotropic Conductivity


For certain shapes of inclusions embedded in a body, the field inside the inclusion is uniform for some boundary condition. We provide a construction scheme for inclusions of general shapes having such a uniformity property in two dimensions based on the conformal mapping technique for the potential problem. Using this complex analysis method, we also design nonelliptical neutral coated inclusions with anisotropic conductivity. Neutral coated inclusions do not perturb a background uniform field when they are inserted into a homogeneous matrix. Although coated inclusions of various shapes are neutral to a single field, only concentric ellipses or confocal ellipsoids can be neutral to all uniform fields. This paper presents our work relating to the construction of nonelliptical coated inclusions with anisotropic conductivity in two dimensions that are neutral to all uniform fields, where the assignment of the flux condition on the boundary of the core depends on the applied background field. Using these neutral inclusions, we obtain cylindrical neutral inclusions in three dimensions, with no flux applied to the boundary of the core and with the anisotropic conductivity function of the shell given in accordance with the background uniform field.


  1. neutral inclusion
  2. antiplane elasticity
  3. anisotropic conductivity
  4. $E_\Omega$-inclusion

MSC codes

  1. 35Q74
  2. 35B30

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1420 - 1440
ISSN (online): 1095-712X


Submitted: 21 February 2019
Accepted: 19 March 2020
Published online: 8 June 2020


  1. neutral inclusion
  2. antiplane elasticity
  3. anisotropic conductivity
  4. $E_\Omega$-inclusion

MSC codes

  1. 35Q74
  2. 35B30



Funding Information

National Research Foundation of Korea : 2016R1A2B4014530, 2019R1F1A1062782
National Science Foundation : DMS-1211359, DMS-1814854

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