Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in AntiPlane Elasticity
 Habib Ammari,
 Giulio Ciraolo,
 Hyeonbae Kang,
 Hyundae Lee,
 Kihyun Yun
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When holes or hard elastic inclusions are closely located, stress which is the gradient of the solution to the antiplane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blowup of the gradient of such an equation. In this paper we show that the blowup of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann–Poincaré type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quantitatively characterize the blowup of the gradient in terms of explicit functions. In electrostatics, our results apply to the electric field, which is the gradient of the solution to the conductivity equation, in the case where perfectly conducting or insulating inclusions are closely located.
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 Title
 Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in AntiPlane Elasticity
 Journal

Archive for Rational Mechanics and Analysis
Volume 208, Issue 1 , pp 275304
 Cover Date
 20130401
 DOI
 10.1007/s0020501205908
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 Habib Ammari ^{(1)}
 Giulio Ciraolo ^{(2)}
 Hyeonbae Kang ^{(3)}
 Hyundae Lee ^{(3)}
 Kihyun Yun ^{(4)}
 Author Affiliations

 1. Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, France
 2. Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy
 3. Department of Mathematics, Inha University, Incheon, 402751, Korea
 4. Department of Mathematics, Hankuk University of Foreign Studies, Younginsi, Gyeonggido, 449791, Korea