Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions |
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Authors: | Habib Ammari Mikyoung Lim |
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Institution: | a Laboratoire Ondes et Acoustique, CNRS UMR 7587 & ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France b Department of Mathematics, Inha University, Incheon 402-751, Republic of Korea c Department of Mathematical Sciences, KAIST, Daejeon 305-701, Republic of Korea d Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea |
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Abstract: | When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution to the conductivity equation blows up in general. In this paper, we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form: one of them has a bounded gradient and the gradient of the other part blows up. Using the decomposition, we derive the best possible estimates for the blow-up of the gradient. We then consider the case when the inclusions have positive permittivities. We show quantitatively that in this case the size of the blow-up is reduced. |
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Keywords: | 35J25 73C40 |
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