Superconductive and insulating inclusions for linear and non-linear conductivity equations

Tommi Brander; Joonas Ilmavirta; Manas Kar

doi:10.3934/ipi.2018004

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2018, Volume 12, Issue 1: 91-123. Doi: 10.3934/ipi.2018004

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Superconductive and insulating inclusions for linear and non-linear conductivity equations

Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland

* Corresponding author: Joonas Ilmavirta
* Corresponding author: Joonas Ilmavirta

Received: April 2016

Revised: August 2017

Published: February 2018

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We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(σ\lvert\nabla u\rvert^{p-2}\nabla u) = 0$ where the measurable conductivity $σ\colonΩ\to[0,∞]$ is zero or infinity in large sets and $1<p<∞$.

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Mathematics Subject Classification: Primary: 35R30, 35J92; Secondary: 35H99.

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