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

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1510.09029

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### Abstract

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 also prove similar results when the underlying equation is the quasilinear $$p$$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $$\operatorname{div}(\sigma\lvert\nabla u\rvert^{p-2}\nabla u)=0$$ where the measurable conductivity $$\sigma\colon\Omega\to[0,\infty]$$ is zero or infinity in large sets and $$1<p<\infty$$.

### Author and article information

###### Journal
2015-10-30
2016-04-04

35R30, 35J92 (Primary), 35H99 (Secondary)
38 pages
math.AP
 ScienceOpen disciplines: Analysis