Superconductive and insulating inclusions for linear and non-linear conductivity equations (bibtex)

by Tommi Brander, Joonas Ilmavirta, Manas Kar

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 $1 < p < \infty$.

Reference:

Superconductive and insulating inclusions for linear and non-linear conductivity equations (Tommi Brander, Joonas Ilmavirta, Manas Kar), Inverse Problems and Imaging, volume 12, number 1, pp. 91–123, 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 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 $1 < p < \infty$.
[arXiv]

Bibtex Entry:

@article{p-inclusion, author = {Tommi Brander and Joonas Ilmavirta and Manas Kar}, title = {{Superconductive and insulating inclusions for linear and non-linear conductivity equations}}, journal = {Inverse Problems and Imaging}, month = jan, volume = {12}, number = {1}, pages = {91--123}, doi = {10.3934/ipi.2018004}, year = {2018}, arxiv = {1510.09029}, gsid = {9722539612223229238}, url={http://users.jyu.fi/~jojapeil/pub/p_cavity.pdf}, 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 $1 < p < \infty$.} }

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