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

TL;DR

This paper develops methods to detect and analyze inclusions with infinite or zero conductivity in both linear and nonlinear conductivity equations, extending techniques to quasilinear p-Laplace equations with complex conductivities.

Contribution

It introduces the use of the enclosure and probe methods for identifying inclusions with extreme conductivities in nonlinear PDEs, including rigorous treatment of the forward problem for the p-Laplace equation with measurable, unbounded conductivities.

Findings

Successfully applied the enclosure method to nonlinear equations.

Established the forward problem for PDEs with zero or infinite conductivities.

Extended boundary measurement techniques to quasilinear p-Laplace equations.

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 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 $\operatorname{div}(\sigma\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $\sigma\colon\Omega\to[0,\infty]$ is zero or infinity in large sets and $1<p<\infty$.