Superconductive and insulating inclusions for linear and non-linear conductivity 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 $\mathrm{div}(\sigma |\nabla u{|}^{p-2}\nabla u)=0$ where the measurable conductivity $\sigma :\Omega \to [0,\infty ]$ is zero or infinity in large sets and $1<p<\infty$.

39 pages

References in corpus (6)

• Monotonicity based shape reconstruction in electrical impedance tomography
• Global uniqueness for the Calderón problem with Lipschitz conductivities
• Enclosure method for the p-Laplace equation
• A new formulation of the probe method and related problems
• Calderón problem for the p-Laplacian: First order derivative of conductivity on the boundary
• Inverse problems for $p$-Laplace type equations under monotonicity assumptions

Cited by in corpus (9)

• Monotonicity and enclosure methods for the p-Laplace equation
• Calderón problem for the p-Laplacian: First order derivative of conductivity on the boundary
• Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term
• Monotonicity Principle in Tomography of Nonlinear Conducting Materials
• Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography
• The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Data
• An inverse boundary value problem for the inhomogeneous porous medium equation
• The $p_0$-Laplace "Signature" for Quasilinear Inverse Problems
• Two uniqueness results in the inverse boundary value problem for the weighted p-Laplace equation