The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Data

TL;DR

This paper introduces a method to approximate nonlinear inverse problems in imaging, specifically Electrical Resistance Tomography, using weighted p-Laplace equations, facilitating the application of linear imaging techniques to nonlinear materials.

Contribution

It demonstrates that nonlinear boundary value problems with large data can be approximated by weighted p-Laplace problems, bridging linear and nonlinear imaging methods.

Findings

Nonlinear problems can be approximated by weighted p-Laplace equations.

Large boundary data allows replacing materials with ideal conductors or insulators.

The approach extends linear imaging algorithms to nonlinear material contexts.

Abstract

This paper is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of developments, although breakthroughs are expected in the not-too-distant future. We consider nonlinear constitutive relationships which, at a given point in the space, present a behaviour for large arguments that is described by monomials of order p and q. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted p-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p-$Laplacian in inverse problems with nonlinear materials. Moreover, when p=2, this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials. The main result of this work is that for "large" Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted p-Laplace problem.