The $p_0$-Laplace "Signature" for Quasilinear Inverse Problems

TL;DR

This paper introduces a novel approach to nonlinear inverse problems in imaging by approximating them with a weighted p_0-Laplace problem, enabling the use of linear methods for nonlinear materials in tomography.

Contribution

It demonstrates that nonlinear inverse problems can be approximated by a weighted p_0-Laplace problem, bridging linear and nonlinear imaging techniques.

Findings

Nonlinear materials can be approximated by a weighted p_0-Laplace problem.

For small Dirichlet data, one material acts as a perfect electric conductor.

The approach extends linear imaging methods to nonlinear materials.

Abstract

This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials in the early stages of developments, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted $p_0$-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p_0$-Laplacian in inverse problems with nonlinear materials. Moreover, when $p_0=2$, this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for "small" Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted $p_0$-Laplace problem.