Piecewise nonlinear materials and Monotonicity Principle

TL;DR

This paper extends the Monotonicity Principle to nonlinear materials with piecewise growth exponents, enabling fast imaging methods for complex multi-material problems including superconductors and insulators.

Contribution

It generalizes the Monotonicity Principle to a broad class of nonlinear, piecewise materials, facilitating advanced imaging techniques in practical scenarios.

Findings

Theoretical extension of MP to nonlinear, piecewise materials.

Numerical examples confirm the validity of the extended MP.

Applicable to materials like superconductors, PEC, and PEI.

Abstract

This paper is focused on the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. This results are relevant because enables the use of a fast imaging method based on MP, to the wide class of problems with two or more materials, where at least one is nonlinear. The treatment is very general and allows to model a wide variety of practical configurations such as, for instance, Superconducting (SC) or Perfect Electrical Conducting (PEC) or Perfect Electrical Insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in [Corbo Esposito et. al, Inverse Problems 2021], where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.