Corrosion detection by identification of a nonlinear Robin boundary condition

TL;DR

This paper addresses an inverse boundary value problem for corrosion detection, demonstrating local identifiability of a nonlinear Robin boundary condition from boundary measurements and adapting linearization methods for nonlinear inverse problems.

Contribution

It proves local identifiability of the nonlinear Robin boundary term from boundary data and suggests a strategy for global identification, adapting methods from semilinear elliptic equations.

Findings

Nonlinear Robin boundary condition can be identified locally from boundary measurements.

A partial result towards global identification is established.

An adaptation of linearization methods for nonlinear inverse problems is developed.

Abstract

We study an inverse boundary value problem in corrosion detection. The model is based on a conductivity equation with nonlinear Robin boundary condition. We prove that the nonlinear Robin term can be identified locally from Cauchy data measurements on a subset of the boundary. A possible strategy for turning a local identification result into a global one is suggested, and a partial result is proved in this direction. The inversion method is an adaptation to this nonlinear Robin problem of a method originally developed for semilinear elliptic equations. The strategy is based on linearization and relies on parametrizing solutions of the nonlinear equation on solutions of the linearized equation.