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Corrosion detection by identification of a nonlinear Robin boundary condition

David Johansson

TL;DR

This work addresses corrosion detection modeled by a conductivity equation with a nonlinear Robin boundary condition, establishing that the nonlinear Robin term $a(x,u)$ can be identified locally from Cauchy data on the accessible boundary. The authors adapt a linearization-based strategy, leveraging Runge approximation to relate nonlinear identifiability to the linearized problem and to the behavior of solutions on the boundary, proving local injectivity of the parameter-to-measurement map and a partial global result. The main contributions are (i) a local identifiability result for $a(x,u)$ on the inaccessible boundary from local Cauchy data, and (ii) a partial global result showing $a$ is determined on the reachable set when global Cauchy data coincide, with openness of the reachable set established via Runge approximation. This advances the understanding of inverse problems for nonlinear Robin types and provides a pathway toward global identifiability under the proposed framework.

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.

Corrosion detection by identification of a nonlinear Robin boundary condition

TL;DR

This work addresses corrosion detection modeled by a conductivity equation with a nonlinear Robin boundary condition, establishing that the nonlinear Robin term can be identified locally from Cauchy data on the accessible boundary. The authors adapt a linearization-based strategy, leveraging Runge approximation to relate nonlinear identifiability to the linearized problem and to the behavior of solutions on the boundary, proving local injectivity of the parameter-to-measurement map and a partial global result. The main contributions are (i) a local identifiability result for on the inaccessible boundary from local Cauchy data, and (ii) a partial global result showing is determined on the reachable set when global Cauchy data coincide, with openness of the reachable set established via Runge approximation. This advances the understanding of inverse problems for nonlinear Robin types and provides a pathway toward global identifiability under the proposed framework.

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.

Paper Structure

This paper contains 6 sections, 14 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Precise problem formulation and main results
  3. Historical overview
  4. Solvability of the nonlinear equation
  5. The inverse problem
  6. Theory for the linear Robin problem

Key Result

Theorem 1.2

Let $n\geq 2$ and $\Omega,\partial\Omega,\Gamma_{A},\Gamma_{I}$ satisfy assumption-1 and assumption-2. Assume that $\gamma$ satisfies assumption-3 and $a_{1},a_{2}\in\mathcal{A}_{ad}$. If there exists solutions $w_{1},w_{2}$ corresponding to $a_{1},a_{2}$ with equal Cauchy data on $\Gamma_{A}$ and c then $w_{1}=w_{2}$ on $\Gamma_{I}$ and it holds, for some $\lambda>0$, that

Theorems & Definitions (29)

  • Definition 1.1: Admissible nonlinearity
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2: Fixed point equation
  • proof
  • Lemma 2.3: Solution map
  • proof
  • proof : Proof of Theorem \ref{['thm-main-result-1']}
  • ...and 19 more