Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem

TL;DR

This paper investigates the unique determination of diffusion and absorption coefficients in a quasilinear elliptic PDE using boundary measurements, introducing a linearization and adjoint approach for coefficient identification.

Contribution

It presents a novel method combining linearization and adjoint techniques to simultaneously identify diffusion and absorption coefficients in a quasilinear elliptic problem.

Findings

Successfully determines $a(0)$ and $c(x)$ from boundary data.

Extends to identify $a(u)$ for all $u$ using an adjoint approach.

Provides a theoretical framework for inverse coefficient problems in PDEs.

Abstract

In this work we consider the identifiability of two coefficients $a(u)$ and $c(x)$ in a quasilinear elliptic partial differential equation from observation of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov [On uniqueness in inverse problems for semilinear parabolic equations. Archive for Rational Mechanics and Analysis, 1993] and special singular solutions to first determine $a(0)$ and $c(x)$ for $x \in \Omega$. Based on this partial result, we are then able to determine $a(u)$ for $u \in \mathbb{R}$ by an adjoint approach.