Determination of quasilinear terms from restricted data and point measurements

TL;DR

This paper introduces a new method to uniquely and stably determine quasilinear terms in elliptic equations using limited boundary measurements and explicit asymptotic analysis, advancing inverse problem solutions.

Contribution

It presents a novel approach based on asymptotic properties to identify quasilinear terms from finite boundary data in elliptic equations.

Findings

Unique determination of quasilinear terms achieved

Stable reconstruction from finite boundary measurements demonstrated

Asymptotic analysis provides new insights into nonlinear elliptic inverse problems

Abstract

We study the inverse problem of determining uniquely and stably quasilinear terms appearing in an elliptic equation from boundary excitations and measurements associated with the solutions of the corresponding equation. More precisely, we consider the determination of quasilinear terms depending simultaneously on the solution and the gradient of the solution of the elliptic equation from measurements of the flux restricted to some fixed and finite number of points located at the boundary of the domain generated by Dirichlet data lying on a finite dimensional space. Our Dirichlet data will be explicitly given by affine functions taking values in $\mathbb R$. We prove our results by considering a new approach based on explicit asymptotic properties of solutions of these class of nonlinear elliptic equations with respect to a small parameter imposed at the boundary of the domain.