Lipschitz and H\"older stable determination of nonlinear terms for elliptic equations

TL;DR

This paper establishes Lipschitz and H"older stability estimates for determining nonlinear terms in elliptic equations from boundary measurements, aiding in the numerical reconstruction of these terms.

Contribution

It provides the first stability estimates for nonlinear terms in elliptic inverse problems using boundary data, combining linearization and singular solutions techniques.

Findings

Proves Lipschitz stability for quasilinear terms

Establishes H"older stability for semilinear terms

Enhances potential for numerical reconstruction

Abstract

We consider the inverse problem of determining some class of nonlinear terms appearing in an elliptic equation from boundary measurements. More precisely, we study the stability issue for this class of inverse problems. Under suitable assumptions, we prove a Lipschitz and a H\"older stability estimate associated with the determination of quasilinear and semilinear terms appearing in this class of elliptic equations from measurements restricted to an arbitrary parts of the boundary of the domain. Besides their mathematical interest, our stability estimates can be useful for improving numerical reconstruction of this class of nonlinear terms. Our approach combines the linearization technique with applications of suitable class of singular solutions.