Lipschitz and Hölder stable determination of nonlinear terms for elliptic equations

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 Hölder stability estimate associated with the determination of quasilinear and semilinear terms appearing in this class of elliptic equations from measurements restricted to an arbitrary part 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.

Stable determination of the nonlinear term in a quasilinear elliptic equation by boundary measurements

“…As we already mentioned we give a proof based on an adaptation of [3, proof of (1.2) of Theorem 1.1] combined with a localization argument borrowed from [13].…”

“…The result in [5] was recently improved in [4]. Both quasilinear and semilinear elliptic inverse problems were considered in [13] where a method exploiting the singularities of fundamental solutions was used to establish stability inequalities. This method was used previously in [3] to obtain a stability inequality at the boundary of the conformal factor in an inverse conductivity problem.…”

“…We show in the present paper how we can modify the proof of [3, (1.2) of Theorem 1.1] to derive the stability inequality stated in Theorem 1. The localization argument was inspired by that in [13].…”