Stability estimate for a semilinear elliptic inverse problem

Abstract

We establish a logarithmic stability estimate for the inverse problem of determining the nonlinear term, appearing in a semilinear boundary value problem, from the corresponding Dirichlet-to-Neumann map. Our result can be seen as a stability inequality for an earlier uniqueness result by Isakov and Sylvester (Commun Pure Appl Math 47:1403-1410, 1994.

On a nonlinear model in domains with cavities arising from cardiac electrophysiology

“…Let us finally point out that recently there has been growing interest towards inverse boundary value problems for semilinear elliptic equations, see for example [15][16][17][18][19]. In particular, we would like to mention [15], where the authors use the nonlinear Dirichlet to Neumann map to recover simultaneously the nonlinear term appearing in the equation and the cavity.…”

Lipschitz and Hölder stable determination of nonlinear terms for elliptic equations

“…In contrast to the important development of this topic in terms of uniqueness results, only few authors considered the stability issue for these problems. For elliptic equations, we are only aware of the works [5,28] where the authors proved a logarithmic stability estimate for the determination of semilinear terms of the form F(u), u ∈ R, with F a sufficiently smooth function or semilinear terms of the form F(x, u) = q(x)u m , x ∈ Ω, u ∈ R, where m is a known integer and the parameter q is unknown. We mention also the recent work of [27] where the authors studied the stable determination of semilinear terms appearing in nonlinear hyperbolic equations.…”

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

“…There are only very few stability inequalities in the literature devoted to the determination of nonlinear terms in quasilinear and semilinear elliptic equations by boundary measurements. The semilinear case was studied in [4] by using a method based on linearization together with stability inequality for the problem of determining the potential in a Schrödinger equation by boundary measurements. 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.…”