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

Abstract

We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.

Lipschitz and Hölder stable determination of nonlinear terms for elliptic equations

“…As far as we know, in all other relevant results related to our inverse problem, that can be found for instance in the articles [9,12,21,28,30], the determination of the quasilinear term a has been considered from Neumann boundary measurements restricted to an open subset of ∂Ω associated with Dirichlet excitations lying in an infinite dimensional space. In Theorem 2.1 and 2.2, we improve these results by restricting the Dirichlet excitations to the space of affine functions of R n taking values in R, which is a space of dimension n + 1, and we consider Neumann measurements restricted to at most n points for the determination of general quasilinear terms depending simultaneously on the solutions and the gradient of the solutions of (1.3).…”

“…Neumann boundary measurements on the whole boundary ∂Ω of the solutions of the equation (1.3) for all possible Dirichlet excitations). In the more recent works [9,21], the authors addressed the stability issue for this last problem for quasilinear terms depending only on the solution from some restriction of the Dirichlet-to-Neumann map associated with (1.3). Similar problems have been investigated for more general class of quasilinear terms depending also on the space variable in the works [5,6,22,31,32] among which the most general one can be found in [6] where the authors addressed the open problem of determining quasilinear terms depending simultaneously on the solutions, the gradient of the solutions and the space variable.…”

Stability estimate for a semilinear elliptic inverse problem

“…Set h j δ = ∂ j H(•, y δ ) and let v j δ denotes the unique weak solution of the BVP div(A∇v) = 0 in Ω 0 , v |∂Ω0 = h j δ . We proceed as in [4] in order to derive the following estimate (2.4)…”

“…In the present work we adapt the analysis in [4] to improve the stability result in [5]. Our construction of special solutions is borrowed from [11].…”

“…We show actually that we have Hölder stability estimate with less boundary measurents and less regular nonlinearities. We establish our stability inequality by following the same method as in [4]. This method consists in constructing special solutions vanishing at a subboundary of the domain.2010 Mathematics Subject Classification.…”