Simultaneous determination of the diffusion and absorption coefficient from boundary data

Bastian Harrach

doi:10.3934/ipi.2012.6.663

Abstract

We consider the inverse problem of determining both an unknown diffusion and an unknown absorption coefficient from knowledge of (partial) Cauchy data in an elliptic boundary value problem. For piecewise analytic coefficients, we prove a complete characterization of the reconstructible information. It is shown to consist of a combination of both coefficients together with the jumps in the leading order diffusion coefficient and its derivative.

Highlights

Let B ⊂ n, n ≥ 2, be a bounded domain with smooth boundary ∂B and outer normal vector ν
We show that we can independently control the terms in the monotony estimates
For the convenience of the reader, we summarize the main steps for the assertion (a)(i) here, the reformulations of (a)(ii’) and (iii) follow analogously

Summary

Introduction

Let B ⊂ n, n ≥ 2, be a bounded domain with smooth boundary ∂B and outer normal vector ν. To be able to combine the coefficients only in a part of the domain, we will use this transformation with a replaced by a more general function α and derive a corresponding monotony relation Three corollaries of this general monotony result will later be used in our uniqueness proof. As in Theorem 2.2, let the coefficients a1, a2, c1, c2 ∈ L∞ + (B) be piecewise analytic functions on a joint partition (Oj, Γ)Jj=1, and let Λa1,c1 , Λa2,c2 be the corresponding local Neumann-to-Dirichlet operators. To reformulate this as a range inclusion we use the following functional analytic lemma.

The boundary part S

Induction basis