Stable Determination of an Inclusion by Boundary Measurements

Abstract

We deal with the problem of determining an inclusion within an electrical conductor from electrical boundary measurements. Under mild a priori assumptions we establish an optimal stability estimate.accounts. Unfortunately, for such a problem, the uniqueness question, not to mention stability, remains a largely open issue.Let us illustrate briefly the main steps of our arguments. We must recall that Isakov's approach to uniqueness is essentially based on two arguments a) the Runge approximation theorem, b) the use of solutions with Green's function type singularities.Also here we shall use singular solutions, and indeed we shall need an accurate study of their asymptotic behavior when the singularity gets close to the set of discontinuity ∂D of the conductivity coefficient 1+(k−1)χ D in (1.1), see Proposition 3.2. On the other hand, it seems that Runge's theorem, which is typically based on nonconstructive arguments, (Lax, [L], Kohn and Vogelius [K-V]) is not suited for stability estimates and therefore we introduced a different approach based on quantitative estimates of unique continuation, see Proposition 3.3.In Section 2 we formulate our main hypotheses and state the stability result, Theorem 2.2. In Section 3 we prove Theorem 2.2 on the basis of some auxiliary Propositions, whose proof is deferred to the following Section 4. The main resultLet us introduce our regularity and topological assumptions on the conductor Ω and on the unknown inclusion D. To this purpose we shall need the following definitions. In places, we shall denote a point x ∈ R n byDefinition 2.1. Let Ω be a bounded domain in R n . Given α, 0 < α ≤ 1, we shall say that a portion S of ∂Ω is of class C 1,α with constants r, L > 0 if, for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 andwhere ϕ is a C 1,α function on B r (0) ⊂ R n−1 satisfying ϕ(0) = |∇ϕ(0)| = 0 and ϕ C 1,α (Br (0)) ≤ Lr.Definition 2.2. We shall say that a portion S of ∂Ω is of Lipschitz class with constants r, L > 0 if for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 andwhere ϕ is a Lipschitz continuous function on B r (0) ⊂ R n−1 satisfying ϕ(0) = 0 and ϕ C 0,1 (Br (0)) ≤ Lr.Remark 2.1. We have chosen to scale all norms in a such a way that they are dimensionally equivalent to their argument. For instance, for any ϕ ∈ C 1,α (B r (0)) we set ϕ C 1,α (Br(0)) = ϕ L ∞ (Br(0)) + r ∇ϕ L ∞ (Br (0)) + r 1+α |∇ϕ| α,Br (0) .For given numbers r, M , δ, L > 0, 0 < α < 1, we shall assume

Unification of the Probe and Singular Sources Methods for the Inverse Boundary Value Problem by the No-Response Test

“…Hence, the first part satisfies D y − a y G a z y · y G a x y dy ≤ c D y − a y − z −2 x − y −2 dy (4.13) For x ∈ D and z ∈ a , arguing as in Alessandrini and Di Cristo (2005), p. 11 inequality (4.12), this last integral is bounded byc ln x − z with some positive constantc. In Alessandrini and Di Cristo (2005), the authors took z on the normal a but their proof still justified also for z ∈ a since the critical point is the inequality (4.9).…”

“…In Alessandrini and Di Cristo (2005), the authors took z on the normal a but their proof still justified also for z ∈ a since the critical point is the inequality (4.9). Using the inequalities x − z ≤ d x D + d z D and ln x − z ≤ c x − z −t , locally for every t > 0, we deduce that D y − a y G a z y · y G a x y dy ≤ c d x D + d z D −t (4.14)…”

“…We consider a function of This inverse boundary value problem was initiated in Isakov (1988) where uniqueness for identifying the inclusion D was proven. In a recent article, Alessandrini and Di Cristo (2005), a stability result concerning this problem of localization of the interface of discontinuity is given. Regarding the reconstruction issue, in Ikehata (1998), a method for identifying the inclusion was proposed.…”

“…The proof of this equivalence is given by freezing and flattening the coefficient x near the point a. To justify these two steps, we combined some recent estimates of the corresponding Green's functions given in Alessandrini and Di Cristo (2005), L p , 1 < p < , and L estimates of solutions for scalar divergence form elliptic equations with discontinuous coefficients, see Seftel (1963) and the recent work Li and Vogelius (2000) respectively.…”

Obstacle and Boundary Determination from Scattering Data

“…Let us consider the first integral. For η ∈ C F (a),θ and ξ on S r , we have the inequality At this point, we make use of an argument from ( [1], pages 209, 210). We decompose the last integral as the sum of…”