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Stable Determination of an Inclusion by Boundary Measurements

G. Alessandrini, M. Di Cristo.

2005
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SIAM Journal on Mathematical Analysis
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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
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... 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 result Let 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 by x = (x , x n ) where x ∈ R n−1 , x n ∈ R. Definition 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 and where 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 and where 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

doi:10.1137/s003614100444191x
fatcat:xajz6nvdmvaznh5avaduulnzum