The problem of detecting corrosion by an electric measurement revisited

We establish a logarithmic stability estimate for the problem of detecting corrosion by a single electric measurement. We give a proof based on an adaptation of the method initiated in \cite{BCJ} for solving the inverse problem of recovering the surface impedance of an obstacle from the scattering amplitude. The key idea consists in estimating accurately a lower bound of the $L^2$-norm, locally at the boundary, of the solution of the boundary value problem used in modeling the problem of detection corrosion by an electric measurement


Introduction
Let Ω be a C n -smooth bounded domain of R n , n = 2, 3. We denote its boundary by Γ and we consider the following boundary value problem (abbreviated to BVP) ∆u = 0, in Ω, ∂ ν u + q(x)u = g, on Γ.
(1.1) In all of this paper we assume that g ∈ H n−3/2 (Γ) and g is non identically equal to zero. For s ∈ R and 1 ≤ r ≤ ∞ , we introduce the vector space B s,r (R n−1 ) := {w ∈ S ′ (R n−1 ); (1 + |ξ| 2 ) s/2 w ∈ L r (R n−1 )}, where S ′ (R n−1 ) is the space of temperate distributions on R n−1 and w is the Fourier transform of w. Equipped with its natural norm w Bs,r (R n−1 ) := (1 + |ξ| 2 ) s/2 w L r (R n−1 ) , B s,r (R n−1 ) is a Banach space (it is noted that B s,2 (R n−1 ) is merely the usual Sobolev space H s (R n−1 )). By local charts and partition of unity, we construct B s,r (Γ) from B s,r (R n−1 ) in the same manner as H s (Γ) is built from H s (R n−1 ).
We shall need that the solution of the BVP (1.1) has some smoothness. In order to give sufficient conditions on data guaranteeing this smoothness, we first define the set of boundary coefficients. Let Q = {q ∈ B n−1/2,1 (Γ); q ≥ 0 and q ≡ 0} where M > 0 is a given constant. By Theorem 2.3 in [Ch1], observing that B n−1/2,1 (Γ) is continuously embedded in B n−3/2,1 (Γ), we have that, for any q ∈ Q, the BVP (1.1) has a unique solution u q ∈ H n (Ω). In addition The constant C above can depend only on Ω, g and M .
Usually in a BVP modeling the problem of detecting corrosion damage by electric measurements the boundary Γ consists in two parts: Γ = Γ a ∪ Γ i , Γ a and Γ i being two disjoint open subsets of Γ. Γ a corresponds to the part of the boundary accessible to measurements and Γ i is the inaccessible part of the boundary where the corrosion damage occurs.
Henceforth, we assume that the current flux g satisfies supp(g) ⊂ Γ a . The function q in (1.1) is known as the corrosion coefficient and it is supported on Γ i . This motivate the introduction of the following set We are interested in the stability issue for the problem consisting in the determination of the boundary coefficient q from the boundary measurement u q|γ , where γ is a subset of the accessible sub-boundary Γ a for which we assume that the following condition holds true: Next, we introduce the notion of mutiply-starshaped domain. We say that D is mutiply-starshaped if there exists a finite number of points in D, say x 1 , . . . , x k , such that any point in D can be connected by a line segment to one of x i . In this case, any two points in D can be connected by a broken line consisting of at most k + 1 line segments. Obviously, the case k = 1 corresponds to the usual notion of starshapedness.
The main result in the present note is the following theorem.
Theorem 1.1. We fix 0 < α < 1 and we assume that Ω is locally convex 1 and Ω is multiply-starshaped. There are three positive constants A and B and σ satisfying for any q ∈ Q 0 M ∩ C α (Γ), we find ǫ = ǫ(q) with the property that for all q ∈ Q 0 with u = u q and u = u q .
The result in Theorem 1.1 can be seen as an improvement of those already established in [CCL] in dimension two and in [BCC] in dimensions two and three. We note that in these above mentioned works the difference of q − q is only estimated in a compact subset of {x ∈ Γ i ; u q (x) = 0}. However there is a counterpart in estimating q − q in the whole Γ. The stability estimates in [CCL] and [BCC] are of logarithmic type, while the estimate in Theorem 1.1 is of double logarithmic type.
There is a wide literature treating the problem of detecting corrosion by electric measurements. We refer to [CFJL,CJ,CCY,Ch2,Ch3,FI,In,Si] where various type of stability estimate are given. We just quote these few references, but of course there are many others.
Unless otherwise specified, all the constants we use in the sequel depend only on data.
2. Lower bound for the local L 2 -norm at the boundary We aim to prove the following theorem.
Theorem 2.1. We assume that Ω is locally convex and Ω is multiply-starshaped. Let M > 0, there is c > 0 such that, for all q ∈ Q 0 M and all x ∈ Γ, we have where r * is a constant that can depend on q.
We need several preliminary results before proving Theorem 2.1. We start by introducing some definitions. As usual, we say that Ω has the uniform exterior ball property (abbreviated to UEBP) if there is ρ > 0 for which, for all x ∈ Γ, we find Next, we recall that Ω has the uniform interior cone property (abbreviated to UICP) if there are R > 0 and θ ∈]0, 2π[ satisfying, for all x ∈ Γ, we find ξ ∈ R n such that |ξ| = 1 and Also, we say that Ω has the uniform interior cone-exterior ball property (abbreviated to UICEBP) if UEBP and UICP are both satisfied at any point x ∈ Γ and in addition where x ′ and ξ are the same as in the definitions of UEBP and UICP respectively. Now let (G) be the following assumption: There exist C > 0 and 0 < r 0 such that for all x ∈ Γ and 0 < r ≤ r 0 , x ∈ Γ, where x ′ and ρ are the same as in the definition of UEBP.
One can easily check that if Ω is locally convex, then Ω possesses both UICESP and (G). For sake of simplicity, we replace in the sequel the assumption that Ω is multiply-starshaped by a stronger one. Precisely, we assume that Ω is starshaped. From the proof of Proposition 2.1 below, one can see that the extension to the case where Ω is multiply-starshaped is obvious.
Therefore the following corollary is immediate from Proposition 2.1.
Note here that δ and r δ may depend also on u.
We recall that B( x, r) = B(x ′ , ρ + r), x ∈ Γ, where x ′ and ρ are the same as in the definition of UEBP. As a peculiar case of Corollary 3.1 in [BCJ], we have Proposition 2.2. There exist two constants C > 0 and 0 < γ < 1/2 with the property that, for any 0 < r ≤ D and any u ∈ H 2 (Ω) satisfying ∆u = 0, the following estimate holds true Also, a slight modification of the first part of the proof of Theorem 4.1 in [BCJ] yields Proposition 2.3. We assume that Ω has UICEBP and we pick x ∈ Γ. For sufficiently small r, we can choose x 0 ∈ Ω, y 0 ∈ Ω two points in the line segment passing through x and directed by ξ such that B(x 0 , r/2) ⊂ B( x, r) ∩ Ω and B(y 0 , κr) ⊂ Ω R/2 , where κ is constant depending only on θ. Let M > 0, there are C > 0, η > 1, and r * > 0, not depending on x 0 and y 0 , such that for all u ∈ H 1 (Ω) satisfying ∆u = 0 in Ω and u H 1 (Ω) ≤ M , e − C r η u H 1 (B(y0,κr)) ≤ u H 1 (B(x0,r)) , 0 < r ≤ r * .

A combination of Corollary 2.2, Proposition 2.2 and Proposition 2.3 gives
Theorem 2.2. Let η > 0, M > 0, x ∈ Γ and assume that Ω has UICEBP and it is starshaped. There is a constant c > 0 such that for all u ∈ H 2 (Ω) satisfying ∆u = 0 in Ω, |u( x)| ≥ η, u H 2 (Ω) ≤ M , and for all where r * can depend on u.
If in addition |∂ ν u| ≤ N |u| on Γ for some constant N , then We are now able to complete the proof of Theorem 2.1.
Proof of Theorem 2.1. We need to prove that there are x ∈ Γ and η > 0 for which |u q ( x)| ≥ η for any q ∈ Q 0 M . We fix Γ 0 an arbitrary nonempty open subset of Γ \ supp(g). By Corollary 1 in [Bo], there is a constant A > 0 such that, for all u ∈ H 2 (Ω) satisfying ∆u = 0 and u H 2 (Ω) ≤ M , we have Let Γ 1 be an open subset of Γ satisfying supp(g) ⊂ Γ 1 ⋐ Γ. Proceeding as previously, we deduce from an usual interpolation inequality This and (2.6) imply Replacing Γ 0 by a smaller subset and proceeding as in the proof of Corollary 2.3, we get Now since H n (Ω) is continuously embedded in C(Ω), we derive from (2.7)

Proof of the stability estimate
First, we paraphrase the proof of Proposition 4.1 in [BCJ] to get that there are B > 0 and σ > 0 such that for any q ∈ Q 0 M , we find ǫ(q) > 0 with the property that for any f ∈ C α (Γ) satisfying Proof of Theorem 1.1. Let v = u − u. Since ∆v = 0, the same argument as in the proof of Theorem 2.1 leads to . Let γ 0 ⋐ γ. Again, by Corollary 1 in [Bo], there is a constant A > 0 for which As we have done previously, we obtain by an interpolation inequality that v H 1 (γ0) ≤ C v 1/3 L 2 (γ) , and since ∂ ν v = 0 on γ, (3.3) implies (3.4) v H 1 (Ω) ≤ A ln(B v L 2 (γ) ) 1/2 .