Determining an anisotropic conductivity by boundary measurements: stability at the boundary

We consider the inverse problem of determining, the possibly anisotropic, conductivity of a body by means of the so called local Neumann to Dirichlet map on a curved portion $\Sigma$ of the boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities proved in \cite{Al-dH-G}, we provide a H\"{o}lder stability estimate on $\Sigma$ when the conductivity is a priori known to be a constant matrix near $\Sigma$.


Introduction
Given ϕ : ∂Ω −→ R, with zero average, consider the Neumann problem where σ = {σ ij (x)} n i,j=1 , with x ∈ Ω, satisfies the uniform ellipticity condition for some positive constant λ.Here, a standard, variational, functional framework is understood.Details will be presented in what follows.Electrical Impedance Tomography (EIT) is the inverse problem of determining the conductivity σ when the Neumann-to-Dirichlet (N-D) map is given, [C].It is well-known that if σ is allowed to be anisotropic, i.e. a full matrix, although symmetric, then it is not uniquely determined by N σ .In fact, if Φ : Ω −→ Ω is a diffeomorphism such that Φ ∂Ω = I, then σ and its push-forward under Φ, give rise to the same N-D map.This construction is due to Tartar, as reported by Kohn and Vogelius .A prominent line of research investigates the determination of σ modulo diffeomorphism which fix the boundary, in this respect we refer to the seminal paper of Lee and Uhlmann [Le-U].From another point of view, anisotropy cannot be neglected in applications, such as medical imaging or geophysics.It is therefore interesting to investigate possible kinds of structural assumptions (physically motivated) under which unique determination of σ from N σ is restored.
In [Al-dH-G] the case of a piecewise constant conductivity σ was treated, and assuming that the interfaces of discontinuity contain portions of curved (non flat) hypersurfaces, uniqueness was proven.Subsequently [Al-dH-G-S], uniqueness was proven also in cases of a layered structure with unknown interfaces.We refer to these two papers for a bibliography on the relevance of anisotropy in applications.
It is still an open problem, to prove, in such settings, estimates of stability.Indeed, a line of research initiated by Alessandrini and Vessella [Al-V] in the isotropic case (i.e.: σ = γI, with γ scalar), suggests that also in the setting of Alessandrini-de Hoop-Gaburro [Al-dH-G] a Lipschitz stability estimate might hold.Such a generalization, however, does not appear to be an easy task, because isotropy intervenes in many steps of the proof in [Al-V].
In this note we treat the first step in a program to prove stability for piecewise constant anisotropic conductivity with curved interfaces.More precisely, assuming σ constant in a neighborhood U of a curved portion Σ of the boundary ∂Ω, we show that σ U depends in a Hölder fashion on the local N-D map N Σ σ (a precise definition will be given in due course).
Since Kohn and Vogelius [Ko-V],  and Alessandrini [A], [Al], it is customary to treat the uniqueness and stability, at the boundary, as a first step towards determination in the interior.And also in the anisotropic case we wish to mention the results of Kang and Yun [K-Y] who proved reconstruction and stability at the boundary up to diffeomorphisms which keep the boundary fixed.
Here, assuming a quantitative formulation of non-flatness of Σ, we are able to stably determine the full conductivity matrix σ (near Σ).More precisely, we shall assume that there exists three points P 1 , P 2 , P 3 ∈ Σ such that the corresponding unit normal vectors to Σ, ν(P 1 ), ν(P 2 ), ν(P 3 ) are quantitatively pairwise distinct.
Our argument here is based on various new features.As noticed in [Al-dH-G], the local N-D map N Σ σ is an integral operator whose kernel K differs from the well-known Neumann kernel N by a bounded correction term.
The common feature of the two kernels is the character of their singularity which in turn encapsulates information on the tangential part of the metric {g ij } n i,j=1 associated to the conductivity σ in dimension n ≥ 3.By testing N Σ σ on suitable combination of mollified δ functions, we achieve a quantitative evaluation of the tangential component of {g ij }.Next, by exploiting the quantitative notion of 'non-flatness' of Σ, we show that the full metric {g ij } can be recovered from three tangential samples at three different points with sufficiently distinct tangent planes, or, equivalently, pairwise distinct normal vectors.
The paper is organized as follows.Section 2 contains the main definitions, including the quantified notion of non-flatness of Σ where we localize the measurements N Σ σ (Section 2.1).This section also contains the statement of our main result of stability at the boundary of anisotropic conductivities σ that are constant near Σ in terms of N Σ σ (Section 2.2).Section 3 is devoted to the construction of mollified delta functions on Σ.The proof of the main result is a two steps procedure.In the first step, contained in Section 4, we stably recover the tangential component of g in terms of N Σ σ .The argument of this proof is based on estimating the asymptotic behaviour of the Neumann Kernel N (•, y) of L = div(σ∇•) and its derivative, near the pole y ∈ Σ, and the sampling N Σ σ on suitable combinations of the mollifiers given in Section 3. In the second step, discussed in Section 5, exploiting the non flatness condition on Σ as well as the structure of the metric g, we derive the stability on the boundary for the full metric g which in turn leads to the stable determination of the conductivity σ on Σ .

Notation and assumptions
In several places in this manuscript it will be useful to single out one coordinate direction.To this purpose, the following notations for points x ∈ R n will be adopted.For n ≥ 3, a point x ∈ R n will be denoted by x = (x ′ , x n ), where x ′ ∈ R n−1 and x n ∈ R.Moreover, given a point x ∈ R n and given a, b ∈ R, we shall denote with B r (x), B r (x ′ ) the open balls in R n , R n−1 centred at x, x ′ , respectively, with radius r and by We shall assume throughout that Ω ⊂ R n , with n ≥ 3, is a bounded domain with Lipschitz boundary, as per definition 2.1 below.
Definition 2.1.We will say that ∂Ω is of Lipschitz class with constants r 0 , L > 0, if for every P ∈ ∂Ω, there exists a rigid transformation of coordinates under which we have P = 0 and where ϕ is a Lipschitz continuous function on B ′ r0 satisfying We fix an open non-empty subset Σ of ∂Ω (where the measurements in terms of the local N-D map are taken).A precise definition of the N-D map and its local version with respect to Σ are given below.
Denoting by Sym n the class of n × n symmetric real valued matrices, we assume that σ ∈ L ∞ (Ω , Sym n ) satisfies the ellipticity condition (1.1).We consider the function spaces The global Neumann-to-Dirichlet map is then defined as follows.
Definition 2.2.The Neumann-to-Dirichlet (N-D) map associated with σ, is characterized as the selfadjoint operator satisfying for every ψ ∈ 0 H − 1 2 (∂Ω), where u ∈ H 1 (Ω) is the weak solution to the Neumann problem For the local version of the N-D map, we consider an open portion of ∂Ω, Σ, and, denoting by ∆ = ∂Ω \ Σ, we introduce the subspace of that is the space of distributions ψ ∈ H − 1 2 (∂Ω) which are supported in Σ and have zero average on ∂Ω.The local N-D map is then defined as follows.
Definition 2.3.The local Neumann-to-Dirichlet map associated with σ, Σ is the operator ), for i = 1, 2, the following equality holds true.
for any ψ i ∈ 0 H − 1 2 (Σ), for i = 1, 2 and u i ∈ H 1 (Ω) being the unique weak solution to the Neumann problem (2.6) Let Ω ⊂ R n be as above.Given α, α ∈ (0, 1), we say that a portion Σ of ∂Ω is of class (2.7) We will also write (2.8) For any P ∈ ∂Ω, we will denote by ν(P ) the outer unit normal to ∂Ω at P .
Definition 2.5.Given Σ as above, we shall say that such a portion of a surface is non-flat (and equivalently the function ϕ) if, there exist three points P 1 , P 2 , P 3 ∈ Σ and a constant C 0 , 0 < C 0 < 1, such that (2.11)

Local stability at the boundary
It will be convenient to define throughout this paper the following quantity (2.12) We will assume that there is a point y ∈ ∂Ω such that, up to a rigid transformation, y = 0, and is a non-flat portion of ∂Ω of class C 2,α with constants ρ > 0, M > 0 and C 0 > 0 as per definitions 2.4, 2.5.
The following notation will also be adopted throughout the manuscript.
i) A constant C is said to be uniform if it depends on the a-priori data only.
ii) We denote by O(t) a function g such that where C, t 0 > 0 are uniform constants.
iii) We set, for where In what follows, • L(B1,B2) will denote the operator norm for linear operators between Banach spaces B 1 , B 2 .
Our main result is stated below.
3 Construction of mollifiers on a graph and their H − 1 2 -norm.
For any two points ξ, x ∈ Σ, with and τ > 0, we denote and we choose C τ in such a way that We define the where we impose the normalization and that for y ∈ ∂Ω, it solves since N Ω 0 (x, •) and h have zero average on ∂Ω.Hence (3.10) We are now in the position to estimate the behaviour of the H − 1 2 (∂Ω) norm of the mollified delta function where C > 0 is a uniform constant.
Proof.By (3.10) and the fact that δ τ is compactly supported on Σ, we have (3.12) The change of variables We recall that for σ(x) = {σ ij (x)} i,j=1,...,n , x ∈ Ω, symmetric, positive definite matrix-valued function satisfying (1.1), we denote by L the operator nd that if in dimension n > 2 we define the matrix on the open set Ω endowed with the Riemannian metric g, see for instance [B-G-M], [U].We emphasize that, being n > 2, the knowledge of σ is equivalent to the knowledge of g.

Stability of the tangential part of g
We start by observing that (3.15), together with the uniform ellipticity assumption (1.1) on σ, implies the following uniform ellipticity of g where λ > 0 has been introduced in (1.1).
We also recall below few facts from [Al-dH-G] about the Neumann kernel to make this manuscript selfcontained.The Neumann kernel N Ω σ for the boundary value problem associated with the operator (3.14) and Ω, for any y ∈ Ω, N Ω σ (•, y), is defined to be the distributional solution to Note that N Ω σ is uniquely determined up to an additive constant.For simplicity we impose the normalization With this convention we obtain by Green's identities that N Ω σ (x, y) extends continuously up to the boundary ∂Ω (provided that x = y) and in particular, when y ∈ ∂Ω, it solves Theorem 4.1.Let y, ρ and Σ be defined by (2.13).If L is the operator (3.14), with coefficients matrix Proof.See [Al-dH-G] for a proof.
Such a uniqueness result obtained in [Al-dH-G], will guide us towards a stability estimate.
In what follows, for y ∈ Σ, we set for i = 1, 2: and Lemma 4.3.Under the same hypotheses of Theorem 4.1, for any y ∈ Σ and any x ∈ Ω \ {y} we have where C > 0 is a uniform constant.
Proof.From Theorem 4.1 for any y ∈ Σ, for any x ∈ Ω, for i = 1, 2, we have Recalling that for y ∈ Σ, N i (•, y) is the distributional solution to (4.15) where η = 1 |∂Ω| ∂Ω η(x)dS(x).Recalling the decomposition (4.12) we obtain (4.16) which leads to Noticing that 2Γ i (•, y) solves the boundary value problem with where C > 0 is a uniform constant and L i;y has been defined in (4.9), therefore and defining we can rewrite (4.17) as where F i (x, y) is zero for x ∈ B ρ (y), therefore bounded for x ∈ Ω. Therefore by combining (4.19) and (4.21), we have that for any By denoting for any x ∈ Ω, for any y ∈ Σ, we have (4.24) and by denoting also we have where C > 0 denotes a uniform constant.
For any z ∈ Ω\{y} and any k ∈ N, we define η k (x) = min{N 1 (x, z), k} with x ∈ Ω.By choosing η(x) = η k (x) in (4.22) and by the dominated convergence theorem we obtain By performing integration by parts on the integral appearing on the left hand side of (4.22) we have where the right hand side of equality (4.27) can be estimated as follows where uniquely determined by imposing the condition where C > 0 is a uniform constant, which leads to where C > 0 is a uniform constant.(4.10) follows from (4.32).To prove (4.11) we observe that By fixing r > 0, with r < ρ 4 and defining we have for i = 1, 2 (4.11) follows by (4.10), (4.36) and an interpolation argument.
Proof.Let d > 0 be such that d < (1 + M )ρ.Given distinct points x, y, w, z ∈ Σ, we recall from [Al-dH-G] the following definition We also recall that knowing N Σ σ is equivalent to knowing K σ , for any x, y, w, z ∈ Σ Lemma 3.8].We also note that, fixing w, z ∈ Σ, K σ , as a function of x, y, has the same asymptotic behaviour of N σ (x, y) Let and let δ τ (•, •) be the approximate Dirac's delta functions on Σ introduced in section 3, centered on the second argument.Then we have The integral appearing on the right hand side of (4.44) depends on x, y, w, z and to estimate how close this We estimate each term on the right hand side of (4.45) as follows where ξ = (1 − t)x + tξ and η = (1 − s)y + sη for some t, s ∈ (0, 1).
Notice that, given the choice of τ in (4.43), we have that The remaining three terms appearing on the right hand side of (4.45)only involve at most one of the poles x, y ∈ Σ at the time and are therefore bounded by CEh.Therefore we have Recalling that . Hence, recalling the definition of τ in (4.43), we obtain the following pointwise estimate for Minimization of (4.51) with respect to h leads to with β = 1 n−1 .Inequality (4.53) is a uniform bound with respect to x, y ∈ Σ. Setting y = 0, ν(0) = −e n , where {e 1 , . . ., e n } denotes the canonical basis of R n , and writing x ∈ Σ as x = (x ′ , ϕ(x ′ )), with x ′ = rξ ′ , with ξ ′ ∈ R n−1 and ||ξ ′ || = 1, (4.3) leads to where we identified ξ ′ with (ξ ′ , 0).By Lagrange's theorem and (4.1), we have and by combining (4.55) with (4.53) together with (4.54), we obtain which concludes the proof, since, as it is well known g (1;n−1) (0) − g (2;n−1) (0) 5 Stability of the full metric g In the following we shall prove that up to a suitable condition on the geometry of Ω and on the structure of the metric g, the knowledge of the Neumann kernel in a neighborhood of Σ allows us to recover the full metric g on Σ.