Stability for the Calderón’s problem for a class of anisotropic conductivities via an ad-hoc misﬁt functional

We address the stability issue in Calderón’s problem for a special class of anisotropic conductivities of the form 𝜎 = 𝛾𝐴 in a Lipschitz domain Ω ⊂ ℝ 𝑛 , 𝑛 ≥ 3 , where 𝐴 is a known Lipschitz continuous matrix-valued function and 𝛾 is the unknown piecewise aﬃne scalar function on a given partition of Ω . We deﬁne an ad-hoc misﬁt functional encoding our data and establish stability estimates for this class of anisotropic conductivity in terms of both the misﬁt functional and the more commonly used local Dirichlet-to-Neumann map.


Introduction
The paper addresses the so-called Calderón's inverse conductivity problem of recovering the conductivity of a body Ω ⊂ ℝ by taking measurements of voltage and electric current on its surface Ω.More specifically, the case when the conductivity is anisotropic and it is a-priori known to be of type = , where is a known Lipschitz continuous matrix valued function on Ω and is a piecewise-affine unknown function on a given partition of Ω, is considered.It is well known that in absence of internal sources or sinks, the electrostatic potential in a conducting body, described by a domain Ω ⊂ ℝ , is governed by the elliptic equation where the symmetric, positive definite matrix ( ) = ( ( )) , =1 , ∈ Ω represents the (possibly anisotropic) electric conductivity.The inverse conductivity problem consists of finding when the so called Dirichlet-to-Neumann (D-N) map is given for any ∈ 1 (Ω) solution to (1.1).Here, denotes the unit outer normal to Ω.If measurements can be taken only on one portion Σ of Ω, then the relevant map is called the local D-N map (Λ Σ ).
This problem arises in many different fields such as geophysics, known as DC method, medicine, known as Electrical Impedance Tomography (EIT) and non-destructive testing of materials.The first mathematical formulation of the inverse conductivity problem is due to Calderón [23], where he addressed the problem of whether it is possible to determine the (isotropic) conductivity = by the D-N map.This seminal paper opened the way to the solution to the uniqueness issue where one is asking whether can be determined by the knowledge of Λ or its local version when measurements are available on a portion of Ω only.
The case when measurements can be taken over the full boundary has been studied extensively in the past and the fundamental papers [2], [44], [45], [57] and [64] had led the way of solving the problem of uniqueness in the isotropic case.We also recall the uniqueness results of Druskin who, independently from Calderón, dealt directly with the geophysical setting of the problem in [28]- [30].His uniqueness result obtained in [29] was for conductivities described by piecewise constant functions (see also [11]).The problem of recovering the conductivity by local measurements has been treated more recently (see [46], [47]).In the present paper, we consider the issue of stability in the inverse conductivity problem, therefore we refer to [22], [24] and [66] for an overview regarding the issues of uniqueness and reconstruction of the conductivity.
Regarding the stability issue, Alessandrini proved in [1] that, in the isotropic case and dimension ≥ 3, assuming a-priori bounds on of the form ‖ ‖ (Ω) ≤ , > 2 + 2, leads to a continuous dependance of in Ω upon Λ of logarithmic type.We also refer to [14], [15] and [51] for subsequent results in this direction.Even though stability at the boundary Ω is of Lipschitz type (see [5], [6]), Mandache [55] showed that in the interior of Ω, the inconvenient logarithmic type of stability is the best possible, in any dimension ≥ 2, under a-priori smoothness assumptions on .It seems therefore reasonable to think that, in order to restore stability in a really (Lipschitz) stable fashion, one needs to replace in some way the a-priori assumptions expressed in terms of regularity bounds with a-priori pieces of information of a different type that suit the underlying physical problem.Alessandrini and Vessella showed in [11] that when is isotropic and piecewise constant on a given partition of Ω, then Lipschitz stability can be restored in terms of the local D-N map (conditional stability).Rondi [59] proved that the Lipschitz constant has an exponential behaviour with rispect to the number of subdomain of the partition.From a medical imaging point of view, the partition of Ω may represent different volumes occupied by different tissues or organs and one can think that their geometrical configuration is given by means of other imaging modalities such as MRI.We also recall [7], [19], [20], [21], [60] and [8], [17], [18], [61] where similar Lipschitz stability results have been obtained for the classical and fractional Calderòn's problem, the Lamé parameters and for a Schrödinger type of equation.
In this paper we address the issue of stability in Calderòn's problem in presence of anisotropy.This choice is motivated by the fact that anisotropy appears quite often in nature.Most tissues in the human body are anisotropic.In the theory of homogenization, anisotropy results as a limit in layered or fibrous structures such as rock stratum or muscle, as a result of crystalline structure or of deformation of an isotropic material.In the geophysical context, in 1920, Conrad Schlumberger [62] recognized that anisotropy may affect geological formations' electrical properties and anisotropic effects when measuring electromagnetic fields in geophysical applications have been studied ever since.Individual minerals are typically anisotropic but rocks composed of them can appear to be isotropic.
From a mathematical point of view, the inverse problem with anisotropic conductivities is an open problem.Since Tartar's observation [43] that any diffeomorphism of Ω which keeps the boundary points fixed has the property of leaving the D-N map unchanged, whereas is modified, different lines of research have been pursued.One direction has been to find the conductivity up to a diffeomorphism which keeps the boundary fixed (see [13], [16], [46], [47], [48], [57] and [63]).Another direction has been the one to formulate suitable a-priori assumptions (possibly fitting some physical context) which constrain the structure of the unknown anisotropic conductivity.For instance, one can formulate the hypothesis that the directions of anisotropy are known while some scalar space dependent parameter is not.Along this line of reasoning, we mention the results in [1], [5], [6], [36], [37], [44] and [49].We also refer to [4], [13], [16], [26], [27], [34], [46] and for related results in the anisotropic case and to [4], [40] and [41] for examples of non-uniqueness.
Here, we follow this second direction by a-priori assuming that the conductivity is of type where ( ) is an unknown affine scalar function on , is a known Lipschitz continuous matrix-valued function on Ω and { } =1 is a given partition of Ω (the precise assumptions on , and { } =1 are given in Subsections 2.1 and 2.2).Allowable partitions for our machinery to work include, in the geophysical setting, models of layered media and bodies with multiple inclusions.The ill-posed nature of the EIT inversion is aggravated the deeper one tries to image inside a body Ω [58], where EIT image resolution becomes quite poor (see [39]), leading to blurry images.Thus, in a geophysical context for example, it becomes difficult to recognise individual thin sediments and rock layers or fractures in the deep subsurface, but the 'average' effect at large scale of fine layering and fracturing are still shown as equivalent anisotropic media.It seems therefore reasonable to model the conductivity within each layer by an anisotropic conductivity to make it up for the finer layering structure within that otherwise might have been neglected the deeper one goes inside Ω due to poor resolution.
In order to introduce the misfit functional, consider two anisotropic conductivities (1) and (2) of type (1.2).If measurements are locally taken on an open portion Σ ⊂ Ω, we conveniently enlarge the physical domain Ω to an augmented domain Ω and consider Green's functions for div( ( ) ∇⋅) in Ω, for = 1, 2, with poles , ∈ Ω ⧵ Ω respectively.Hence we express the error in the measurements corresponding to (1) and (2) by means of the misfit functional  ( (1) , (2) where , are suitably chosen sets compactly contained in Ω ⧵ Ω and  0 ( , ) is defined by the surface integral We have obtained the following stability estimate of Hölder type: where > 0 is a constant that depends on the a-priori information only.The augmented domain Ω is chosen in such a way that 1 (⋅, ) are supported in Σ in the trace sense, hence belonging to the domain of the local D-N maps Λ Σ , = 1, 2 (see Section 2.3 for the formal definitions of the local D-N map and the appropriate spaces).Therefore, not only (1.5), together with the well-known Alessandrini's identity [2], implies a Lipschitz stability estimate of in terms of the more commonly used local D-N map in the mathematical literature, but it also indicates that the set of measurements , with , ∈ Ω ⧵ Ω is enough to stably determine .A Lipschitz stability estimate in terms of Λ Σ was obtained in [37] for the case = , with piecewise constant instead.The piecewise affine parametrizations considered in the present work tie in well with the finite elements method for computations.With the stability estimate (2.13) at hand, one can apply certain iterative methods for reconstruction within a subspace of piecewise affine functions with a starting model at a distance less than the radius of convergence to the unique solution [9], [33], [31] and [32].This radius is known to be roughly inversely proportional to the stability constant appearing in the estimate.More importantly, we can iteratively construct the best piecewise affine approximation for a given domain partition.Since the stability constant will grow at least exponentially with the number of subdomains in the partition [59], the radius of convergence shrinks accordingly.One can expect accurate piecewise affine approximations with relatively less subdomains (compared to the piecewise constant case of [37]) to describe the subsurface, noting that the domain partition need not be uniform and may show a local refinement, and hence our result provides the necessary insight for developing a practical approach with relatively minor prior information.
To the best of our knowledge a first stability estimate in terms of an ad-hoc misfit functional was achieved in the mathematical literature in [9] in the context of the Full Waveform Inversion.Such an estimate proved to be key for the implementation and reliability of a reconstruction procedure (see [9,33]) based on the use of Cauchy data only, being the latter independent on the availability of the Dirichlet to Neumann map.In the more recent result in [35] an ad-hoc misfit functional has been introduced in the context of imaging elastic media.
We also observe that another advantage of choosing the misfit functional over the local D-N map (even if available) to model the measurements error in EIT is motivated by its potentially simpler numerical implementation, compared to the computation of the norm of bounded linear operators between 1 2 spaces and their duals.Moreover, the misfit functional could also provide, again in the context of a possible numerical reconstruction of , additional features compared to the more traditional least-squares approach, allowing, in particular, for a distinction between the computational and the observational measurements.This is due to the introduction of the possibly distinct sets and that can almost be arbitrarily chosen outside the physical domain Ω.For example, could be an arbitrarily chosen set for the numerical data acquisition for the sake of the simulations, where could model a more realistic set that fits the geometric disposition of the electrodes in the actual measurements acquisition.Hence, in the discrete setting, such distinction can potentially require minimal information about the observational acquisition geometry of the electrodes employed for the observational measurements.This is due to the definition of the misfit functional that does not compare simulations and observations directly, but it rather compares products of observed and simulated measurements.Note also that with a slight modification, our arguments can apply when the local Neumann-to-Dirichlet (N-D) map is available instead, see for instance the discussion in [6].
The paper is organized as follows.In Section 2 we introduce the main assumptions on the domain Ω and the anisotropic conductivity .Section 2 contains the formal definitions of the local D-N map (subsection 2.3), the misfit functional (sunsection 2.4) and the statement of our main result (Theorem 2.1).A Lipschitz stability estimate in terms of the local D-N map follows as a straightforward consequence (Corollary 2.2).Section 3 is devoted to the introduction of some technical tools of asymptotic estimates for the Green function (Proposition 3.1) and propagation of smallness (Proposition 3.2) needed for the machinery of the proof of Theorem 2.1.The proof of Theorem 2.1 and Corollary 2.2 are also contained in this section.Section 4 contains the proofs of Proposition 3.1 and Proposition 3.2.

Misfit functional and the main result 2.1 Assumptions about the domain Ω
For ≥ 3, a point ∈ ℝ will be denoted by = ( ′ , ), where ′ ∈ ℝ −1 and ∈ ℝ.Moreover, given a point ∈ ℝ , we will denote with ( ), ′ ( ′ ) the open balls in ℝ , ℝ −1 respectively centred at and ′ with radius and by ( ) the cylinder ( ) = ′ ( ′ ) × ( − , + ).Set = (0), = (0), the positive real half space ℝ + = {( ′ , ) ∈ ℝ ∶ > 0}, the positive semisphere centred at the origin + = ∩ ℝ + , the positive semicylinder + = ∩ ℝ + .Similar definitions for ℝ − , − and − .Let us recall a couple of definitions concerning the regularity of the boundary of the domain.DEFINITION 2.1.Let Ω be a bounded domain in ℝ .A portion Σ of Ω is of Lipschitz class with constants 0 , > 0 if for each point ∈ Σ there exists a rigid transformation of coordinates under which coincides with the origin and where is a Lipschitz function on Let Ω be a domain in ℝ .A subset Σ of Ω is a flat portion of size 0 if for each point ∈ Σ there exists a rigid transformation of coordinates under which coincides with the origin and From now on, we will consider Ω ⊂ ℝ , ≥ 3 as a bounded, measurable domain with boundary Ω of Lipschitz class with positive constants 0 , as in Definition 2.1 and satisfying where |Ω| denotes the Lebesgue measure of Ω.Moreover, we assume that there exists a partition of bounded subdomains = { } =1 contained in Ω such that the following conditions hold: 1.
for = 1, … , are connected, pairwise non-overlapping subdomains with boundaries which are of Lipschitz class with constants 0 , 2. Ω = ⋃ =1 ; 3. (Chain of subdomains.)First, we assume that there exists one region, let us call it 1 , such that the intersection 1 ∩ Σ contains a flat portion Σ 1 of size 0 ∕3 (see Definition 2.2) and that for every ∈ {2, … , } there exists a collection of indices 1 , … , ∈ {1, … , } such that 1 = 1 and = and the subdomains are pairwise disjoint.Secondly, we assume that, for every fixed sub-index = 1, … , of the chain, the intersection Finally, for each of these flat sub-portions Σ +1 , = 1, … , −1, there exist a point +1 ∈ Σ +1 and a rigid transformation of coordinates under which +1 coincides with the origin and Later, we will add a domain 0 ⊂ ℝ ⧵ Ω so that, when indexing the chain of subdomains, we agree that 0 = 0 .

A-priori information on the anisotropic conductivity
Our stability result for the Calderón inverse problem concerns a special family of anisotropic conductivities .Let us describe in details their form.The conductivities ( ) = { ( )} are real-valued, symmetric × matrices such that ∈ ∞ (Ω, ) and have the form where the scalars ∈ ℝ and the vectors ∈ ℝ , = 1, … , are the unknowns, ( ) is a known fixed matrix and = { } =1 is the known partition of Ω introduced in Section 2.1.Furthermore, a) the scalar functions are bounded, piecewise linear and there is a positive constant ̄ > 1 such that for any = 1, … , for any ∈ Ω; (2.3) b) the matrix ( ) satisfies the following Lipschitz continuity condition: there exists a constant ̄ > 0 such that c) The matrix is positive definite and there exists a constant > 1 such that The set of positive constants { , 0 , , , ̄ , ̄ , } with ∈ ℕ and the space dimension ≥ 3, is called the a-priori data.
In the paper several constants depending on the a-priori data will appear.In order to simplify our notation, we will denote them by , 1 , 2 … , avoiding in most cases to point out their specific dependence on the a priori data which may vary from case to case.

The local Dirichlet-to-Neumann map
By now, assume simply that Ω is a bounded domain with Ω of Lipschitz class.Since Dirichlet data are different from zero on a small portion Σ ⊂ Ω, we introduce a suitable trace space for the formulation of the local Dirichlet-to-Neumann map.
The trace space where is the unit outward normal of Ω and ∈ 1 (Ω) is the weak solution to the boundary value problem div The map (2.6) can be identified with the bilinear form

Misfit functional
To begin with, we introduce the Green function in an augmented domain Ω as follows.From the assumptions on the domain Ω (Section 2.1) there is a point 1 ∈ Σ that coincides with the origin, up to a rigid transformation of coordinates.For simplicity, let us assume that the locally flat portion Σ 1 coincides with the entire portion Σ.Let us define the domain 0 ⊂ ℝ ⧵ Ω as and such that 0 ∩ Ω ⊂⊂ Σ.We define the augmented domain Ω as the set (2.9) It turns out that Ω is of Lipschitz class with constants 0 3 and ̃ , where ̃ depends on only.Denote Finally, we introduce two sets contained in 0 : the sets and which are compactely supported in 0 , i.e. , ⊂⊂ 0 .In the following sections, we might identify these sets with the set ( 0 ) , but in general, thay can be freely chosen in 0 .
Consider two anisotropic conductivities ( ) , = 1, 2 as in Section 2.2.Without loss of generality, we can extend them to the augmented domain Ω by setting their value equal to the identity matrix on 0 , so that they are of the form We denote with the same symbol the extended conductivity.
For every ∈ 0 , the Green's function (⋅, ) associated to = div( ( ) (⋅)∇⋅) and Ω with pole , is the weak solution to the Dirichlet problem div( where (⋅ − ) is the Dirac distribution centred at .We recall the following properties for the Green's functions (see [52]): For ( , ) ∈ × , define the following surface integral (2.12) We define the misfit functional as the quantity (2.13)

Stability estimate
In previous works (see [7], [11], [37]), Lipschitz stability estimates have been established for piecewise constant and piecewise linear isotropic conductivities and a certain class of anisotropic conductivities respectively, in terms of the local Dirichlet-to-Neumann map.Here, we extend these results to the class of anisotropic conductivities defined in Section 2.2.First, we determine a bound to the ∞ -norm of the difference between two anisotropic conductivities in terms of the square root of the misfit functional introduced above.Then, we derive a Lipschitz stability result in terms of the local D-N map.
THEOREM 2.1.Let Ω be a bounded domain as in assumptions 2.1.Let (1) and (2) be two anisotropic conductivities as in assumptions 2.2, i.e. of the form where = { } =1 is the chain of subdomains as in assumptions 2.1, ( ) is the known Lipschitz matrix and ( ) ( ) are the piecewise-affine functions given by the formula for ( ) ∈ ℝ and ( ) ∈ ℝ .Then there exists a positive constant such that where depends on the a priori data only.
From this result, it follows a Lipschitz stability estimate in terms of the local D-N maps.
COROLLARY 2.2.Assume that the hypothesis of Theorem 2.1 hold, then where > 0 is a constant depending on the a-priori data only.REMARK 2.3.From now on, as we deal with two different anisotropic conductivities ( ) , = 1, 2, we will simply denote with the symbol Λ the local DN map Λ Σ ( ) .

Proof of the main result
The proof of Theorem 2.1 is based on an argument that combines asymptotic estimates for the Green's function of the elliptic operator div( (⋅)∇⋅) (Proposition 3.1), together with a result of unique continuation (Proposition 3.2).In this section we introduce these technical results (proved in Section 4), then we prove Theorem 2.1 and Corollary 2.2.

Behaviour of Green's function near interfaces
We shall denote with the fundamental solution for the Laplace operator (here denotes the volume of the unit ball in ℝ ).Let { } =0 , ∈ {1, … , } be the chain of subdomains as in assumptions 2.1, {Σ } =1 be the corresponding sequence of flat portions with special points 1 , … , .Moreover, let ( +1 ) denotes the unit normal to at the point +1 pointing outside .) where is the positive definite matrix = √ ( +1 ) −1 .
Fix a chain of subdomains { } =0 as in assumptions 2.1 for the domain Ω.Set DEFINITION 3.1.For any , ∈  , define the singular solution The set {  ( , )} =0 is a family of real-valued functions which satisfies the following inequality: where ( ) = dist( ,  ) and is a positive constant depending on and only.One can prove (see [11]) that for every , ∈  with = 0, … , , the functions  (⋅, ),  ( , ⋅) belongs to 1 ( ) and are weak solutions, respectively, to div (1) We introduce the following parameters: Notice that = 0 , 0 < < 1. Choose ∈ ℕ, fix a point ̄ ∈ Σ +1 , then define where is a point into the domain near the interface Σ +1 .For a given ∈ (0, 1 ] define the function For successive estimates, it is important to point out the following inequality: The following estimate for  ( , ) holds true, for any = 1, … , .
PROPOSITION 3.2.(Estimates of unique continuation) Suppose that for a positive number 0 and > 0 we have then the following inequalities hold true for every ∈ (0, 1 ] for any , = 1, … , , where , with h( ) as above, ( +1 ) is the exterior unit normal to at the point +1 pointing outside and 1 , 2 > 0 depend on the a-priori data only.

Proof of Theorem 2.1 and the Corollary 2.2
Proof of Theorem 2.1.First, notice that where ̄ is the Lipschitz constant from assumptions 2.2.Let be the subdomain of Ω such that Then, inequality (2.15) will follow from for > 1 a positive constant depending on a priori estimates.
To prove (3.16), we find convenient , as previously stated, to work in the augmented domain Ω as in (2.9), where 0 is the domain defined in (2.8), on which we have defined the extended conductivity ( ) for = 1, 2 by setting Recalling that is the subdomain of Ω where the maximum of | (1) − (2) | is reached, let 0 , 1 , … , be the chain of subdomains as in Section 2.1 and let Σ 1 , … , Σ be the corresponding flat portions.Set 0 =  ( (1) , (2) Given a differentiable function on a domain Ω, we can split its differential as where is the − 1 dimensional vector of the tangential partial derivatives of on Σ and denotes the normal partial derivative of on Σ , respectively for = 1, 2, … , .

11).
Set = = , split the right hand side of (3.21) into the sum of two integrals 1 ( ) and 2 ( ): where The integral 2 ( ) can be easily estimate using [11,Proposition 3.1] as Let us estimate 1 ( ) from below in terms of ‖ which leads to and , by adding and subtracting the fundamental solution ̃ ( ) Γ we have Now, up to a change of coordinate we can suppose that 1 is the origin .Let us apply the asymptotic estimate (3.4) to (3.27) for = √ −1 (0), it follows that where the > 0 depends on the a-priori data only.By definition (3.1), we can express explicitly the fundamental solution Γ inside the integrals and obtain: By estimating the integrals in (3.28) with respect to the parameter , we can bound | 1 ( )| from below as follows: By (3.23) and (3.24), it follows that which leads to the following estimate for the conductivity: Dividing by 2− both sides and for → 0 + , we obtain 22), for = = as above, we split again the ℎ partial derivative of the singular solution as follows: where With a similar argument as in (3.24) one can determine an upper bound for ̄ 2 of the form where depends on the a-priori data.Notice that for any point ∈ 0 ( 1 ) ∩ 1 , the following equality holds 1 )( 1))( − 1 ) , Proceeding as in (3.25) and (3.26), Up to a rigid transformation, we can assume that 1 coincides with the origin of the coordinate system.Using a similar technique as in (3.27) and by Theorem 3.1, this leads to By (3.30), we derive the following lower bound: Finally, optimizing the right hand side with respect to , the estimate is given by the following inequality so that (3.20) is proved.

Interior estimates
We show that from the case = 1 we obtain the following estimate for the case = 2: Since the proofs of (3.36) and (3.37) are similar, we prove (3.37), assuming that (3.36) holds.
Let 0 = 0 ∕ 4 , where 4 is the constant introduced in Theorem 3.1.Pick ∈ (0, 0 ∕6).Fix the point = ( 2 ) = 2 − ( 2 ) where = h−1 1 .We split the integral solution into two parts: where As in the boundary estimates, we can bound from above 2 ( ) as follows: Now, let us estimate from below the integral 1 ( ) in terms of the quantity | ( (1) 2 − (2)  2 )( 2 )|.First, notice that for any ∈ 0 ( 2 ) ∩ Σ 2 we can rewrite ( ) 2 as By (3.41), (2) Up to a rigid transformation of coordinates, we can assume that 2 coincides with the origin of the coordinate system.By Theorem 3.1, We can estimate the two last terms of the right hand side by (3.36).Then where the constant > 0 depends on the a-priori data and on .This leads to Secondly, by (3.39) and (3.40), .
By unique continuation (Proposition 3.2), we can estimate the integral solution as Since h is a function of , we have to estimate h and 1 h in terms of .Recalling (3.12), it turns out that One can show that the following inequality holds: Then, combining (3.45) together with (3.44), Finally, optimizing with respect to , (3.37) follows.Proceeding as above, for = 3, … , , one can show that the following inequalities hold: and respectively, the procedure is similar to the one seen above.We just point out that, for ( , , then we can bound from above the integral solution by unique continuation (3.14) and (3.15).
Notice that From the property (3.5) it follows that 1∕ (1) . By the estimates (3.46) and (3.47) it follows that This leads to the following estimate for = Since the function 1∕ is invertible for 0 0 + < −2 (otherwise the statement is proven), it follows that ≤ − Hence, (3.50) is proven.

Proof of technical propositions
In this section we give the proof of the propositions needed for the proof of the main result (Theorem 2.1).
Then, there exist positive constants 0 < ′ ≤ 1, > 0 depending on ̄ , , 0 and only, such that for any ≤ 2 and for any ∈ −2 , the following estimate holds Proof.For the proof we refer to Li-Vogelius [54], where piecewise 1, ′ estimates for solutions to elliptic equations in divergence form with piecewise Hölder continuous coefficients have been demonstrated.
By Theorem 4.1 it follows that Hence by (4.3) and (4.8) we From the following interpolation inequality together with (4.7) we obtain . Now, we look for a pointwise bound for ∇ ∇ ( , ).Define the cylinder ̂ = ′  By the representation formula for Γ, and by (4.12) and (4.13), Arguing as above, the following estimate holds: By the following interpolation inequality and by (4.15) and (4.14), we conclude that where 2 = 1 ′ 1+ ′ .

Propagation of smallness
In order to prove Theorem 3.2, we state and prove a preliminary Proposition 4.2, where we determine a pointwise bound for the weak solution to the conductivity equation in the interior of Ω. where ∈ {0, … , − 1}.Suppose there exist , > 0 such that Then, for every ∈ (0, 1 ], where > 1 depends only on a-priori data.
Proof of Proposition 4.2.We adapt the proof in [11,Proposition 4.4] to the case of the anisotropic conductivity.
Up to a rigid transformation of coordinate, we can suppose that 1 = − 32 .From (4.20), Choose an arbitrary point ̄ ∈ Σ 1 , possibly different from 1 .Let be a Jordan curve joining 1 to 1 ( ̄ ) such that ⊂ ( 0 ) ̄ , where ̄ = min{dist( 1 , Σ 1 ), dist( 1 ( ̄ ), Σ 1 )}, and ( 0 ) ̄ is connected.Notice that 1 ( ̄ ) ∈ ( 0 ) ̄ .Let us define a set of points { }, = 1, … , through the following process: Apply the three sphere inequality in the case of pure principal part (see [ and > 1 is a constant which depends on , , max 4 0 , 1 . Notice that ( 2 ) ⊂ 3 ( 1 ) = 3 ( 1 ) so that the 2 -norm of on ( 2 ) can be easily estimated applying the three sphere inequality for the spheres of rays , 3 , 4 centred at 2 .Moreover, by [38,Theorem 8.17], since is a weak solution to (4.18), it follows that where depends on , and |Ω|.By iterating this process, we can estimate the ∞ -norm of along the chain of spheres centred at points of the curve .In conclusion, Fix ∈ (0, 1 ].Recalling the parameters introduced in (3.9), the following inclusions hold: for any = 1, 2, … .Notice that 1 < for a suitable , then 1 ( 1 ( ̄ )) ⊂ ( 1 ( ̄ )).We proceed by moving from one centre to the successive one along the axis of the cone ̄ , ( ̄ ), 1 , 0 ∕3 allowing to get closer and closer to the vertex ̄ and stop this process when we reach the sphere of radius h.Then, from (4.27),In order to prove our claim, we need to estimate the gradient of .Recalling that 0 = | 0 and 1 = | 1 and 0 is harmonic in 0 , from the three sphere inequality applied to ∇ 0 and the results of [54], one can recover the following estimates: Now we can apply the following estimate due to Trytten [65]: In order to bound the lefthand side of (4.33), we have to estimate the following quantities: iii) ∫ 1 ∩ ∕4 ( 1 ) |∇ 1 | 2 .For i), we can just use (4.30).For ii), since ∇ 1 = ∇ 1 + (∇ 1 ⋅ ) , The first integral on the righthand side can be estimated using (4.32).For the other term, one uses the transmission conditions

DEFINITION 2 . 4 .
Let Σ be a non-empty (flat) open portion of Ω.The subspace of 1∕2 ( Ω) of trace functions which are compactly supported in Σ is defined as