Optimal depth-dependent distinguishability bounds for electrical impedance tomography in arbitrary dimension
OOPTIMAL DEPTH-DEPENDENT DISTINGUISHABILITY BOUNDS FORELECTRICAL IMPEDANCE TOMOGRAPHY IN ARBITRARY DIMENSION
HENRIK GARDE AND NUUTTI HYV ¨ONEN
Abstract.
The inverse problem of electrical impedance tomography is severely ill-posed. Inparticular, the resolution of images produced by impedance tomography deteriorates as thedistance from the measurement boundary increases. Such depth dependence can be quantifiedby the concept of distinguishability of inclusions. This paper considers the distinguishability ofperfectly conducting ball inclusions inside a unit ball domain, extending and improving knowntwo-dimensional results to an arbitrary dimension d ≥ Keywords: electrical impedance tomography, Kelvin transformation, depth dependence, distin-guishability.2010 Mathematics Subject Classification: 35P15, 35R30, 47A30.1.
Introduction
The inverse conductivity problem in
Electrical Impedance Tomography (EIT) is known to behighly ill-posed, and under reasonable assumptions it only allows conditional log-type stabilityestimates [2, 29]. Such general estimates are uniform over the examined domain, although Lipschitzstability has been observed at the measurement boundary [2, 3, 4, 12, 36, 23, 33, 32, 34]. Thissuggests the stability of the inverse conductivity problem is actually depth-dependent, which is alsoreflected in the quality of numerical reconstructions. Using the notion of distinguishability , someresults characterizing this depth dependence have recently been obtained in two dimensions [5,18]. Motivated by the inherent three-dimensionality of EIT, this work extends and improves thedepth-dependent distinguishability bounds for perfectly conducting inclusions presented in [18]to an arbitrary spatial dimension d ≥
2. Although we focus here solely on the nonlinear inverseconductivity problem, it should be acknowledged that there also exist previous works tacklingdepth-dependent sensitivity for its linearized version [6, 31].The main tool in our analysis is the Kelvin transformation [24] that takes the role played byM¨obius transformations in [18]. The Kelvin transformation is a traditional tool in, e.g., potentialtheory for problems in unbounded domains, and it is typically defined using the inversion in theunit sphere [9, 10, 30, 37]. However, we need to consider inversions with respect to arbitraryspheres in our analysis, leading to the employment of translated and dilated versions of the classicKelvin transformation (cf. [8, 20]). A central property of all Kelvin transformations, relating themto EIT and also explaining their use in potential theory, is that a function is harmonic if andonly if its Kelvin transformation is harmonic. Compared to the use of M¨obius transformationsfor d = 2 in [18], a complication related to Kelvin transformations for d > B denote the Euclidean unit ball in R d for any integer d ≥
2. We consider the settingwhere a single perfectly conducting concentric ball B (0 , r ) of radius 0 < r < B invariant, onto a nonconcentric ball B ( C, R ) ⊂ B centered at C ∈ B .It turns out that for any ball B ( C, R ) ⊂ B there exists a unique inversion, that relates it in thismanner, to a concentric ball B (0 , r ) with r = r ( | C | , R ) ∈ (0 , C and R as functions of r ∈ (0 ,
1) and a vector a ∈ B \ { } parametrizing all inversions mapping B onto a r X i v : . [ m a t h . A P ] O c t H. GARDE AND N. HYV ¨ONEN itself. To be more precise, a ∈ B ( C, R ) is the image of the origin under the considered inversion,and thus ρ := | a | ∈ (0 ,
1) can be interpreted as a parameter controlling the ‘nonconcentricity’ or‘depth’ of B ( C, R ). Indeed, a small ρ corresponds to an almost concentric inclusion, whereas ρ close to 1 indicates that B ( C, R ) lies close to ∂ B .Let Λ ,r and Λ C,R denote the
Dirichlet-to-Neumann (DN) maps on ∂ B when a perfectly con-ducting inclusion is placed on B (0 , r ) and on B ( C, R ), respectively. In both cases, the remainderof B is characterized by unit conductivity, i.e. by the Laplace equation. Furthermore, let Λ bethe DN map on ∂ B for the inclusion-free problem with unit conductivity. It is well known thatΛ ,r − Λ and Λ C,R − Λ are smoothening [27] and belong in particular to L ( L ( ∂ B )), the spaceof bounded linear operators on L ( ∂ B ). The distinguishability of the inclusion B ( C, R ) is definedto be (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) (see, e.g., [5, 13, 18, 22]), and it can be motivated as follows: AssumeΛ is a DN map on ∂ B corrupted by an additive self-adjoint noise perturbation E ∈ L ( L ( ∂ B )).If the size of this perturbation, (cid:107) E (cid:107) L ( L ( ∂ B )) , is larger than the distinguishability of B ( C, R ) andwithout further information on the structure of E , it is impossible to determine if the datum inhand corresponds to B ( C, R ) embedded in B or to an inclusion-free B .The main result of this paper (Theorem 4.4) relates the distinguishabilities of B (0 , r ) and B ( C, R ) as follows: 1 − ρ ρ ≤ (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) ≤ − ρ ρ . (1.1)Observe that the ratio of the operator norms in (1.1) can be interpreted as a function of r ∈ (0 , ρ = | a | is given; fixing ρ determines the employed Kelvin transformation, or inversion, upto a rotation of B , but the sizes of the considered inclusions B (0 , r ) and B ( C ( a, r ) , R ( ρ, r )) arestill controlled by r ∈ (0 , r ∈ (0 ,
1) exactly give the upper and lower bounds in(1.1), respectively. To be more precise, the lower bound is reached when r → − and the upperbound when r → + .Both the upper and the lower bound in (1.1) converge to zero as ρ → − , which characterizeshow a perfectly conducting inclusion B ( C, R ) becomes more distinguishable to EIT when it ap-proaches the measurement boundary — even though R = R ( ρ, r ) converges to zero when ρ → − for any fixed r ∈ (0 , d >
2. In particular, note that the bounds in (1.1) are independent of the dimension d , as are theformulas for C and R as functions of a and r . Consult Theorem 2.6 in Section 2.1 for the explicitrelation between a , r , C , and R .The Riemann mapping theorem of harmonic morphisms does not have a counterpart for d > B and gives acharacterization of an inversion that maps a given ball inside B onto a ball concentric with B whileleaving B itself invariant (Theorem 2.6). The behavior of the associated Kelvin transformationson ∂ B is also investigated. In Section 3, we relate the DN maps corresponding to concentric andnonconcentric perfectly conducting inclusions via Kelvin transformations (Theorem 3.2). Finally,Section 4 proves our main result (Theorem 4.4). The paper is completed by two appendices. Ap-pendix A analyzes the relation between Kelvin and M¨obius transformations in two dimensions, inorder to ease the comparison of our results and techniques with those in [18]. Appendix B givesa representation formula for the Kelvin transformed DN map, which may be useful for numericalsimulations.1.1. Notational remarks.
We denote by L ( X, Y ) the space of bounded linear operators betweenBanach spaces X and Y , and introduce the shorthand notation L ( X ) := L ( X, X ). PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 3
For any scalar-valued function g , we denote by g k , k ∈ Z , the composition of the k ’th powerand g . That is, g k ( x ) := g ( x ) k for any k ∈ Z and all x in the domain of g for which g ( x ) k is defined.In particular, g − = 1 /g is not the inverse of g . The ‘power notation’ has its usual meaning forlinear operators and matrices. In particular, A − is the inverse of the matrix/operator A .The open Euclidean ball and sphere with center C ∈ R d and radius R > B ( C , R ) and S ( C , R ) := ∂B ( C , R ), respectively. We use the shorthand notation B := B (0 ,
1) forthe unit ball and denote by d S the standard (unnormalized) spherical measure on ∂ B .The Euclidean norm is denoted by | · | , and { e j } dj =1 is the canonical basis for R d . For a ∈ R d \{ } ,we set e a := a | a | ∈ ∂ B . 2. Kelvin transformations
The inversion in the sphere S ( C , R ), denoted I C , R : R d \ {C} → R d \ {C} , is defined as I C , R ( x ) := R x − C| x − C| + C = g C , R ( x )( x − C ) + C , where g C , R ( x ) := R / | x − C| . Note that I C , R maps the points in the interior of S ( C , R ) to itsexterior and vice versa. Furthermore, I C , R ( x ) is the unique point on the half-line from C through x satisfying | I C , R ( x ) − C|| x − C| = R (2.1)for any x (cid:54) = C . This clearly indicates the fixed points of I C , R are precisely S ( C , R ). Moreover, I C , R ◦ I C , R = id, i.e., I C , R is an involution. The inversion I C , R maps spheres and hyperplanes ontospheres and hyperplanes, but not necessarily respectively and with possibly C removed [8]. Inparticular, as I C , R clearly maps any bounded set E ⊂ R d with dist( E, C ) > C to spheres. We use the special notation ˆ x := I , ( x ), x ∈ R d / { } ,for the inversion in the unit sphere ∂ B = S (0 , C ∈ R d , choose any open Ω ⊆ R d \ {C} , and set Ω ∗ := I C , R (Ω). The composition ofa function with I C , R can be interpreted as a linear operator from L (Ω) to L (Ω ∗ ) or viceversa, i.e., we define I C , R f := f ◦ I C , R (2.2)for all f ∈ L (Ω) or f ∈ L (Ω ∗ ). In particular, I C , R is an involution. We also introduce a linearmultiplication operator G C , R via G C , R f := g C , R f. Since C / ∈ Ω, it is obvious that G C , R maps L (Ω) onto itself, and its inverse G − C , R : L (Ω) → L (Ω) is defined as the multiplication by g − C , R . The same conclusions also apply on L (Ω ∗ )because C / ∈ Ω ∗ as well. A straightforward calculation leads to the identities G C , R I C , R = I C , R G − C , R and G − C , R I C , R = I C , R G C , R , (2.3)which will be frequently utilized in what follows. Definition 2.1.
The general
Kelvin transformation is defined as a linear map K C , R : L (Ω) → L (Ω ∗ ), or K C , R : L (Ω ∗ ) → L (Ω), by setting K C , R := G d − C , R I C , R = I C , R G − d C , R . (2.4) Remark . The latter equality in Definition 2.1 follows from (2.3) and demonstrates that K C , R is also an involution. Moreover, the Kelvin transformation obviously inherits the identities G C , R K C , R = K C , R G − C , R and G − C , R K C , R = K C , R G C , R (2.5)from the associated inversion operator, cf. (2.3).As preparation for the following result: for any y ∈ R d \{ } , let P y be the orthogonal projectiononto the line span { y } , and let Q y ∈ R d × d be an arbitrary matrix whose first row is y T / | y | andthe remaining d − { y } ⊥ . Clearly Q y is orthogonal andsatisfies Q y y = | y | e . H. GARDE AND N. HYV ¨ONEN
Proposition 2.3.
The Jacobian matrix J C , R of I C , R on R d \ {C} is J C , R ( x ) = g C , R ( x )(id − P x −C ) (2.6)= g C , R ( x ) Q − x −C diag( − , , , . . . , Q x −C , (2.7) As direct consequences,(i) det( J C , R ) = − g d C , R ,(ii) J T C , R = J C , R , J C , R = g C , R id , and J − C , R = g − C , R J C , R .Proof. The expression (2.6) follows directly from ∂∂x j I C , R ( x ) = R (cid:18) | x − C| e j − x − C ) j | x − C| ( x − C ) (cid:19) , x ∈ R d \ {C} , which readily yields J C , R ( x ) y = g C , R ( x ) (cid:18) y − y · ( x − C ) | x − C| ( x − C ) (cid:19) = g C , R ( x )(id − P x −C ) y for any y ∈ R d and x ∈ R d / {C} .As Q x −C rotates the coordinate system so the direction of x − C corresponds to the first coor-dinate, we have P x − C = Q − x −C diag(1 , , , . . . , Q x −C . Hence, J C , R ( x ) = g C , R ( x )(id − P x −C ) = g C , R ( x ) Q − x −C (cid:0) id − diag(2 , , , . . . , (cid:1) Q x −C , x ∈ R d / {C} , which proves (2.7). Finally, (i) and (ii) immediately follow from (2.7) since Q x −C is orthogonal. (cid:3) Obviously, I C , R maps compact subsets of Ω to compact subsets of Ω ∗ and vice versa. Since I C , R is smooth on R d \ {C} , it holds that I C , R ( C ∞ c (Ω)) ⊆ C ∞ c (Ω ∗ ) and I C , R ( C ∞ c (Ω ∗ )) ⊆ C ∞ c (Ω).By applying I C , R to these inclusions, the reversed ones C ∞ c (Ω) ⊆ I C , R ( C ∞ c (Ω ∗ )) and C ∞ c (Ω ∗ ) ⊆I C , R ( C ∞ c (Ω)) follow. Since g C , R is smooth and positive on R d \ {C} , all these inclusions also holdwhen I C , R is replaced by G k C , R I C , R for any k ∈ Z . Altogether, we have thus established G k C , R I C , R ( C ∞ c (Ω)) = C ∞ c (Ω ∗ ) = K C , R ( C ∞ c (Ω)) , k ∈ Z , which naturally also holds if the roles of Ω and Ω ∗ are reversed.In consequence, we may extend both K C , R and I C , R to continuous linear maps on distributionsfrom D (cid:48) (Ω) to D (cid:48) (Ω ∗ ) by setting (cid:104) K C , R u, φ (cid:105) := (cid:104) u, G C , R K C , R φ (cid:105) , (2.8) (cid:104)I C , R u, φ (cid:105) := (cid:104) u, G d C , R I C , R φ (cid:105) , (2.9)for any u ∈ D (cid:48) (Ω), all φ ∈ C ∞ c (Ω ∗ ), and with (cid:104)· , ·(cid:105) denoting the dual pairing of ( D (cid:48) , C ∞ c ). Thedefinition (2.8) coincides with that of the distributional Kelvin transformation in [20], and it canbe motivated as follows: for any u ∈ L (Ω) and φ ∈ C ∞ c (Ω ∗ ), we have (cid:90) Ω ∗ K C , R u φ d x = (cid:90) Ω ∗ I C , R u G d − C , R φ d x = (cid:90) Ω u ( I C , R G d − C , R φ ) | det( J C , R ) | d x = (cid:90) Ω u G d C , R I C , R G d − C , R φ d x = (cid:90) Ω u G C , R K C , R φ d x, (2.10)where we used Proposition 2.3(i), (2.3), and (2.4). In other words, (2.8) coincides with the standardKelvin transformation (2.4) on L (Ω). The same conclusion also applies to (2.9). It is easy tocheck that K C , R : D (cid:48) (Ω) → D (cid:48) (Ω ∗ ) and I C , R : D (cid:48) (Ω) → D (cid:48) (Ω ∗ ) are involutions satisfying (2.5)and (2.3), respectively.If Ω is also bounded and satisfies dist(Ω , C ) >
0, it holds G k C , R I C , R ( H m (Ω)) = H m (Ω ∗ ) = K C , R ( H m (Ω)) , k ∈ Z , PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 5 for any m ∈ Z and also with the roles of Ω and Ω ∗ reversed. To see this, note first that Ω is boundedif and only if dist(Ω ∗ , C ) >
0. Hence, the derivatives of I C , R and g C , R , up to an arbitrary order | m | , are uniformly bounded by a constant (depending on | m | ) in both Ω and Ω ∗ . Therefore, forany k ∈ Z , G k C , R I C , R ( H m (Ω)) ⊆ H m (Ω ∗ ) and G k C , R I C , R ( H m (Ω ∗ )) ⊆ H m (Ω). Applying G k C , R I C , R to the latter inclusion yields H m (Ω ∗ ) ⊆ G k C , R I C , R ( H m (Ω)), which proves the claim. In fact, (cid:107) G k C , R I C , R u (cid:107) H m (Ω ∗ ) ≤ C (Ω , R , C , k, m ) (cid:107) u (cid:107) H m (Ω) for all u ∈ H m (Ω) , (2.11)demonstrating, in particular, the boundedness of K C , R : H m (Ω) → H m (Ω ∗ ). The same conclusionnaturally applies to the inverse K C , R : H m (Ω ∗ ) → H m (Ω) as well. Definition 2.4.
The usual translation and dilation operators are defined on locally integrablefunctions by T a f ( x ) := f ( x − a ) and D b f ( x ) := f ( b − x ) for a ∈ R d and b > K C , R : L (Ω) → L (Ω ∗ ) satisfies K C , R = T C D R K , D R − T −C , (2.12)and thus all Kelvin transformations are dilated and translated variants of K , . Take note that(2.12) also holds for the distributional Kelvin transformation, if the the dilation and translationare defined on distributions via the dual pairing in the natural manner, i.e., in the way that theextended definitions coincide with the original ones for locally integrable functions.We now arrive at the following fundamental theorem. Theorem 2.5.
The commutation relation ∆ K C , R = G C , R K C , R ∆ holds on D (cid:48) ( R d \ {C} ) .Proof. The result is well known to hold for K , on C ( R d \ { } ) (see., e.g., [8, Theorem 1.6.3] or[9, Proposition 4.6]), that is, ∆ K , u = G , K , ∆ u for all u ∈ C ( R d \ { } ). Employing (2.12) and the identity G C , R T C D R = T C D R G , , we obtain∆ K C , R = ∆ T C D R K , D R − T −C = R − T C D R ∆ K , D R − T −C = R − T C D R G , K , ∆ D R − T −C = G C , R T C D R K , D R − T −C ∆= G C , R K C , R ∆on C ( R d \ {C} ). To finalize the proof, let u ∈ D (cid:48) ( R d \ {C} ) and φ ∈ C ∞ c ( R d \ {C} ) be arbitrary.We apply the definition of distributional differentiation and (2.8) to deduce (cid:104) ∆ K C , R u, φ (cid:105) = (cid:104) u, G C , R K C , R ∆ φ (cid:105) = (cid:104) u, ∆ K C , R φ (cid:105) = (cid:104) ∆ u, G C , R K C , R G C , R φ (cid:105) = (cid:104) G C , R K C , R ∆ u, φ (cid:105) , where we also used (2.5) in the third equality. (cid:3) Kelvin transformations on the unit ball.
From now on, we restrict our attention to theunit ball, i.e. choose Ω = B , and concentrate on such inversions that also Ω ∗ = B . However, itwould be straightforward to generalize our results to a ball of any radius. We will employ thesymbols C, R , instead of C , R , to annotate a ball B ( C, R ) embedded in B .Let a ∈ B \ { } and write a = ρe a with ρ = | a | and e a ∈ ∂ B . Recall the notation ˆ a = I , ( a ),notice that | ˆ a | = ρ − >
1, and define b := ( ρ − − / = ρ − (1 − ρ ) / . We introduce a special H. GARDE AND N. HYV ¨ONEN
Kelvin transformation on L ( B ) by setting K a := K ˆ a,b ; similarly, we also define g a := g ˆ a,b , G a := G ˆ a,b , I a := I ˆ a,b , and I a := I ˆ a,b . More explicitly, K a f ( x ) = G d − a I a f ( x ) = (cid:18) b | x − ˆ a | (cid:19) d − f (cid:18) b x − ˆ a | x − ˆ a | + ˆ a (cid:19) , f ∈ L ( B ) , and this definition naturally extends to D (cid:48) ( B ) through (2.8).The following theorem shows that I a leaves B invariant for any a ∈ B \ { } , and it also givesa characterization of how any nonconcentric ball inside B can be mapped to a concentric oneby I a with a suitable a . It is worth noting that the formulas (2.13) and (2.14) below generalizethe two-dimensional result in [18, Proposition 2.1]; see Appendix A. Moreover, an equivalentcharacterization for three spatial dimensions can be found in [20]. Theorem 2.6.
Assume a ∈ B \ { } . The inversion I a leaves B invariant, that is, I a ( B ) = B and I a ( ∂ B ) = ∂ B . In particular, I a | ∂ B ( x ) = (id − P x − ˆ a ) x, x ∈ ∂ B , with P y denoting the orthogonal projection onto the line spanned by y ∈ R d \ { } .The following two items completely characterize how a concentric ball embedded in B is deformedunder a given I a , as well as which I a maps a given ball embedded in B to a concentric one:(i) Let r ∈ (0 , and a = ρe a with ρ ∈ (0 , and e a ∈ ∂ B . Then I a ( B (0 , r )) = B ( C, R ) and I a ( S (0 , r )) = S ( C, R ) with C = ρ ( r − ρ r − e a , R = r ( ρ − ρ r − . (2.13) (ii) Let R ∈ (0 , − c ) and C = ce a with c ∈ (0 , and e a ∈ ∂ B . Then I a ( B ( C, R )) = B (0 , r ) and I a ( S ( C, R )) = S (0 , r ) with r = 1 + R − c − (cid:112) ((1 − R ) − c )((1 + R ) − c )2 R , a = C − Rr . (2.14)
Proof.
Assuming | x | = 1, we obtain I a ( x ) = ( | ˆ a | − x − ˆ a ) | x − ˆ a | + ˆ a = ( | ˆ a | − x + 2(1 − x · ˆ a )ˆ a | x − ˆ a | , as well as (id − P x − ˆ a ) x = x − x · ( x − ˆ a ) | x − ˆ a | ( x − ˆ a ) = ( | ˆ a | − x + 2(1 − x · ˆ a )ˆ a | x − ˆ a | . These formulas verify the claimed representation for I a on ∂ B .Let us then prove that I a maps the closure of B onto itself. Clearly, I a sends the points on theline spanned by a to that same line, with the exception of ˆ a / ∈ B that is mapped to infinity andis the only point of discontinuity for I a . Let B a := { te a | t ∈ ( − , } . As I a ( ± e a ) = ∓ e a and I a (0) = a ∈ B a , it follows from the continuity of I a that I a ( B a ) = B a . Since I a is the inversionin the sphere S (ˆ a, b ), it is symmetric about the line spanned by a . In particular, I a maps anysphere centered on the line spanned by a , and not intersecting ˆ a , onto another sphere centeredon that very same line. Hence, I a ( ∂ B ) = ∂ B because I a ( ∂ B ) is known to contain ± e a . As I a is acontinuous involution away from ˆ a / ∈ B and I a (0) ∈ B , it must in fact hold I a ( B ) = B .Because (2.14) in part (ii) of the assertion follows by a straightforward but tedious calculationbased on (2.13) and I a being an involution (cf. [18]), we only need to consider the proof of part (i).In the same manner as above, it can be argued that I a maps S (0 , r ), with 0 < r <
1, onto somesphere S ( ce a , R ) ⊂ B and that B a ∩ S ( ce a , R ) = { I a ( − re a ) , I a ( re a ) } , where I a ( re a ) = ( ρ − − r − ρ − ( ρ − − r ) e a + ρ − e a = ρ − r − ρr e a ,I a ( − re a ) = − ( ρ − − r + ρ − ( ρ − + r ) e a + ρ − e a = ρ + r ρr e a . PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 7
Hence, C = ce a = ( I a ( − re a )+ I a ( re a )) / R = | I a ( − re a ) − C | , giving the expressions in (2.13).As mentioned above, the two relations between a , r , C , and R in (2.13) are equivalent tothose in (2.14), and so we can also assume the knowledge of the latter in the following. Since B ( ce a , R ) ⊆ I a ( B ) = B , it must hold 0 < c < − R . With the help of the latter equality in (2.14),we thus get | a − ce a | = | ρ − c | = (cid:12)(cid:12)(cid:12) c − Rr − c (cid:12)(cid:12)(cid:12) = cr − Rr R < − R − Rr R < R as 0 < r <
1. In other words, I a (0) = a ∈ B ( ce a , R ). As I a is a continuous involution, it thusmaps the whole of B (0 , r ) onto B ( ce a , R ) and the proof is complete. (cid:3) As expected, the Jacobian matrix of I a is denoted J a := J ˆ a,b , with J ˆ a,b explicitly given inProposition 2.3. The following corollary provides information about the behavior of J a on ∂ B ,enabling substitutions corresponding to I a in integrals over ∂ B . In particular, it enables theintroduction of the distributional Kelvin transformation on D (cid:48) ( ∂ B ), in the sense of distributionson a smooth manifold (cf. e.g. [21, Chapter 6.3]). Corollary 2.7.
For x ∈ ∂ B , J a ( x ) x = g a ( x ) I a ( x ) and J a ( x ) I a ( x ) = g a ( x ) x .Proof. The first equality follows from a combination of Proposition 2.3 and Theorem 2.6 thatrelate both I a ( x ) and J a ( x ) to id − P x − ˆ a when x ∈ ∂ B . By applying the inverse of J a ( x ) to thisfirst equality, one obtains x = g a ( x ) J − a ( x ) I a ( x ) = g − a ( x ) J a ( x ) I a ( x ) , x ∈ ∂ B , where we used Proposition 2.3(ii) in the second step. This completes the proof. (cid:3) The linear maps K a , I a , and G a can obviously also be interpreted as operators on L ( ∂ B ) = L ( ∂ B ), and G a as such on D (cid:48) ( ∂ B ) as well. In the spirit of (2.8) and (2.9), we introduce theextensions K a , I a : D (cid:48) ( ∂ B ) → D (cid:48) ( ∂ B ) via (cid:104) K a f, ϕ (cid:105) := (cid:104) f, G a K a ϕ (cid:105) , (cid:104)I a f, ϕ (cid:105) := (cid:104) f, G d − a I a ϕ (cid:105) , for any f ∈ D (cid:48) ( ∂ B ) and all ϕ ∈ C ∞ c ( ∂ B ) = C ∞ ( ∂ B ). As in the case of (2.8) and (2.9), thesedefinitions make sense because C ∞ ( ∂ B ) = G ka I a ( C ∞ ( ∂ B )) for any k ∈ Z , and it is also easy tocheck that the extensions still satisfy (2.3) and (2.5). Remark . The extended operators coincide with the standard ones (2.4) and (2.2) on L ( ∂ B ).Indeed, one can prove this by performing similar calculations as in (2.10), with the exception thatthis time around the (boundary) Jacobian determinant reads | det J a ( x ) || J a ( x ) x | = g da ( x ) | g a ( x ) I a ( x ) | = g d − a ( x ) , x ∈ ∂ B , (2.15)due to Corollary 2.7 and since ν ( x ) = x is the exterior unit normal at x ∈ ∂ B . For K a such acalculation is actually explicitly carried out in the proof of Lemma 2.9 below. As for the Sobolevspaces over the domain Ω in (2.11), it follows straightforwardly that (cid:107) G ka I a f (cid:107) H s ( ∂ B ) ≤ C ( a, k, s ) (cid:107) f (cid:107) H s ( ∂ B ) (2.16)for any s, k ∈ Z . The standard theory on interpolation of Sobolev spaces demonstrates that (2.16)actually holds for any s ∈ R ; see, e.g., [1, 28]. Finally, it follows via a density argument that K a and I a are involutions on H s ( ∂ B ) for any s ∈ R .We have now gathered enough tools to explicitly evaluate certain operator norms of K a andclosely related operators. Lemma 2.9.
The following results hold in L ( L ( B )) and L ( L ( ∂ B )) :(i) G a K a is an isometry in L ( L ( B )) and G a K a is an isometry in L ( L ( ∂ B )) .(ii) K ∗ a = G a K a in L ( L ( B )) and K ∗ a = G a K a in L ( L ( ∂ B )) . H. GARDE AND N. HYV ¨ONEN (iii) There are the following operator norm equalities: (cid:107) K a (cid:107) L ( L ( B )) = (cid:107) G a K a (cid:107) L ( L ( B )) = (cid:107) K a (cid:107) L ( L ( ∂ B )) = (cid:107) G a K a (cid:107) L ( L ( ∂ B )) = (cid:107) G ± a (cid:107) L ( L ( B )) = (cid:107) G ± a (cid:107) L ( L ( ∂ B )) = 1 + ρ − ρ . Proof.
Proof of part (i). It follows directly from (2.4), the change of variables formula, Proposi-tion 2.3(i), and (2.3) that (cid:90) B | G a K a f | d x = (cid:90) B G da I a | f | d x = (cid:90) B G da I a G da I a | f | d x = (cid:90) B | f | d x (2.17)for all f ∈ L ( B ), which proves the first half of the claim. To prove the second half, observe that(2.15) yields (cid:90) ∂ B I a g d S = (cid:90) ∂ B G d − a g d S and (cid:90) ∂ B g d S = (cid:90) ∂ B G d − a I a g d S (2.18)for all g ∈ L ( ∂ B ). As in (2.17), one thus obtains (cid:90) ∂ B | G a K a f | d S = (cid:90) ∂ B | f | d S for all f ∈ L ( ∂ B ), which completes the proof of part (i).Proof of part (ii). We only prove the result in L ( L ( ∂ B )) since the proof for L ( L ( B )) followsfrom the same line of reasoning. For any f, g ∈ L ( ∂ B ), (cid:90) ∂ B K a f g d S = (cid:90) ∂ B I a f G d − a g d S = (cid:90) ∂ B f G d − a I a G d − a g d S = (cid:90) ∂ B f G da I a g d S = (cid:90) ∂ B f G a K a g d S, where we employed (2.18) and (2.3).Proof of part (iii). Since ˆ a (cid:54)∈ B , it is straightforward to see that both the maximum and theminimum of g a over the compact set B are attained on ∂ B . More precisely, the maximum (orthe minimum) is obviously found at the point closest to (or furthest from) ˆ a , i.e. at e a (or − e a ),which leads to sup x ∈ B g a ( x ) = ρ − − | e a − ˆ a | = ( ρ − − ρ − + 1)( ρ − − = 1 + ρ − ρ , inf x ∈ B g a ( x ) = ρ − − | e a + ˆ a | = ( ρ − − ρ − + 1)( ρ − + 1) = 1 − ρ ρ . In particular, as the norm of a multiplication operator on L is given by the essential supremumof the multiplier, we have (cid:107) G ± a (cid:107) L ( L ( B )) = (cid:107) G ± a (cid:107) L ( L ( ∂ B )) = (cid:18) ρ − ρ (cid:19) / . Due to the duality results of part (ii), what remains to be shown is (cid:107) K a (cid:107) L ( L ( B )) = (cid:107) G a (cid:107) L ( L ( B )) and (cid:107) K a (cid:107) L ( L ( ∂ B )) = (cid:107) G a (cid:107) L ( L ( ∂ B )) . Again, the proofs of these identities are analogous, and weonly show the latter. By virtue of part (i) and (2.5), (cid:107) K a f (cid:107) L ( ∂ B ) = (cid:107) G a K a G a f (cid:107) L ( ∂ B ) = (cid:107) G a f (cid:107) L ( ∂ B ) for all f ∈ L ( ∂ B ), which concludes the proof. (cid:3) For d >
2, this follows from the maximum principle and the fact that g d − a is harmonic. PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 9
In addition to Theorem 2.5, we must also consider the commutation of K a and ∇ , as thisis needed for handling Neumann boundary values. Unfortunately, the resulting expression issomewhat more complicated than that in Theorem 2.5. First of all, note that ∇ (cid:0) g ma ( x ) (cid:1) = mg m − a ( x ) ∇ g a ( x ) = − mg ma ( x ) x − ˆ a | x − ˆ a | , m ∈ N . Hence, for any u ∈ C ∞ ( B ), ∇ K a u ( x ) = g d − a ( x ) ∇I a u ( x ) + I a u ( x ) ∇ g d − a ( x )= J T a ( x ) K a ∇ u ( x ) + (2 − d ) x − ˆ a | x − ˆ a | K a u ( x ) , (2.19)where K a is separately applied to each component of ∇ u . As ˆ a / ∈ B , the expression (2.19) is validfor all x ∈ B ; recall that C ∞ ( B ) consists of the restrictions of the elements in C ∞ ( R d ) to B .Observe that ν ( x ) = x is the exterior unit normal to ∂ B for any x ∈ ∂ B . According to (2.19), ν · ∇ K a u ( x ) = G da I a ( ν · ∇ u )( x ) + (2 − d ) x · x − ˆ a | x − ˆ a | K a u ( x )= G a K a ( ν · ∇ u )( x ) + (2 − d ) H a K a u ( x ) , (2.20)for all x ∈ ∂ B and u ∈ C ∞ ( B ). Here, the multiplication operator H a ∈ L ( H s ( ∂ B )), s ∈ R , isdefined by H a f ( x ) := x · x − ˆ a | x − ˆ a | f ( x ) , and the first equality in (2.20) follows from the identity x · J T a ( x ) K a ∇ u ( x ) = g da ( x ) I a ( x ) · I a ∇ u ( x ) = g da ( x ) I a ( ν · ∇ u )( x ) , x ∈ ∂ B , that is based on Corollary 2.7.Let U = W \ ∂ B where W ⊆ B is a relatively open neighborhood of ∂ B . The identity (2.20)extends by continuity to all u in H ( U ) := { v ∈ H ( U ) | ∆ v ∈ L ( U ) } equipped with the graph norm, that is,( ν · ∇ K a u ) | ∂ B = G a K a (cid:0) ( ν · ∇ u ) | ∂ B (cid:1) + (2 − d ) H a K a ( u | ∂ B ) , u ∈ H ( U ) . (2.21)This result follows from C ∞ ( U ) being dense in H ( U ) [28], the Neumann trace extending to abounded operator H ( U ) → H − / ( ∂ B ) [14, Lemma 1, p. 381], the standard trace theorem, andthe boundedness of K a on H ± / ( ∂ B ) and H ( U ) guaranteed by (2.16), (2.11), and Theorem 2.5.The most essential message of (2.21) is that a Neumann condition on ∂ B is transformed by K a into a Robin condition. Luckily, the Robin condition transforms back to a Neumann condition fordifference measurements of EIT, as revealed in the next section.3. Application to electrical impedance tomography
Let Ω
C,R := B \ B ( C, R ) for C ∈ B and R ∈ (0 , − | C | ). We only consider the case of aperfectly conducting inclusion B ( C, R ) (formally with conductivity ∞ ) and assume that Ω C,R ischaracterized by unit conductivity. Hence, if the electric potential on the exterior boundary is setto f ∈ H / ( ∂ B ), then the interior potential u ∈ H (Ω C,R ) is the unique solution to∆ u = 0 in Ω C,R ,u = (cid:40) ∂B ( C, R ) ,f on ∂ B . (3.1)We define the DN map associated to the inclusion B ( C, R ) asΛ
C,R : f (cid:55)→ ν · ∇ u | ∂ B , H / ( ∂ B ) → H − / ( ∂ B ) , and note that it is well known to be bounded due to the continuous dependence of the solution to(3.1) on the Dirichlet data and a suitable Neumann trace theorem (cf., e.g., [14, Lemma 1, p. 381]).We also define Λ : H / ( ∂ B ) → H − / ( ∂ B ) to be the DN map for the inclusion-free problem∆ w = 0 in B ,w = f on ∂ B . For each C ∈ B \ { } and R ∈ (0 , − | C | ), we choose the unique a = a ( C, R ) ∈ B \ { } suchthat I a ( B ( C, R )) = B (0 , r ) and I a ( B (0 , r )) = B ( C, R ) for some
R < r <
1, which is possible byvirtue of Theorem 2.6. We consistently use this connection between
C, R and a, r in what follows.The accordingly Kelvin-transformed potential ˜ u := K a u ∈ H (Ω ,r ) is the unique solution of∆˜ u = 0 in Ω ,r , ˜ u = (cid:40) ∂B (0 , r ) , ˜ f on ∂ B , (3.2)for ˜ f := K a f . Indeed, the first equation is a direct consequence of Theorem 2.5. To prove thesecond one, observe that obviously ( K a v ) | ∂ Ω ,r = K a ( v | ∂ Ω C,R )for all v ∈ C ∞ (Ω C,R ), and this equality extends by density for any v ∈ H (Ω C,R ) due to (obviousgeneralizations of) the estimates (2.11), (2.16), and the continuity of the Dirichlet trace mapon both H (Ω C,R ) and H (Ω ,r ). Since K a : H (Ω ,r ) → H (Ω C,R ) is obviously the inverse of K a : H (Ω C,R ) → H (Ω ,r ), the solution of (3.1) can alternatively be written as K a ˜ u with ˜ u beingthe solution to (3.2). Remark . The physically correct condition at a perfectly conducting inclusion, is that thepotential equals such a constant on its boundary that the corresponding normal current densityhas zero mean, under the sound assumption that there are no sinks or sources inside the inclusion.Notice that this constant may depend on both f and the inclusion itself. Let ˜Λ C,R be the DNmap corresponding to the boundary conditions of such a physically accurate setting. Since weonly have a single connected inclusion, ˜Λ C,R = Λ
C,R | Y C,R , where Y C,R is a linear subspace of H / ( ∂ B ), which again may depend on the inclusion. Due to this inconvenience — in particular,for the inverse conductivity problem where the inclusion is not known a priori — the DN operatorwith a larger domain Λ C,R is often investigated instead of ˜Λ
C,R (cf., e.g., [7, 11, 16, 26, 25, 35]).This is also the choice in this work.We are now ready to prove an explicit relation between the DN maps for the concentric andnonconcentric geometries.
Theorem 3.2.
Let B ( C, R ) = I a ( B (0 , r )) for a ∈ B \ { } and r ∈ (0 , . Then, Λ C,R = G a K a Λ ,r K a + (2 − d ) H a . Furthermore, Λ C,R − Λ = G a K a (Λ ,r − Λ ) K a , or equivalently, Λ ,r − Λ = G a K a (Λ C,R − Λ ) K a . Proof.
Let ˜ g := ( ν · ∇ ˜ u ) | ∂ B ∈ H − / ( ∂ B ) be the normal current density for the solution of (3.2).We obtain directly from (2.21) thatΛ C,R f = ( ν · ∇ u ) | ∂ B = ( ν · ∇ K a ˜ u ) | ∂ B = G a K a ˜ g + (2 − d ) H a K a ˜ f = G a K a Λ ,r ˜ f + (2 − d ) H a K a ˜ f = (cid:0) G a K a Λ ,r K a + (2 − d ) H a (cid:1) f for all f ∈ H / ( ∂ B ). This proves the first part of the claim. In general, different constants appear on each connected component of a perfectly conducting inhomogeneity.
PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 11
Following exactly the same line of reasoning as above,Λ = G a K a Λ K a + (2 − d ) H a , which shows the claimed representation for Λ C,R − Λ . As G a K a and K a are involutions on C ∞ ( ∂ B ) (cf. (2.5)), they are also such on any H s ( ∂ B ), s ∈ R , by density and (2.16). Hence, therepresentation for Λ ,r − Λ follows directly from that of Λ C,R − Λ , and the proof is complete. (cid:3) Before determining the spectrum of Λ ,r − Λ needed in proving our main result, we brieflyreview a few facts about spherical harmonics; see, e.g., [9, 15, 19] for additional details. Denoting R d \ { } (cid:51) x = ηθ with η = | x | and θ = x | x | ∈ ∂ B , it is well known that the Laplace operator canbe written in polar coordinates (cf. [19, Section 4.5]) as∆ = 1 η d − ∂ η ( η d − ∂ η ) + 1 η ∆ ∂ B , (3.3)where ∆ ∂ B is the Laplace–Beltrami operator on ∂ B with respect to θ .A polynomial p on R d is called homogeneous of degree n if p ( x ) = (cid:80) | α | = n c α x α with scalarcoefficients c α , or equivalently p ( tx ) = t n p ( x ) for t ∈ R ; following the standard notation, α ∈ N d is here a multi-index, | α | := (cid:80) dj =1 α j , and x α := Π dj =1 x α j j . The complex vector space H n,d , of spherical harmonics of degree n , comprise the harmonic polynomials homogeneous of degree n restricted to ∂ B , i.e., H n,d := (cid:26) p | ∂ B | p ( x ) = (cid:88) | α | = n c α x α , x ∈ R d , ∆ p = 0 (cid:27) . The corresponding dimension α n,d := dim( H n,d ) is given by α n,d = (cid:18) n + d − d − (cid:19) − (cid:18) n + d − d − (cid:19) , where we use the convention (cid:0) mk (cid:1) = 0 for m < k .The eigenvalues of ∆ ∂ B are ˜ λ n := − n ( n + d − n ∈ N , with the algebraic and geometricmultiplicity α n,d . The eigenspace corresponding to the eigenvalue ˜ λ n is H n,d , spanned by theorthonormal n th degree spherical harmonics { f n,j } α n,d j =1 . The set of f n,j for all n ∈ N and j ∈{ , . . . , α n,d } is an orthonormal basis for L ( ∂ B ), i.e. L ( ∂ B ) = (cid:76) ∞ n =0 H n,d .Using separation of variables and classic Sturm–Liouville theory for (3.3), it is known that anyharmonic function u on B or Ω ,r can be written as u ( x ) = ∞ (cid:88) n =0 α n,d (cid:88) j =1 c n,j R n ( η ) f n,j ( θ )for ( c n,j ) ∈ (cid:96) . Here R n is a solution of η R (cid:48)(cid:48) n ( η ) + ( d − ηR (cid:48) n ( η ) + ˜ λ n R n ( η ) = 0 , (3.4)with suitable boundary conditions at η = 1 and either at η = 0 or η = r depending on thoserequired from u . If u | ∂ B = f n,j , then R n (1) = 1 and ( ν · ∇ u ) | ∂ B = R (cid:48) n (1) f n,j , i.e., R (cid:48) n (1) is the n theigenvalue of the associated DN map corresponding to the eigenfunction f n,j . In particular, thealgebraic and geometric multiplicity of R (cid:48) n (1) is also α n,d . Based on these observations, we canexplicitly determine the eigenvalues of Λ ,r , Λ , and Λ ,r − Λ . Proposition 3.3.
The eigenvalues of Λ are { n } n ∈ N and those of Λ ,r are, for n ∈ N , ˆ λ n := n + ( n + d − r n + d − − r n + d − , d > ∨ n ≥ , − r ) , d = 2 ∧ n = 0 , both with the algebraic and geometric multiplicity α n,d and the eigenspace H n,d . As a consequence,the eigenvalues of Λ ,r − Λ are λ n := n + d − r − d − n − , d > ∨ n ≥ , − r ) , d = 2 ∧ n = 0 , also with the algebraic and geometric multiplicity α n,d and the eigenspace H n,d .Proof. Note that (3.4) is a second order Cauchy–Euler equation. Its indicial equation m n + ( d − m n + ˜ λ n = 0has the solutions m n = (cid:40) n =: m + n , − d − n =: m − n . If d >
2, then m + n (cid:54) = m − n for all n ∈ N , which also holds for d = 2 when n ≥
1. Hence, allsolutions to (3.4) are of the form R n ( η ) = (cid:40) c n η n + ˜ c n η − d − n , d > ∨ n ≥ ,c + ˜ c log( η ) , d = 2 ∧ n = 0 . Starting with Λ , we see that the boundary conditions for (3.4) are lim sup η → + | R n ( η ) | < ∞ and R n (1) = 1, which immediately imply ˜ c n = 0 and c n = 1 for all n ∈ N . The eigenvalues of Λ arethus R (cid:48) n (1) = n for n ∈ N , as claimed.Now considering Λ ,r , the boundary conditions for (3.4) become R n ( r ) = 0 and R n (1) = 1. Forthe special case d = 2 and n = 0, we arrive at R ( η ) = 1 − log( η )log( r ) . For d > n ≥
1, we obtain R n ( η ) = η n − r n + d − + η − d − n − r − d − n . Evaluating R (cid:48) n (1) provides the sought-for representation for ˆ λ n . Furthermore, as the eigenfunctionsof Λ ,r and Λ coincide, we obtain the representation for λ n by evaluating the difference R (cid:48) n (1) − n for n ∈ N . (cid:3) Remark . The eigenvalues λ n , n ∈ N , of Λ ,r − Λ given in Proposition 3.3 decay strictly in n .Indeed, the derivative of the function y (cid:55)→ yr − y − reveals that the claim holds for n ∈ N if d ≥ n ∈ N if d = 2, because1 + log( r n + d − ) < r n + d − when 2 n + d − > , since r ∈ (0 , d = 2, we see that λ > λ if and only if 1 + log( r − ) < r − , which holds as r − >
1. Note that the observed strict decay is in contrast to the case of an inclusion with finiteconductivity, where the eigenvalues may exhibit an initial increase in magnitude before decayingwith respect to the ordering of the spherical harmonics [18, Remark 3.2].For completeness, the following lemma shows that the limit behavior lim n →∞ λ n = 0 guaranteesthat the difference map Λ C,R − Λ extends to a compact self-adjoint operator on L ( ∂ B ). However,it is actually well known that Λ C,R − Λ is smoothening because Λ C,R and Λ are pseudodifferentialoperators (modulo a smoothing operator) with the same symbol [27]. Lemma 3.5.
For each ball satisfying B ( C, R ) ⊂ B , the operator Λ C,R − Λ continuously extendsto a compact self-adjoint operator in L ( L ( ∂ B )) .Proof. The proof is completely analogous to [18, Lemma 3.3] that should be consulted for furtherdetails.For r ∈ (0 , { λ n } n ∈ N of Λ ,r − Λ are bounded and satisfy lim n →∞ λ n = 0.Combined with the corresponding eigenfunctions { f n,j } forming an orthonormal basis for L ( ∂ B )and Λ ,r − Λ being symmetric in the L ( ∂ B )-inner product, it follows that Λ ,r − Λ continuouslyextends to a compact self-adjoint operator in L ( L ( ∂ B )). PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 13
For any C ∈ B \ { } and R ∈ (0 , − | C | ), we may choose a ∈ B \ { } such that I a ( B ( C, R )) = B (0 , r ) for some r ∈ (0 , C,R − Λ inheritscompactness and self-adjointness from Λ ,r − Λ . (cid:3) Depth-dependent norm bounds
In this section we can finally present some depth-dependent norm estimates. However, it isconvenient to first introduce certain weighted L -spaces on ∂ B . Definition 4.1.
For a ∈ B \ { } and s ∈ R , we denote by L a,s ( ∂ B ) the weighted L ( ∂ B )-spaceequipped with the inner product and norm (cid:104) f, g (cid:105) a,s := (cid:104) G sa f, G sa g (cid:105) L ( ∂ B ) and (cid:107) f (cid:107) a,s := (cid:113) (cid:104) f, f (cid:105) a,s , f, g ∈ L ( ∂ B ) , respectively. Furthermore, we denote by (cid:107) · (cid:107) a,s,t the operator norm of L ( L a,s ( ∂ B ) , L a,t ( ∂ B )) for s, t ∈ R .As ˆ a / ∈ B , the function g sa is bounded away from zero and infinity on ∂ B . Hence, it is obviousthat the topologies of L ( ∂ B ) and L a,s ( ∂ B ) are the same for any s ∈ R . However, using thesenewly defined weighted norms, one obtains useful relations between the norms of concentric andnonconcentric DN maps. Theorem 4.2.
Let B ( C, R ) = I a ( B (0 , r )) for a ∈ B \ { } and r ∈ (0 , . For any s, t ∈ R , (cid:107) Λ C,R − Λ (cid:107) a,s,t = (cid:107) Λ ,r − Λ (cid:107) a, − s, − − t , or equivalently, (cid:107) Λ ,r − Λ (cid:107) a,s,t = (cid:107) Λ C,R − Λ (cid:107) a, − s, − − t . Proof.
The result follows from a direct calculation utilizing Theorem 3.2, (2.5), Lemma 2.9(i), and K a being an involution on L ( ∂ B ). To be more precise, (cid:107) Λ C,R − Λ (cid:107) a,s,t = (cid:107) G a K a (Λ ,r − Λ ) K a (cid:107) a,s,t = sup f ∈ L ( ∂ B ) \{ } (cid:107) G ta G a K a (Λ ,r − Λ ) K a f (cid:107) L ( ∂ B ) (cid:107) G sa f (cid:107) L ( ∂ B ) = sup f ∈ L ( ∂ B ) \{ } (cid:107) G a K a G − − ta (Λ ,r − Λ ) f (cid:107) L ( ∂ B ) (cid:107) G sa K a f (cid:107) L ( ∂ B ) = sup f ∈ L ( ∂ B ) \{ } (cid:107) G − − ta (Λ ,r − Λ ) f (cid:107) L ( ∂ B ) (cid:107) G a K a G − sa f (cid:107) L ( ∂ B ) = sup f ∈ L ( ∂ B ) \{ } (cid:107) G − − ta (Λ ,r − Λ ) f (cid:107) L ( ∂ B ) (cid:107) G − sa f (cid:107) L ( ∂ B ) = (cid:107) Λ ,r − Λ (cid:107) a, − s, − − t . This also proves the second part of the claim by replacing s and t with 1 − s and − − t . (cid:3) Remark . Some natural choices in Theorem 4.2 are s = 1 / t = − / s = 1 and t = − (cid:107) Λ C,R − Λ (cid:107) a, / , − / = (cid:107) Λ ,r − Λ (cid:107) a, / , − / , (cid:107) Λ C,R − Λ (cid:107) a, , − = (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) , (cid:107) Λ ,r − Λ (cid:107) a, , − = (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) . In addition, the choice s = 1 and t = − C,R − Λ ; seeAppendix B for the precise formulation of this result.Finally, it is time to present our optimal depth-dependent distinguishability bounds. Theorem 4.4.
Assume B ( C, R ) = I a ( B (0 , r )) for a ∈ B \{ } and r ∈ (0 , . Let { λ n } n ∈ N denotethe set of eigenvalues for Λ ,r − Λ , cf. Proposition 3.3. Then, − ρ ρ ≤ (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) ≤ (1 − ρ ) d (1 + ρ ) d + 4 ρ λ λ (cid:16) λ λ + 2 (cid:17) / ≤ − ρ ρ . (4.1) Furthermore, these bounds are optimal in the sense that inf r ∈ (0 , (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) = 1 − ρ ρ , sup r ∈ (0 , (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) = 1 − ρ ρ . (4.2) Proof.
Since g a is bounded from below and above by positive constants on ∂ B , it follows that G − a ( L ( ∂ B ) \ { } ) = L ( ∂ B ) \ { } . Hence, applying Theorem 4.2 with s = 1 and t = − (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) = (cid:107) Λ ,r − Λ (cid:107) a, , − = sup f ∈ L ( ∂ B ) \{ } (cid:107) G − a (Λ ,r − Λ ) f (cid:107) L ( ∂ B ) (cid:107) G a f (cid:107) L ( ∂ B ) = sup f ∈ L ( ∂ B ) \{ } (cid:107) G − a (Λ ,r − Λ ) G − a f (cid:107) L ( ∂ B ) (cid:107) f (cid:107) L ( ∂ B ) = (cid:107) G − a (Λ ,r − Λ ) G − a (cid:107) L ( L ( ∂ B )) . (4.3)In consequence, we immediately obtain the lower bound (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) ≥ (cid:107) G a − (cid:107) − L ( L ( ∂ B )) = 1 − ρ ρ , where the equality follows from Lemma 2.9(iii). We postpone proving the optimality of thisestimate till the end of this proof.Let us then consider the upper bound in (4.1). If x ∈ ∂ B , then g − a ( x ) = c + c f ( x ) , (4.4)with c := b − (1 + ρ − ) = (1 + ρ ) / (1 − ρ ), c := − b − = − ρ / (1 − ρ ), and f ( x ) := x · ˆ a .In particular, g − a is a weighted sum of spherical harmonics of degree zero and one. Hence, wededuce (cid:107) G − a (Λ ,r − Λ ) G − a (cid:107) L ( L ( ∂ B )) = sup f ∈ L ( ∂ B ) \{ } (cid:107) G − a (Λ ,r − Λ ) G − a f (cid:107) L ( ∂ B ) (cid:107) f (cid:107) L ( ∂ B ) ≥ (cid:107) g − a (Λ ,r − Λ ) g − a (cid:107) L ( ∂ B ) (cid:107) g − a (cid:107) L ( ∂ B ) = (cid:107) ( c + c f ) / ( c λ + c λ f ) (cid:107) L ( ∂ B ) (cid:107) g − a (cid:107) L ( ∂ B ) . (4.5)By symmetry, or using integration formulas for polynomials on the unit sphere [17], we have (cid:107) g − a (cid:107) L ( ∂ B ) = (cid:90) ∂ B ( c + c x · ˆ a ) d S ( x ) = c | ∂ B | . To also simplify the numerator on the right-hand side of (4.5), we write (cid:107) ( c + c f ) / ( c λ + c λ f ) (cid:107) L ( ∂ B ) = (cid:90) ∂ B ( c + c x · ˆ a )( c λ + c λ x · ˆ a ) d S ( x )= c λ | ∂ B | + c c λ ( λ + 2 λ ) (cid:90) ∂ B ( x · ˆ a ) d S ( x )= c λ | ∂ B | + c c ρ − λ ( λ + 2 λ ) (cid:90) ∂ B x d S ( x )= c λ | ∂ B | + c c ρ − λ ( λ + 2 λ ) | ∂ B | d − , (4.6) PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 15 where the odd powers of x · ˆ a under the integral vanish due to symmetry, and the third equalityfollows by renaming the line spanned by a as the first coordinate axis. However, evaluating theintegral of x over ∂ B to arrive at (4.6) requires some extra calculations:We recall first a few facts about the gamma function, namely Γ( m + 1) = m ! for m ∈ N , | ∂ B | = 2 π d/ / Γ( d ), Γ( ) = π / , Γ( ) = π / , and Legendre’s duplication formula for z ∈ C with Re( z ) >
0, Γ( z + ) = 2 − z π / Γ(2 z )Γ( z ) . (4.7)Applying (4.7) twice, we get Γ( z )2Γ( z + 1) = Γ(2 z )Γ(2 z + 1) . (4.8)Using the appropriate formula from [17] together with (4.8) finally gives (cid:90) ∂ B x d S ( x ) = 2Γ( )Γ( ) d − Γ( d + 1) = π d/ Γ( d + 1) = | ∂ B | Γ( d )2Γ( d + 1) = | ∂ B | Γ( d )Γ( d + 1) = | ∂ B | d − , which completes the proof of (4.6).Now we are finally ready to derive the upper bound in (4.1). Since λ = (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) by Remark 3.4, the formulas (4.3)–(4.6) yield (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) ≤ (cid:107) g − a (cid:107) L ( ∂ B ) λ (cid:107) ( c + c f ) / ( c λ + c λ f ) (cid:107) L ( ∂ B ) = λ c λ + c ρ − λ ( λ + 2 λ ) d − = dc d + c ρ − λ λ (cid:16) λ λ + 2 (cid:17) . Taking the square root and inserting c = (1 + ρ ) / (1 − ρ ) and c = − ρ / (1 − ρ ), we finallyarrive at (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) ≤ (1 − ρ ) d (1 + ρ ) d + 4 ρ λ λ (cid:16) λ λ + 2 (cid:17) / ≤ − ρ ρ . What remains to be shown is that the derived bounds are optimal in the sense of (4.2); in fact,we will demonstrate that the upper bound is reached when r → + and the lower bound at theopposite extreme r → − . To begin with, note that the ratio λ n /λ , n ≥
1, can be written as afunction of r ∈ (0 ,
1) and d ∈ N \ { } as λ n λ = − nr n log( r )1 − r n , d = 2 , (2 n + d − r n (1 − r d − )( d − − r n + d − ) , d ≥ . In particular, λ n /λ is an increasing function of r with lim r → + λ n /λ = 0 and lim r → − λ n /λ = 1for n ≥
1. The spectral decomposition of Λ ,r − Λ thus revealsΛ ,r − Λ (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) = P + T r , where P is the orthogonal projection in L ( L ( ∂ B )) onto constant functions, and T r is a positivesemi-definite operator with (cid:107) T r (cid:107) L ( L ( ∂ B )) = λ λ = o ( r ). To be more precise, T r = ∞ (cid:88) n =1 α n,d (cid:88) j =1 λ n λ (cid:104) · , f n,j (cid:105) L ( ∂ B ) f n,j . Moreover, both G − a P G − a and G − a T r G − a are positive semi-definite, and clearly it also holds (cid:107) G − a T r G − a (cid:107) L ( L ( ∂ B )) = o ( r ). Hence, (4.3) leads tosup r ∈ (0 , (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) = sup r ∈ (0 , (cid:107) G − a P G − a + G − a T r G − a (cid:107) − L ( L ( ∂ B )) = sup r ∈ (0 , inf (cid:107) f (cid:107) L ∂ B ) =1 (cid:0) (cid:104) G − a P G − a f, f (cid:105) L ( ∂ B ) + (cid:104) G − a T r G − a f, f (cid:105) L ( ∂ B ) (cid:1) − = inf (cid:107) f (cid:107) L ∂ B ) =1 (cid:104) G − a P G − a f, f (cid:105) − L ( ∂ B ) = (cid:107) G − a P G − a (cid:107) − L ( L ( ∂ B )) . Because of (4.4), G − a P G − a g − a = c g − a , i.e., c is the only nonzero eigenvalue of the self-adjointrank one operator G − a P G − a . In particular, c equals the operator norm of G − a P G − a , whichgives sup r ∈ (0 , (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) = c − = 1 − ρ ρ , proving one half of (4.2).The second half of (4.2) follows from a similar line of reasoning. Since λ n λ (cid:37) r → − ,it follows that T r → id − P in the strong operator topology as r → − by virtue of dominatedconvergence. Moreover, (cid:104) T r f, f (cid:105) L ( ∂ B ) is obviously nondecreasing with respect to r ∈ (0 ,
1) foreach f ∈ L ( ∂ B ). This givesinf r ∈ (0 , (cid:107) Λ ,r − Λ (cid:107) L ( L ( ∂ B )) (cid:107) Λ C,R − Λ (cid:107) L ( L ( ∂ B )) = inf r ∈ (0 , inf (cid:107) f (cid:107) L ∂ B ) =1 (cid:0) (cid:104) G − a P G − a f, f (cid:105) L ( ∂ B ) + (cid:104) G − a T r G − a f, f (cid:105) L ( ∂ B ) (cid:1) − = inf r ∈ (0 , inf (cid:107) f (cid:107) L ∂ B ) =1 (cid:0) (cid:104) P G − a f, G − a f (cid:105) L ( ∂ B ) + (cid:104) T r G − a f, G − a f (cid:105) L ( ∂ B ) (cid:1) − = inf (cid:107) f (cid:107) L ∂ B ) =1 (cid:0) (cid:104) P G − a f, G − a f (cid:105) L ( ∂ B ) + (cid:104) (id − P ) G − a f, G − a f (cid:105) L ( ∂ B ) (cid:1) − = inf (cid:107) f (cid:107) L ∂ B ) =1 (cid:104) G − a f, f (cid:105) − L ( ∂ B ) = (cid:107) G − a (cid:107) − L ( L ( ∂ B )) = 1 − ρ ρ , where the last equality follows from Lemma 2.9(iii). This completes the proof. (cid:3) The r -dependent upper bound in (4.1) satisfies C d ( ρ ) := (cid:18) (1 − ρ ) d (1 + ρ ) d + 12 ρ (cid:19) / ≤ (1 − ρ ) d (1 + ρ ) d + 4 ρ λ λ (cid:16) λ λ + 2 (cid:17) / , where C d ( ρ ) annotates the least upper bound , attained when r → − , for a given dimension d . Itis evident that C d ( ρ ) → (1 − ρ ) / (1 + ρ ) as d → ∞ , i.e., the effect that r ∈ (0 ,
1) has on the r -dependent upper bound diminishes as the dimension grows. Figure 4.1 compares the least upperbound C d ( ρ ) with the r -independent lower and upper bounds from (4.1) for d = 2 , . . . , Remark . By comparing (4.1) to the numerical studies in [18], it is observed that the upperbound − ρ ρ in Theorem 4.4 also seems to be tight for inclusions of finite conductivity in twospatial dimensions. Remark . By choosing f = g a instead of f = g − a in (4.5), one may exploit the fact that thefirst eigenfunction of Λ ,r − Λ is constant to deduce another lower bound for the operator norm PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 17
Figure 4.1.
The lower bound − ρ ρ and the upper bound − ρ ρ from (4.1) com-pared with the least upper bound C d ( ρ ) for d = 2 , . . . ,
15 as functions of ρ ∈ (0 , G − a (Λ ,r − Λ ) G − a . Using the slice integration formula in [9, Corollary A.5], this leads to adifferent (i.e. worse) upper bound (cf. (4.1)) (cid:107) g a (cid:107) L ( L ( ∂ B )) (cid:107) g a − (cid:107) L ( L ( ∂ B )) = 1 − ρ (cid:112) ρ (cid:18) ( d − V d − dV d (cid:90) − (1 − y ) ( d − / ρ − ρy d y (cid:19) / , (4.9)where V m denotes the volume of the unit ball in R m for m ∈ N . The integral in (4.9) allows anexplicit expression, which can be found by several applications of the binomial theorem, Ruffini’srule, and the connection between gamma and beta functions. The bound (4.9) improves as d increases — and tends from above towards the upper bound of Theorem 4.4. Most notably, for d = 2 it gives ( − ρ ρ ) / which is the (nonoptimal) upper bound found in [18, Theorem 3.5] fortwo spatial dimensions. Acknowledgments.
This work was supported by the Academy of Finland (decision 312124) andthe Aalto Science Institute (AScI).
Appendix A. Comparison with M¨obius transformations in two dimensions
To ease the comparison of our results and techniques with those in [18] for the two-dimensionalsetting, we briefly analyze the relation between the M¨obius transformations and inversions thatmap the unit disk onto itself. Since K a = I a for d = 2, this should be enough for convincing thereader that the two approaches are essentially the same in two dimensions.We identify B with the unit disk in the complex plane. In particular, ˆ a = a/ | a | = 1 /a for a ∈ B \ { } ⊂ C . All M¨obius transformations that map the unit disk onto itself are of the form M a ( x ) := x − aax − a ( x − a )( x − ˆ a ) | x − ˆ a | = (ˆ a − a ) ˆ a ( x − ˆ a ) | x − ˆ a | + ˆ a, a ∈ B \ { } , up to rotations. We can also rewrite I a in a similar form, I a ( x ) = ( | ˆ a | − x − ˆ a | x − ˆ a | + ˆ a = (ˆ a − a ) ˆ a ( x − ˆ a ) | x − ˆ a | + ˆ a. In particular, for ρ = | a | it holds I ρ ( x ) = M ρ ( x ). On the other hand, if a = ρ e i ζ for some ζ ∈ R ,it is easy to verify that M a ( x ) = e i ζ M ρ (e − i ζ x ) and I a ( x ) = e i ζ I ρ (e − i ζ x ) , x ∈ B . These two observations immediately lead to the identity e − i ζ I a = e − i ζ M a . Since complex con-jugation corresponds to reflection with respect to the real axis, we have proved the identity I a = Ref a ◦ M a , where Ref a denotes the reflection with respect to the line spanned by a .Obviously, | I a ( x ) | = | M a ( x ) | = | x − a || ax − | = | x − a || x − ˆ a | | ˆ a | =: r x,a , x ∈ B , and (2.1) indicates | I a ( x ) − ˆ a | = b | x − ˆ a | =: ˜ r x,a , x ∈ B . Due to the symmetry of I a ( x ) and M a ( x ) about the line span { a } = span { ˆ a } , it is easy to ge-ometrically deduce that M a ( x ) and I a ( x ) are the two intersections of the circles S (0 , r x,a ) and S (ˆ a, ˜ r x,a ). See Figure A.1 for a visualization of this geometric interpretation. In particular, theconcept of ‘depth’, characterized through ρ = | a | , is equivalent for the two transformations. -1 -0.5 0 0.5 1 1.5 2-1.5-1-0.500.51 Figure A.1.
Geometric interpretations of I a ( x ) and M a ( x ) in two dimensions. Appendix B. A representation formula for a Kelvin-transformed DN map
In this appendix, we present a diagonalization of Λ
C,R − Λ that we consider interesting inits own right, even though it is not needed when proving the main result of this work. Beforeproceeding further, we recommend reviewing the summary on spherical harmonics in Section 3and the definition of the weighted L -spaces on ∂ B introduced in Section 4.Observe that { φ n,j } := { K a f n,j } is an orthonormal basis for L a, ( ∂ B ) and { ψ n,j } := { G a K a f n,j } is such for L a, − ( ∂ B ) due to Lemma 2.9(ii) and K a being an involution. In particular, both ofthese are (non-orthonormal) bases for the standard space L ( ∂ B ). This leads to the followingrepresentation of Λ C,R − Λ . Proposition B.1.
Assume B ( C, R ) = I a ( B (0 , r )) for a ∈ B \ { } and r ∈ (0 , . Let { λ n } n ∈ N denote the set of eigenvalues for Λ ,r − Λ , cf. Proposition 3.3. Then diag( { λ n } ) , with each λ n repeated according to its multiplicity α n,d , is a matrix representation for Λ C,R − Λ with respectto the bases { φ n,j } and { ψ n,j } of L ( ∂ B ) , that is, (cid:104) (Λ C,R − Λ ) φ m,j (cid:48) , ψ n,j (cid:105) a, − = λ m δ m,n δ j (cid:48) ,j . In particular, (Λ C,R − Λ ) f = ∞ (cid:88) n =0 α n,d (cid:88) j =1 λ n (cid:104) f, ψ n,j (cid:105) L ( ∂ B ) ψ n,j (B.1) for any f ∈ L ( ∂ B ) . PTIMAL DISTINGUISHABILITY BOUNDS FOR EIT 19
Proof. As K a is an involution, ( λ m , φ m,j (cid:48) ) is an eigenpair of K a (Λ ,r − Λ ) K a . Hence, due toTheorem 3.2 and Lemma 2.9(ii), (cid:104) (Λ C,R − Λ ) φ m,j (cid:48) , ψ n,j (cid:105) a, − = (cid:104) G a K a (Λ ,r − Λ ) K a φ m,j (cid:48) , G a K a f n,j (cid:105) L ( ∂ B ) = λ m (cid:104) G a K a f m,j (cid:48) , K a f n,j (cid:105) L ( ∂ B ) = λ m (cid:104) f m,j (cid:48) , f n,j (cid:105) L ( ∂ B ) , which proves the first part of the claim as { ψ n,j } is an orthonormal basis for L a, − ( ∂ B ) and { f n,j } is such for L ( ∂ B ).To show (B.1), we simply write up expansions for (Λ C,R − Λ ) f and f in terms of { ψ n,j } and { φ m,j (cid:48) } , respectively, and apply the above result,(Λ C,R − Λ ) f = ∞ (cid:88) n =0 α n,d (cid:88) j =1 (cid:104) (Λ C,R − Λ ) f, ψ n,j (cid:105) a, − ψ n,j = ∞ (cid:88) n =0 ∞ (cid:88) m =0 α n,d (cid:88) j =1 α m,d (cid:88) j (cid:48) =1 (cid:104) (Λ C,R − Λ ) φ m,j (cid:48) , ψ n,j (cid:105) a, − (cid:104) f, φ m,j (cid:48) (cid:105) a, ψ n,j = ∞ (cid:88) n =0 α n,d (cid:88) j =1 λ n (cid:104) f, φ n,j (cid:105) a, ψ n,j , which completes the proof as (cid:104) f, φ n,j (cid:105) a, = (cid:104) f, ψ n,j (cid:105) L ( ∂ B ) . (cid:3) Remark
B.2 . It is worth noting that (B.1) is not a spectral decomposition of Λ
C,R − Λ since { ψ n,j } in Proposition B.1 is not an orthonormal basis for L ( ∂ B ). In fact, as seen in the proof, itprovides a spectral decomposition of G − a (Λ C,R − Λ ) with eigenpairs ( λ n , φ n,j ). References [1] R. A. Adams and J. J. F. Fournier.
Sobolev spaces , volume 140 of
Pure and Applied Mathematics (Amsterdam) .Elsevier/Academic Press, Amsterdam, second edition, 2003.[2] G. Alessandrini. Stable determination of conductivity by boundary measurements.
Appl. Anal. , 27:153–172,1988.[3] G. Alessandrini and R. Gaburro. Determining conductivity with special anisotropy by boundary measurements.
SIAM J. Math. Anal. , 33(1):153–171, 2001.[4] G. Alessandrini and R. Gaburro. The local Calder´on problem and the determination at the boundary of theconductivity.
Comm. PDE , 34(7-9):918–936, 2009.[5] G. Alessandrini and A. Scapin. Depth dependent resolution in electrical impedance tomography.
J. InverseIll-Posed Probl. , 25(3), 2017.[6] H. Ammari, J. Garnier, and K. Sølna. Partial data resolving power of conductivity imaging from boundarymeasurements.
SIAM J. Math. Anal. , 45(3):1704–1722, 2013.[7] H. Ammari and H. Kang.
Polarization and Moment Tensors: with Applications to Inverse Problems andEffective Medium Theory , volume 162 of
Applied Mathematical Sciences . Springer-Verlag, New York, 2007.[8] D. H. Armitage and S. J. Gardiner.
Classical Potential Theory . Springer London, 2001.[9] S. Axler, P. Bourdon, and W. Ramey.
Harmonic Function Theory . Springer New York, 2001.[10] K. Bogdan and T. ˙Zak. On Kelvin Transformation.
J. Theor. Probab. , 19(1):89–120, 2006.[11] D. Borman, D. B. Ingham, B. T. Johansson, and D. Lesnic. The method of fundamental solutions for detectionof cavities in EIT.
J. Integral Equ. Appl. , 21(3):383–406, 2009.[12] R. M. Brown. Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwiseresult.
J. Inverse Ill-Posed Probl. , 9(6):567–574, 2001.[13] M. Cheney and D. Isaacson. Distinguishability in impedance imaging.
IEEE T. Bio-Med. Eng. , 39(8):852–860,1992.[14] R. Dautray and J.-L. Lions.
Mathematical Analysis and Numerical Methods for Science and Technology ,volume 2. Springer, 1988.[15] C. Efthimiou and C. Frye.
Spherical Harmonics in p Dimensions . World Scientific, 2014.[16] K. Erhard and R. Potthast. The point source method for reconstructing an inclusion from boundary measure-ments in electrical impedance tomography and acoustic scattering.
Inverse Problems , 19:1139–1157, 2003.[17] G. B. Folland. How to integrate a polynomial over a sphere.
Amer. Math. Monthly , 108(5):446–448, 2001.[18] H. Garde and K. Knudsen. Distinguishability revisited: Depth dependent bounds on reconstruction quality inelectrical impedance tomography.
SIAM J. Appl. Math. , 77(2):697–720, 2017. [19] D. Gurarie.
Symmetries and Laplacians , volume 174 of
North-Holland Mathematics Studies . North-HollandPublishing Co., Amsterdam, 1992. Introduction to harmonic analysis, group representations and applications.[20] M. Hanke, L. Harhanen, N. Hyv¨onen, and E. Schweickert. Convex source support in three dimensions.
BITNumer. Math. , 52(1):45–63, 2011.[21] L. H¨ormander.
The analysis of linear partial differential operators I . Classics in Mathematics. Springer-Verlag,Berlin, 2003.[22] D. Isaacson. Distinguishability of conductivities by electric current computed tomography.
IEEE T. Med.Imaging , 5(2):91–95, 1986.[23] H. Kang and K. Yun. Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator.
SIAM J. Math. Anal. , 34:719–735, 2002.[24] W. Thomson (Lord Kelvin). Extraits de deux lettres adress´ees `a M. Liouville.
J. Math. Pures Appl. , 12:256–264,1847.[25] R. Kress. Conformal mapping and impedance tomography.
J. Phys.: Conf. Ser. , 290:012009, 2011.[26] R. Kress. Inverse problems and conformal mapping.
Complex Var. Elliptic , 57(2-4):301–316, 2012.[27] J. Lee and G. Uhlmann. Determining anisotropic real-analytic conductivities by boundary measurements.
Comm. Pure Appl. Math. , 42(2):1097–1112, 1989.[28] J.-L. Lions and E. Magenes.
Non-homogeneous boundary value problems and applications, Vol. 1 . Springer–Verlag, New York–Heidelberg, 1972.[29] N. Mandache. Exponential instability in an inverse problem for the Schr¨odinger equation.
Inverse Problems ,17(5):1435–1444, 2001.[30] K. Michalik and M. Ryznar. Kelvin transform for α -harmonic functions in regular domains. DemonstratioMath. , 45(2), 2012.[31] S. Nagayasu, G. Uhlmann, and J.-N. Wang. A depth-dependent stability estimate in electrical impedancetomography.
Inverse Problems , 25(7):075001, 2009.[32] G. Nakamura and K. Tanuma. Direct determination of the derivatives of conductivity at the boundary fromthe localized Dirichlet to Neumann map.
Comm. Korean Math. Soc. , 16:415–425, 2001.[33] G. Nakamura and K. Tanuma. Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map.
Inverse Problems , 17:405–419, 2001.[34] G. Nakamura and K. Tanuma. Formulas for reconstructing conductivity and its normal derivative at theboundary from the localized Dirichlet to Neumann map. In
Recent development in theories & numerics , pages192–201. World Sci. Publ., River Edge, NJ, 2003.[35] J. K. Seo, O. Kwon, and S. Kim. Location search techniques for a grounded conductor.
SIAM J. Appl. Math. ,62(4):1383–1393, 2002.[36] J. Sylvester and G. Uhlmann. Inverse boundary value problems at the boundary — continuous dependence.
Comm. Pure Appl. Math. , 41:197–219, 1988.[37] J. Wermer.
Potential Theory . Springer Berlin Heidelberg, 1974.(H. Garde)
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg,Denmark.
E-mail address : [email protected] (N. Hyv¨onen) Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100,02150 Espoo, Finland.
E-mail address ::