An inverse conductivity problem in multifrequency electric impedance tomography
aa r X i v : . [ m a t h . A P ] M a r An inverse conductivity problem inmultifrequency electric impedance tomography
Dedicated to Masahiro Yamamoto for his sixtieth birthday
Jin Cheng , Mourad Choulli , and Shuai Lu Shanghai Key Laboratory for Contemporary Applied Mathematics,Key Laboratory of Mathematics for Nonlinear Sciences andSchool of Mathematical Science, Fudan University, 200433 Shanghai, [email protected], [email protected] Université de Lorraine, 34 cours Léopold, 54052 Nancy cedex, France [email protected]
Abstract.
We deal with the problem of determining the shape of aninclusion embedded in a homogenous background medium. The multifre-quency electrical impedance tomography is used to image the inclusion.For different frequencies, a current is injected at the boundary and theresulting potential is measured. It turns out that the potential solvesan elliptic equation in divergence form with discontinuous leading coef-ficient. For this inverse problem we aim to establish a logarithmic typestability estimate. The key point in our analysis consists in reducing theoriginal problem to that of determining an unknown part of the innerboundary from a single boundary measurment. The stability estimateis then used to prove uniqueness results. We also provide an expansionof the solution of the BVP under consideration in the eigenfunction ba-sis of the Neumann-Poincaré operator associated to the Neumann-Greenfunction.
We firstly proceed with the mathematical formulation of the problemunder consideration. In order to specify the BVP satisfied by the poten-tial, we consider Ω and D two Lipschitz domains of R n , n ě
2, so that D Ť Ω . Fix k ą k P p , k u , the function a D on Ω by a D p k q “ k ` p k ´ k q χ D . Here χ D is the characteristic function of the inclusion D , k is the con-ductivity of the background medium and k is the conductivity of theinclusion.It is well known that in the present context the potential solves the BVP " div p a D p k q ∇ u q “ Ω,k B ν u “ f on B Ω, (1) Jin Cheng, Mourad Choulli and Shuai Luwhere B ν is the derivative along the unit normal vector field ν on B Ω pointing outward Ω .Let V “ " u P H p Ω q ; ż B Ω u p x q dσ p x q “ * . Note that V is a Hilbert space when it is endowed with the scalar product a p u, v q “ ż Ω ∇ u ¨ ∇ v, u, v P V. We leave to the reader to check that the norm induced by this scalarproduct is equivalent to the H p Ω q -norm on V .Pick f P L pB D q and k P p , k u . Then it is not hard to see that,according to Lax-Milgram’s lemma, the BVP (1) possesses a unique vari-ational solution u D p k q P V . That is u D p k q is the unique element of V satisfying ż Ω a D p k q ∇ u D p k q ¨ ∇ vdx “ ż B Ω fvdσ p x q , v P V. (2)Recall that the trace operator t : w P C p Ω q ÞÑ u |B Ω P C pB Ω q isextended to a bounded operator, still denoted by t , from H p Ω q into L pB Ω q .Since we will consider conductivities varying with the frequency ω , weintroduce the map k : ω P p ,
8q ÞÑ k p w q P p , k u . We assume inthe sequel the condition lim ω Ñ8 k p ω q “ 8 . Let J be a given subset of p , . In the present work, we are mainlyinterested in determining the unknown subdomain D from the boundarymeasurements t u D p k p ω qq , ω P J . Prior to state our main result, we introduce some notations and defini-tions. Let R n ` “ t x “ p x , x n q P R n ; x n ą u ,Q “ t x “ p x , x n q P R n ; | x | ă | x n | ă u ,Q ` “ Q X R n ` ,Q “ t x “ p x , x n q P R n ; | x | ă x n “ u . Fix 0 ă α ă
1. We say that the bounded domain U of R n is of class C ,α with parameters ̺ ą ℵ ą x P B U there exists a bijectivemap φ : Q Ñ B : “ B R n ´ p x, ̺ q , satisfying φ P C ,α p Q q , φ ´ P C ,α p B q and } φ } C ,α p Q q ` } φ ´ } C ,α p B q ď ℵ , so that φ p Q ` q “ U X B and φ p Q q “ B X B U . If we substitute in this definition C ,α by C , then we obtain the defi-nition of a Lipschitz domain with parameters ̺ and ℵ .nverse conductivity problem 3Let D p ̺, ℵ q denote the set of subdomains D so that D Ť Ω , and Ω z D is of class C ,α with parameters ̺ ą ℵ ą D Ť Ω Ť Ω and δ ą
0. Consider then D p ̺, ℵ , δ q the set ofsubdomains D Ť Ω Ť Ω satisfying D Ť D , Ω z Ω and Ω z D aredomains of class C ,α with parameters ̺ ą ℵ ą p D, B Ω q ě δ, and the following assumption holds B p x , d p x , D qq Ă Ω z D, x P Ω z D. (3)Henceforward, we make the assumption that Ω , D and δ are chosen insuch a way that D p ̺, ℵ , δ q is nonempty.Define also C p ̺, ℵ , ̺ , ℵ , δ q as the set of couples p D , D q so that D j P D p ̺, ℵ , δ q , j “ ,
2, and Ω z D Y D is a domain of class C , withparameters ̺ ą ℵ ą d Ug on a bounded domain U of R n by d Ug p x, y q “ inf t ℓ p ψ q ; ψ : r , s Ñ U is Lipschitz path joining x to y u , where ℓ p ψ q “ ż ˇˇˇ ψ p t q ˇˇˇ dt is the length of ψ .Note that, according to Rademacher’s theorem, any Lipschitz continuousfunction ψ : r , s Ñ D is almost everywhere differentiable with ˇˇˇ ψ p t q ˇˇˇ ď L a.e. t P r , s , where L is the Lipschitz constant of ψ . Therefore, ℓ p ψ q is well defined.From [14, Lemma 3.3], we know that d Ug P L p U ˆ U q whenever U is ofclass C , . Fix then b ą C p ̺, ℵ , ̺ , ℵ , δ, b q as the subset ofthe couples p D , D q P C p ̺, ℵ , ̺ , ℵ , δ q so that d Ω z D Y D g ď b . Recall that the Hausdorff distance for compact subsets of R n is given by d H p D , D q “ max ˜ max x P D d p x, D q , max x P D d p x, D q ¸ and, following [1], we define the modified distance d m by d m p D , D q “ max ˆ max x PB D d p x, D q , max x PB D d p x, D q ˙ . As it is pointed out in [1], d m is not a distance. To see it, consider D “ B p , qz B p , { q and D “ B p , q . In that case simple computationsshow that 0 “ d m p D , D q ă d H p D , D q “ . In this example D Ă D , but D Ć D . However, we can enlarge slightly D in order to satisfy D Ă D . Indeed, if D “ B p , { q then D Ă D and 14 “ d m p D , D q ă d H p D , D q “ . Jin Cheng, Mourad Choulli and Shuai LuIn dimension two, take D “ tp r, θ q ; η ă θ ă π ´ η, { ă r ă u ,0 ă η ă π . Smoothing the angles of D we get a C simply connecteddomain so that, if again D “ B p , { q ,14 ď d m p D , D q ă d H p D , D q “ . In all these examples a small translation in the direction of one of thecoordinates axes for instance enables us to construct examples with D z D ‰ H , D z D ‰ H and d m p D , D q ă d H p D , D q .What these examples show is that it is difficult to give sufficient geometricconditions ensuring that the following equality holds d m p D , D q “ d H p D , D q . (4)It is obvious to check that (4) is satisfied whenever D and D are ballsor ellipses, but not only.The subset of couples p D , D q P C p ̺, ℵ , ̺ , ℵ , δ, b q satisfying (4) willdenoted C p ̺, ℵ , ̺ , ℵ , δ, b q .We fix in all of this text f P C ,α pB Ω q non negative and non identicallyequal to zero.We aim in this paper to establish the following result. Theorem 1.
Let d “ p ̺, ℵ , ̺ , ℵ , δ, b q . There exist two constant C “ C p d q and ă Λ ˚ “ Λ ˚ p d q ă e ´ e so that, for any p D , D q P C p d q with B D X B D ‰ H and any sequence p k j q P p , satisfying k j Ñ 8 , wehave d H p D , D q ď C p ln ln | ln Λ |q ´ , provided that ă Λ : “ sup j } t u D p k j q ´ t u D p k j q} L pB Ω q ă Λ ˚ . As an immediate consequence of this theorem we have the followingcorollary.
Corollary 1.
Let d “ p ̺, ℵ , ̺ , ℵ , δ, b q . There exist two constant C “ C p d q and ă Λ ˚ “ Λ ˚ p d q ă e ´ e so that, for any p D , D q P C p d q with B D X B D ‰ H and any sequence of frequencies p ω j q P p , satisfying ω j Ñ 8 , we have d H p D , D q ď C p ln ln | ln Λ |q ´ , provided that ă Λ : “ sup j } t u D p k p ω j qq ´ t u D p k p ω j qq} L pB Ω q ă Λ ˚ . The inverse problem we consider in the present paper was already stud-ied, in the case of smooth star-shaped subdomains with respect to somefixed point, by Ammari and Triki [2]. For a similar problem with a singleboundary measurement we refer to [9] where a Lipschitz stability esti-mate was established for a non monotone one-parameter family of un-known subdomains. The literature on the determination of an unknownnverse conductivity problem 5part of the boundary is rich, but we just quote here [1,8,16] (see alsothe references therein). We note that another multifrequency mediumproblem considers single observation for varying multiple wavenumbers[4,6]. To have an overview, we recommend a recent review paper [5]which nicely summarizes the theoretical and numerical results in multi-frequency inverse medium and source problems for acoustic Helmholtzequations and Maxwell equations.The key step in our proof consists in reducing the original inverse problemto the one of determining an unknown part of the inner boundary froma single boundary measurement. For this last problem we provided inSection 2 a logarithmic stability estimate. This intermediate result isthen used in Section 3 to prove Theorem 1. Section 4 contains uniquenessresults obtained from Theorem 1.The idea of reducing the original problem to the one of recovering theshape of an unknown inner part of the boundary was borrowed from thepaper by Ammari and Triki [2].Our analysis combines both ideas from [1,2] together with some recentresults related to quantifying the uniqueness of continuation in varioussituations [11,12,13].This paper is completed by a last section in which we give an expansionof the solution of the BVP (1) in the eigenfunction basis of the Neumann-Poincaré operator (shortened to NP operator in the rest of this paper)related to the Neumann-Green function.
Pick D P D p ̺, ℵ q and let ˜ u D P H p Ω q satisfying ˜ u D “ D and it isthe variational solution of the BVP $&% ∆ ˜ u “ Ω z D, ˜ u “ B D, B ν ˜ u “ f on B Ω. As ˜ u D P C p Ω z D q by the usual interior regularity of harmonic functions,we can apply, for an arbitrary ω , D Ť ω Ť Ω , both the Hölder regularitytheorem for Dirichlet BVP in ω z D ([17, Theorem 6.14 in page 107]) andthe Hölder regularity theorem for Neumann BVP in Ω z ω ([17, Theorem6.31 in page 128]). Therefore ˜ u D P C ,α p Ω z D q .Also, note that, according to the maximum principle and Hopf’s maxi-mum principle, ˜ u D ě u D “ ˜ u D ´ |B Ω | ż B Ω ˜ u D dσ p x q . Lemma 1.
Let D , D P D p ̺, ℵ q and set m j “ |B Ω | ż B Ω ˜ u D dσ p x q . If B D X B D ‰ H then | m ´ m | ď } ˜ u D ´ ˜ u D } L pBp D Y D qq . (5) Jin Cheng, Mourad Choulli and Shuai Lu Proof. As B D X B D Ă Bp D Y D q and˜ u D ´ ˜ u D “ m ´ m on B D X B D , the expected inequality follows easily. [\ Under the assumptions and notations of Lemma 1, we have from (5) } ˜ u D ´ ˜ u D } L pBp D Y D qq ď } ˜ u D ´ ˜ u D } L pBp D Y D qq . (6)Let d “ p ̺, ℵ , ̺ , ℵ , b q . Then, checking carefully the result in [11, Sec-tion 2.4] we find that there exist three constants C “ C p d q ą c “ c p d q ą β “ β p d q so that, for any 0 ă ǫ ă D , D P C p d q ,we have C } ˜ u D ´ ˜ u D } L pBp D Y D qq (7) ď ǫ β } ˜ u D ´ ˜ u D } C ,α p Ω zp D Y D qq ` e c { ǫ } ˜ u D ´ ˜ u D } H pB Ω q . Let us note that [11, Proposition 2.30] holds for an arbitrary boundedLipschitz domain (we refer to [7] or [12] for a detailed proof of thisimprovement).
Proposition 1.
Set d “ p ̺, ℵ , δ q . There exists C “ C p d q so that, forany D P D p d q , we have } ˜ u D } C ,α p Ω z D q ď C. Proof.
Let 0 ď g P C ,α pB Ω q and denote by w P C ,α p Ω z D q the solutionof the BVP $&% ∆w “ Ω,w “ g on B D , B ν w “ f on B Ω. The existence of such function is guaranteed by the usual elliptic regu-larity for both Dirichlet and Neumann BVP’s for the Laplace operator.Similar argument will be discussed hereafter.In light of the fact that r u D ´ w is harmonic in Ω z D , r u D ´ w “ ´ w ď B D and B ν p r u D ´ w q “ B Ω and, we find by applying the twicethe maximum principle and Hopf’s lemma that p ďq r u D ď w in Ω z D .Whence } ˜ u D } C p Ω z D q ď } w } C p Ω z D q . Let µ “ inf p δ, dist p Ω, Ω qq{ U “ t x P Ω z D ; dist p x, Ω q ď µ u . By [10, Lemma 3.11 in page 118], we have } ˜ u D } C ,α p U q ď µ ´ C p n q} ˜ u D } C p Ω z D q ď µ ´ C p n q} w } C p Ω z D q . (8)Let u P C ,α p Ω z D q be the solution of the BVP " ∆u “ Ω z D,u “ ˜ u D on B Ω Y B D, nverse conductivity problem 7and u P C ,α p Ω z Ω q be the solution of the BVP " ∆u “ Ω z Ω , B ν u “ B ν ˜ u D on B Ω Y B Ω . A careful examination of the classical Schauder estimates in [17, Chapter6], for both Dirichlet and Neumann problems, we see that the differentconstants only depend on C ,α parameters of the domain. Therefore } u } C ,α p Ω z D q ď C p d q} u } C ,α pB Ω YB D q , (9) } u } C ,α p Ω z Ω q ď C p d q}B ν u } C ,α pB Ω YB Ω q . (10)We put (8) in (9) and (10) in order to get } ˜ u D } C ,α p Ω z D q ď C. The expected inequality follows then by noting that } ˜ u D } C ,α p Ω z D q ď } ˜ u D } C ,α p Ω z D q . The proof is then complete. [\ Let w P H p Ω z Ω q . We have from the usual interpolation inequalitiesand trace theorems, with d “ p ̺, ℵ q , } w } H pB Ω q ď C p d q} w } { L pB Ω q } w } { H { pB Ω q ď C p d q C Ω z Ω } w } { L pB Ω q } w } { H p Ω z Ω q , the constant C Ω z Ω only depends on Ω z Ω .In light of this inequality, Proposition 1 and inequalities (6) and (7), wecan state the following result Theorem 2.
Set d “ p ̺, ℵ , ̺ , ℵ , δ, b q . There exist three constants C “ C p d q ą , c “ c p d q ą and β “ β p d q so that, for any ă ǫ ă and p D , D q P C p d q with B D X B D ‰ H , we have C } ˜ u D ´ ˜ u D } L pBp D Y D qq ď ǫ β ` e c { ǫ } ˜ u D ´ ˜ u D } { L pB Ω q . Theorem 3.
Let D P D p ̺, ℵ q . For any sequence p k j q in p , suchthat lim j Ñ8 k j “ 8 , we have lim j Ñ8 } t u D p k j q ´ t ˜ u D } L pB Ω q “ . In particular, p t u D p k j qq P ℓ p L pB Ω qq .Proof. Let W “ " w P H p Ω z D q ; ż B Ω w p x q dσ p x q “ * .W is a closed subspace of H p Ω z D q and the norm } ∇ w } L p Ω z D q is equiva-lent on W to the norm } w } H p Ω z D q . Moreover, for any w P V , w | Ω z D P W . Jin Cheng, Mourad Choulli and Shuai LuClearly, v D p k q “ u D p k q ´ ˜ u D is the variational solution of the BVP " div pp k ` p k ´ k q χ D q ∇ v q “ ´ div pp k ` p k ´ k q χ D q ∇ ˜ u D q in Ω, B ν v “ B Ω. As ż B Ω u D p k q dσ p x q “ ż B Ω ˜ u D dσ p x q “ , we have v D p k q P V .By Green’s formula, for any w P V , we get ´ k ż Ω z D ∇ v D p k q ¨ ∇ wdx ´ k ż D ∇ v D p k q ¨ ∇ wdx (11) “ k ż Ω z D ∇ ˜ u D ¨ ∇ wdx ´ ż B Ω fwdσ p x q . Take in this identity w “ v D p k q and make use Cauchy-Schwarz’s identityin order to obtain k } ∇ v D p k q} L p D q ` k } ∇ v D p k q} L p Ω z D q (12) ď k } ∇ v D p k q} L p Ω z D q } ∇ ˜ u } L p Ω z D q ` } f } L pB Ω q } t v D p k q} L pB Ω q . But the trace operator w P H p Ω z D q ÞÑ w |B Ω P L pB Ω q is bounded.Hence there exists a constant C ą
0, depending on Ω and D so that } w } L pB Ω q ď C } ∇ w } L p Ω z D q , w P W. Therefore } ∇ v D p k q} L p Ω z D q ď } ∇ ˜ u } L p Ω z D q ` C k ´ } f } L pB Ω q : “ M. (13)This in (12) entails } ∇ v D p k q} L p D q ď M ? k . (14)Pick p k j q a sequence in p , k u so that k j Ñ 8 as j Ñ 8 . Under thetemporary notation v j “ v D p k j q , we have from (13) and (14) that v j isbounded in H p Ω q and ∇ v j Ñ L p D q when j Ñ 8 . Subtracting ifnecessary a subsequence, we may assume that v j Ñ v P H p Ω q , stronglyin H { p Ω q and weakly in H p Ω q . In consequence, ∇ v “ D .Define W “ t w P W ; w “ B D u . An extension by 0 of a function in W enables us to consider W as aclosed subspace of V .It is not hard to see that (11) yields ´ k ż Ω z D ∇ v j ¨ ∇ wdx “ k ż Ω z D ∇ ˜ u D ¨ ∇ wdx ´ ż B Ω fwdσ p x q , w P W . Passing to the limit when j Ñ 8 , we obtain ´ k ż Ω z D ∇ v ¨ ∇ wdx “ k ż Ω z D ∇ ˜ u D ¨ ∇ wdx ´ ż B Ω fwdσ p x q , w P W . nverse conductivity problem 9Taking in this identity an arbitrary w P C p Ω z D q , we find that v isharmonic in Ω z D . Applying then generalized Green’s function to deducethat B ν v “
0. On the other hand we know that v j converges strongly to v in H { pB Ω q (thank to the continuity of the trace operator). Therefore v P V and hence v is identically equal to zero, implying in particularthat v j converges strongly to 0 in L pB Ω q . [\ We get by combining Theorems 2 and 3 the following result.
Theorem 4. If d “ p ̺, ℵ , ̺ , ℵ , δ, b q , then there exist three constants C “ C p d q ą , c “ c p d q ą and β “ β p d q so that, for any ă ǫ ă and p D , D q P C p d q with B D X B D ‰ H , and for any sequence p k j q in p , such that lim j Ñ8 k j “ 8 , we have C } ˜ u D ´ ˜ u D } L pBp D Y D qq ď ǫ β ` e c { ǫ « sup j } u D p k j q ´ u D p k j q} L pB Ω q ff { . If U is a bounded domain of R n , we set U δ “ t x P U ; dist p x, B U q ą δ u , δ ą . Define then κ p U q “ sup t δ ą U δ ‰ Hu . We endow C ,α p U q with norm | u | ,α : “ } ∇ u } L p U q n ` r ∇ u s α , where r ∇ u s α “ sup " | ∇ u p x q ´ ∇ u p y q|| x ´ y | α ; x, y P U , x ‰ y * . For 0 ă α ă η ą M ą
0, define S p U q “ S p U , α, η, M q by S p U q “ t u P C ,α p U q ; | u | ,α ď M, } ∇ u } L p Γ q ě η and ∆u “ u . Denote by U “ U p ̺, ℵ , h , b q the set of bounded domains U of R n thatare of class C , , with parameters ̺ ą ℵ ą
0, and satisfy κ p U q ě h ą d U g ď b . Theorem 5.
Let ˜ d “ p ̺, ℵ , h , b , α, η, M q .There exist two constants c “ c p ˜ d q ą so that, for any U P U , ă δ ă h , x P U δ and u P S p U q , wehave e ´ e c { δ ď } u } L p B p x ,δ qq . Proof.
We mimic the proof of [13, Theorem 2.1] in which we substitutethe three-ball inequality of u by a three-ball inequality for ∇ u . If oneexamines carefully the proof of [13, Theorem 2.1], he can see that thedifferent constants do not depend on U P U but only on p ̺, ℵ , h , b q . Weobtain e ´ e c { δ ď } ∇ u } L p B p x ,δ qq . This and Caccioppoli’s inequality yield the expected inequality. [\ Corollary 2.
Set d “ p ̺, ℵ , δ q . Then there exist two constants c “ c p d q ą so that, for any D P D p d q , x P Ω z D , we have e ´ e c { d ď } ˜ u D } L p B p x ,d { qq , where d “ d p x , D q .Proof (of Theorem 1). Set ˜ u j “ ˜ u D j , j “ , D z D | ˜ u | “ max D z D | ˜ u ´ ˜ u | “ max Bp D z D q | ˜ u ´ ˜ u | ď max Bp D Y D q | ˜ u ´ ˜ u | . (15)Pick x P B D so thatmax x PB D d p x, D q “ d p x , D q : “ d . Noting that ˜ u ě
0, we apply Harnack’s inequality (see [17, Proof ofTheorem 2.5 in page 16]) in order to getmax B p x ,d { q ˜ u ď n min B p x ,d { q ˜ u ď n max D z D | ˜ u | . Whence ż B p x ,d { q ˜ u p x q dx ď ˇˇ S n ´ ˇˇ ´ d { ¯ n ˜ max D z D | ˜ u | ¸ . This and the estimate in Corollary 2 yield, by changing if necessary theconstant c , e ´ e c { d ď max D z D | ˜ u | . (16)We have similarly e ´ e c { d ď max D z D | ˜ u | ď max Bp D Y D q | ˜ u ´ ˜ u | , (17)where d : “ max x PB D d p x, D q . In light of (15), (16) and (17) entail e ´ e c { d ď max Bp D Y D q | ˜ u ´ ˜ u | . (18)Here d “ max p d , d q “ d H p D , D q .Set Λ : “ sup j } u D p k j q ´ u D p k j q} L pB Ω q . nverse conductivity problem 11Then estimates in Theorem 4 in (18) give e ´ e c { d ď ǫ β ` e c { ǫ Λ { , ă ǫ ă . (19)Since the function ǫ P p , q Ñ ǫ β e ´ cβ is increasing, if Λ ă e ´ c : “ Λ then we can take ǫ in (19) so that ǫ β e ´ cβ “ Λ { . A straightforwardcomputation shows that ǫ ď p c ` β q| ln Λ | ´ . Modifying if necessary c in (19), we obtain e ´ e c { d ď | ln Λ | ´ . (20)Therefore, if Λ ă min p Λ , e ´ e q then (20) implies d ď c p ln ln | ln Λ |q ´ . The proof is then complete. [\ We first observe that the following uniqueness result is an immediateconsequence of Theorem 1.
Corollary 3.
Assume that Ω is of class C ,α . Let D Ť D j Ť Ω of class C ,α , j “ , , so that B D XB D ‰ H and d m p D , D q “ d H p D , D q . If t u D p k ℓ q “ t u D p k ℓ q for some sequence p k ℓ q in p , k u with k ℓ Ñ 8 when ℓ Ñ 8 , then D “ D . It is worth mentioning that this uniqueness result together with the ana-lyticity of the mapping k ÞÑ t u D p k q enable us to establish an uniquenessresult when k varies in a subset of p , k u possessing an accumulationpoint. Prior to that, we prove the following Lemma. Lemma 2.
Assume that Ω and D are two Lipschitz domains of R n sothat D Ť Ω . Then the mapping k P p , k u ÞÑ t u D p k q P L pB Ω q is real analytic.Proof. Let k P p , k u and | ℓ | ď k { k ` ℓ p , k u . Since u D p k ` ℓ q is the solution of the variational problem ż Ω a D p k ` ℓ q ∇ u D p k ` ℓ q ¨ ∇ vdx “ ż B Ω fvdσ p x q , v P V, we have min p k { , k q} u D p k ` ℓ q} V ď } t }} f } L pB Ω q . (21)Here } t } denotes the norm of t as bounded operator acting from V into L pB Ω q .Next, let w D p k q P V be the solution of the variational problem ż Ω a D p k q ∇ w D p k q ¨ ∇ v “ ´ ż Ω χ D ∇ u D p k q ¨ ∇ v, v P V. ż Ω a D p k q ∇ r u D p k ` ℓ q ´ u D p k q ´ ℓw D p k qs ¨ ∇ v “ ℓ ż Ω χ D ∇ r u D p k q ´ u D p k ` ℓ qs ¨ ∇ v, v P V. Hencemin p k, k q} u D p k ` ℓ q ´ u D p k q ´ ℓw D p k q} V ď | ℓ |} u D p k q ´ u D p k ` ℓ q} V . (22)But ż Ω a D p k q ∇ r u D p k ` ℓ q ´ u D p k qs¨ ∇ v “ ´ ℓ ż Ω χ D ∇ u D p k ` ℓ q¨ ∇ v, v P V. This entailsmin p k, k q} u D p k ` ℓ q ´ u D p k q} V ď | ℓ |} u D p k ` ℓ q} V , which, combined with (21), yields } u D p k ` ℓ q ´ u D p k q} V ď C p k q| ℓ | , (23)with C p k q “ r min p k { , k qs ´ } t }} f } L pB Ω q .Putting (23) into (22), we get } u D p k ` ℓ q ´ u D p k q ´ ℓw D p k q} V ď C p k q ℓ , where C p k q “ r min p k, k qs ´ C p k q .In other words, we proved that the mapping k P p , k u ÞÑ u D p k q P V is differentiable and its derivative u D p k q is the solution of the variationalproblem ż Ω a D p k q ∇ u D p k q ¨ ∇ v “ ´ ż Ω χ D ∇ u D p k q ¨ ∇ v, v P V. Therefore, we have the a priori estimate } u D p k q} V ď Cφ p k q , (24)where we set C “ } t }} f } L pB Ω q and φ p k q “ r min p k, k qs ´ .Now an induction argument in j shows that k P p , k u ÞÑ u D p k q P V is j -times differentiable and u p j q D p k q is the solution of the variationalproblem ż Ω a D p k q ∇ u p j q D p k q ¨ ∇ v “ ´ j ż Ω χ D ∇ u p j ´ q D p k q ¨ ∇ v, v P V. In light of this identity and (24) we show, again by using an inductionargument in j , that } u p j q D p k q} V ď Cj ! φ p k q j ` . nverse conductivity problem 13Consequently, if | k ´ ℓ | ă φ p k q ´ “ min p k, k q , the series ÿ j ě j ! } u p j q D p k q} V p k ´ ℓ q j converges and hence, thank to the completeness of V , the series ÿ j ě j ! u p j q D p k qp k ´ ℓ q j converges in V . That is we proved that k P p , k u ÞÑ u D p k q P V is real analytic and, since t P B p V, L pB Ω qq , we conclude that k Pp , k u ÞÑ t u D p k q P L pB Ω q is also real analytic. [\ In light of Corollary 3 and the fact that a real analytic function F : p , k u Ñ V cannot vanish in a subset of p , k u possessingan accumulation point in p , k u without being identically equal tozero, we get the following uniqueness result. Corollary 4.
Assume that Ω is of class C ,α . Let D Ť D j Ť Ω of class C ,α , j “ , so that B D X B D ‰ H and d m p D , D q “ d H p D , D q .If t u D “ t u D in some subset of p , k u having an accumulationpoint in p , k u , then D “ D . The uniqueness results in Corollaries 3 and 4 are different from thoseexisting in the literature in the case of single measurement (comparewith [18, Theorems 4.3.2 and 4.3.5]).
We introduce some definitions and results that will be proved in Ap-pendix A.Let N be the Neumann-Green function on Ω . That is N obeys to thefollowing properties, where y P Ω is arbitrary, ∆N p¨ , y q “ δ y , B ν N p¨ , y q |B Ω “ . We normalize N so that ż B Ω N p x, y q dσ p x q “ , y P Ω. Denote by E the usual fundamental solution of the Laplacian in thewhole space. That is E p x q “ " ´ ln | x |{p π q if n “ . | x | ´ n {pp n ´ q ω n q if n ě . Here ω n “ | S n ´ | .We establish in Appendix A that the Neumann-Green function is sym-metric and has the form N p x, y q “ E p x ´ y q ` R p x, y q , x, y P Ω, x ‰ y. R P C p Ω ˆ Ω q and R p¨ , y q P H { p Ω q , y P Ω .Consider the integral operators acting on L pB D q as follows S D f p x q “ ż B D N p x, y q f p y q dσ p y q , f P L pB D q , x P Ω. We will see later that S D is extended to a bounded operator from H ´ { pB D q into the space H D “ t u P H p Ω q ; ∆u “ Ω zB D and B ν u “ B Ω u endowed with the norm } ∇ u } L p Ω q . Note that according to the usualtrace theorems B ν u is an element of H ´ { pB Ω q .We define the NP operator K D as the integral operator acting on L pB D q with weakly singular kernel L p x, y q “ p x ´ y q ¨ ν p y q| x ´ y | n ` B ν p y q R p x, y q , x, y P B
D, x ‰ y. Finally, we establish in Appendix A that S D “ S D |B D defines an iso-morphism from H ´ { pB D q onto H { pB D q . Theorem 6.
Define successively, as long as the maximum is positive,the energy quotients λ ` j p D q “ max g Kt g ` D, ,...,g ` D,j ´ u } ∇ S D g } L p Ω z D q ´ } ∇ S D g } L p D q } ∇ S D g } L p Ω q . Here the orthogonality is with respect to the scalar product p ∇ S D ¨| ∇ S D ¨q L p Ω q .The maximum is attained at g ` D,j P H { pB D q .Define similarly λ ´ j p D q “ min g Kt g ´ D, ,...,g ´ D,j ´ u } ∇ S D g } L p Ω z D q ´ } ∇ S D g } L p D q } ∇ S D g } L p Ω q . The minimum is attained at g ´ D,j P H { pB D q .The potentials S D g ˘ D,j together with all S D h , h P ker p K D S D q Ă H ´ { pB D q ,are mutually orthogonal and complete in H D . This eigenvalue variational problem is correlated to the eigenvalue prob-lem of the NP operator K D . Precisely, we have Corollary 5.
The spectrum of K D consists in the eigenvalues µ ˘ j p D q “´ λ ˘ j p D q{ , j ě , multiplicities included, together with possibly the pointzero. The extremal functions g ˘ D,j are exactly the eigenvalues of K D . Set ϕ ˘ D,j “ S D g ˘ D,j {} ∇ S D g ˘ D,j } L p Ω q , j ě H ˘ D “ span t ϕ ˘ D,j ; j ě u . Let ϕ D,j , 1 ď j ď ℵ if 0 ă ℵ ă 8 and j ě ℵ “ 8 , be an orthonormalbasis of H D “ t ψ “ S D h ; h P ker p K D S D qu . Here ℵ P r , is thedimension of H D .nverse conductivity problem 15For simplicity convenience we only treat the case ℵ “ 8 . The results inthe case ℵ ă 8 are quite similar.The preceding theorem says that H D “ H ` D ‘ H ´ D ‘ H D .We are now ready to give the expansion of the solution of the BVP (1)in the basis t ϕ ǫD,j , j P I, ǫ
P t` , ´ , uu , where we set I “ t j ě u . Proposition 2.
Let u D p k q be the solution of the BVP (1) .Then(i) u D admits the following expansion, where ˜ u D is as in the beginningof Section 2, u D p k q “ ˜ u D ` ÿ j P I ǫ Pt` , ´ , u A ǫD,j p k q ϕ ǫD,j , the coefficients A ǫD,j p k q satisfies, for j P I and ǫ P t` , ´ , u , p k ´ k q ÿ i P I η Pt` , ´ , u A ηD,i p k qp ∇ ϕ ηD,i | ∇ ϕ ǫD,j q L p D q ` k A ǫD,j p k q “ k B ǫD,j , where B ǫD,j “ p f | ϕ ǫD,j q L pB Ω q ´ p ∇ ˜ u D | ∇ ϕ ǫD,j q L p Ω z D q . (ii) k P p , k u Ñ A ǫD,j p k q , j P I and ǫ P t` , ´ , u , is real analytic.Moreover, for any ˜ k P p , k u , there exists δ ą so that, for any j P I , ǫ P t` , ´ , u and | k ´ ˜ k | ă δ , the series ÿ ℓ P N ℓ ! d ℓ dk ℓ A ǫD,j p ˜ k qp k ´ ˜ k q ℓ converges.Proof. (i) Recall that u D has the following decomposition u D p k q “ ˜ u D ` v D p k q . Observing that v D p k q satisfies B ν v D p k q “ H ´ { pB Ω q ) anddiv p a D p k q ∇ v D p k qq “ ´ div p a D p k q ˜ u D q in D p Ω q , we find by using the generalized Green’s formula k p ∇ v D p k q| ∇ ϕ q L p D q ` k p ∇ v D p k q| ∇ ϕ q L p Ω z D q (25) “ k p f | ϕ q L pB Ω q ´ k p ∇ ˜ u D | ∇ ϕ q L p Ω z D q , ϕ P H p Ω q . We expand v D p k q in the basis t ϕ ǫD,j , j P I, ǫ
P t` , ´ , uu : v D p k q “ ÿ j P I ǫ Pt` , ´ , u A ǫD,j p k q ϕ ǫD,j . Taking ϕ “ ϕ ǫD,j in (25), we obtain p k ´ k qp ∇ v D p k q| ∇ ϕ ǫD,j q L p D q ` k A ǫD,j p k q “ k B ǫD,j , (26)for j P I and ǫ P t` , ´ , u .6 Jin Cheng, Mourad Choulli and Shuai LuWhence p k ´ k q ÿ i P I η Pt` , ´ , u A ηD,i p k qp ∇ ϕ ηD,i | ∇ ϕ ǫD,j q L p D q ` k A ǫD,j p k q “ k B ǫD,j , (27)for j P I and ǫ P t` , ´ , u .(ii) We know from the preceding section that k P p , k u Ñ v D p k q P V is real analytic. Then so is k P p , k u Ñ A ǫD,j p k q P C and k Pp , k u Ñ p ∇ v D p k q| ∇ ϕ ǫD,j q L p D q P C , j P I , ǫ P t` , ´ , u .We get by taking successively the derivative in (26) with respect to k p k ´ k qp ∇ v p ℓ q D p k q| ∇ ϕ ǫD,j q L p D q ` k d ℓ dk ℓ A ǫD,j p k q (28) “ ´ ℓ p ∇ v p ℓ ´ q D p k q| ∇ ϕ ǫD,j q L p D q , for j P I , ǫ P t` , ´ , u and ℓ P N zt u .The choice of k “ ˜ k in (28) entails k d ℓ dk ℓ A ǫD,j p ˜ k q “ ´p ˜ k ´ k qp ∇ v p ℓ q D p ˜ k q| ∇ ϕ ǫD,j q L p D q ` (29) ´ ℓ p ∇ v p ℓ ´ q D p ˜ k q| ∇ ϕ ǫD,j q L p D q , for j P I , ǫ P t` , ´ , u and ℓ P N zt u .We have ˇˇˇ p ∇ v p ℓ q D p ˜ k q| ∇ ϕ ǫD,j q L p D q ˇˇˇ ď } ∇ v p ℓ q D p ˜ k q} L p Ω q , ℓ P N . On the other hand, the series ÿ ℓ P N ℓ ! v p ℓ q D p ˜ k qp k ´ ˜ k q ℓ converges in V provided that | k ´ ˜ k | ď δ , for some δ . Therefore, in lightof (29), we can assert that the series ÿ ℓ P N ℓ ! d ℓ dk ℓ A ǫD,j p ˜ k qp k ´ ˜ k q ℓ also converges whenever | k ´ ˜ k | ď δ . The proof is then complete. [\ Remark 1.
Unfortunately, computing all the terms p ∇ ϕ ηD,i | ∇ ϕ ǫD,j q L p D q seems to be not possible, especially for η ‰ ǫ . Let us compute those equalto zero. As ϕ ˘ D,j is a solution of a minimisation problem, we obtain in astandard way λ ˘ k p D q ż Ω ∇ ϕ ˘ D,j ¨ ∇ ϕdx “ ż Ω z D ∇ ϕ ˘ D,j ¨ ∇ ϕdx ´ ż D ∇ ϕ ˘ D,j ¨ ∇ ϕdx, for any ϕ P span t ϕ ˘ D, , . . . ϕ ˘ D,j ´ u K .In particular,0 “ ż Ω z D ∇ ϕ ˘ D,j ¨ ∇ ϕ ˘ D,i dx ´ ż D ∇ ϕ ˘ D,j ¨ ∇ ϕ ˘ D,i dx, i ą j. (30)nverse conductivity problem 17But 0 “ ż Ω z D ∇ ϕ ˘ D,j ¨ ∇ ϕ ˘ D,i dx ` ż D ∇ ϕ ˘ D,j ¨ ∇ ϕ ˘ D,i dx, i ą j. This and (30) yield p ∇ ϕ ˘ D,i | ∇ ϕ ˘ D,j q L p D q “ p ∇ ϕ ˘ D,i | ∇ ϕ ˘ D,j q L p Ω z D q “ , i ą j. (31)We have similarly0 “ ż Ω z D ∇ ϕ D,j ¨ ∇ ϕdx ´ ż D ∇ ϕ D,j ¨ ∇ ϕdx, for any ϕ P H D . Hence0 “ ż Ω z D ∇ ϕ D,j ¨ ∇ ϕ D,i dx ´ ż D ∇ ϕ D,j ¨ ∇ ϕ D,i dx, i, j P I. (32)As before, we deduce from (32) p ∇ ϕ D,i | ∇ ϕ D,j q L p D q “ p ∇ ϕ D,i | ∇ ϕ D,j q L p Ω z D q “ , i, j P I, i ‰ j. (33) Appendix A: Spectral analysis of the NPoperator
Prior to proceed to the spectral analysis, we define some integral operatorwith weakly singular kernels. Let D Ť Ω be a bounded domain of R n , n ě
2, of class C ,α , for some 0 ă α ă ν the unit normal outward vector field on B Ω . Then a slightmodification of the proof of [15, Lemma 3.15, page 124] yields |p x ´ y q ¨ ν p x q| ď C | x ´ y | ` α , x, y P B
D, x ‰ y. Here the constant C only depends on D . Hence ˇˇˇˇ p x ´ y q ¨ ν p x q| x ´ y | n ˇˇˇˇ ď C | x ´ y | n ´ ´ α , x, y P B D. (34)Define the integral operator K ˚ D : L pB D q Ñ L pB D q by K ˚ D f p x q “ ż B D p x ´ y q ¨ ν p x q| x ´ y | n f p y q dσ p y q , f P L p D q . Estimate (34) says that the kernel of K ˚ D is weakly singular and thereforeit is compact (see for instance [22, Section 2.5.5, page 128]).Note that K ˚ D is nothing but the adjoint of the operator K : L pB D q Ñ L pB D q given as follows K D f p x q “ ż B D p x ´ y q ¨ ν p y q| x ´ y | n f p y q dσ p y q , f P L p D q . K ˚ D , K D is also an integral operator with weakly singular kernel andthen it is also compact.Denote by E the usual fundamental solution of the Laplacian in thewhole space. That is E p x q “ " ´ ln | x |{p π q if n “ . | x | ´ n {pp n ´ q ω n q if n ě . Here ω n “ | S n ´ | .Recall that the single layer potential S D is the integral operator withkernel E p x ´ y q : S D f p x q “ ż B D E p x ´ y q f p y q dσ p y q , f P L pB D q , x P R n zB D. Before stating a jump relation satisfied by S D f , we introduce the nota-tions, where w P C p R n q and x P B D , w |˘ p x q “ lim t Œ w p x ˘ tν p x qq , B ν w p x q |˘ “ lim t Œ ∇ w p x ˘ tν p x qq ¨ ν p x q . For any f P L pB D q , B ν S D f p x q |˘ exists as an element of L pB D q and thefollowing jump relation holds B ν S D f p x q |˘ p x q “ ˆ ˘ ` K ˚ D ˙ f p x q a.e. x P B Ω. (35)We refer for instance to [3, Theorem 2.4 in page 16] and its comments.For y P Ω , consider the following BVP " ∆R “ Ω, B ν R “ ´B ν E p¨ ´ y q on B Ω. (36)As B ν E p¨´ y q P L pB Ω q , in light of [19, Theorem 2, page 204 and Remarks(b) page 206], the BVP (36) has a unique solution R p¨ , y q P H { p Ω q sothat ż B Ω R p x, y q dx “ κ p y q , where the constant κ p y q is to be determined hereafter.Define then N by N p x, y q “ E p x ´ y q ` R p x, y q , x, y P Ω, x ‰ y. The function N obeys to the following properties, where y P Ω is arbi-trary, ∆N p¨ , y q “ δ y , B ν N p¨ , y q |B Ω “ . We fix in the rest of this text κ p y q in such a way that ż B Ω N p x, y q dσ p x q “ . (37)The function N is usually called the Neumann-Green function.nverse conductivity problem 19Mimicking the proof of [3, Lemma 2.14, page 30], we get N p x, y q “ N p y, x q , x, y P Ω , x ‰ y . Hence R p x, y q “ R p y, x q , x, y P Ω . By interiorregularity for harmonic functions R p¨ , y q , y P Ω , belongs to C p Ω q andconsequently R p x, ¨q , x P Ω is also in C p Ω q implying that R P C p Ω ˆ Ω q .Consider the integral operators acting on L pB D q as follows S D f p x q “ ż B D N p x, y q f p y q dσ p y q , f P L pB D q , x P Ω. Clearly S D “ S D ` S D , where S D is the integral with (smooth) kernel R , i.e. S D f p x q “ ż B D R p x, y q f p y q dσ p y, q f P L pB D q , x P Ω. Using (35) we find that S D obeys to the following the jump condition: B ν S D f p x q |˘ p x q “ ˆ ˘ ` K ˚ D ˙ f p x q a.e. x P B Ω. (38)Here K ˚ D “ K ˚ D ` K ˚ D, , where K ˚ D, is the integral operator with kernel B ν p x q R , which is the dual of the integral operator K D, whose kernel is B ν p y q R (thank to the symmetry of R ).We get in particular that K ˚ D : L pB D q Ñ L pB D q is compact. Morespecifically, K ˚ D : L pB D q Ñ H pB D q is bounded (see for instance [3,Theorem 2.11, page 28]).We defined in Section 5 S D “ S D |B D that we consider as a boundedoperator on L pB D q and set H D “ t u P H p Ω q ; ∆u “ D p Ω zB D qu . Define on H D the positive hermitian form p u | v q H D “ ż Ω ∇ u ¨ ∇ vdx. The corresponding semi-norm is denoted by } u } H D “ p u | u q { H D . Let u ´ “ u | D and u ` “ u | Ω z D . As u ´ P H p D q and ∆u “ D p D q ,we know from the usual trace theorems that B ν u ´ P H ´ { p D q . Simi-larly, we have B ν u ` P H ´ { pB D q X H ´ { pB Ω q . Therefore, according togeneralized Green’s formula, for any u, v P H D , we have ż D ∇ u ´ ¨ ∇ v ´ “ xB ν u ´ | v ´ y { , B D , (39) ż Ω z D ∇ u ` ¨ ∇ v ` “ ´xB ν u ` | v ` y { , B D ` xB ν u ` | v ` y { , B Ω . (40)The symbol x¨|¨y { ,Γ denotes the duality pairing between H { p Γ q andits dual H ´ { p Γ q , with Γ “ B D or Γ “ B Ω .0 Jin Cheng, Mourad Choulli and Shuai LuTaking the sum side by side in inequalities (39) and (40), we find p u | v q H D “ xB ν u ´ ´ B ν u ` | v y { , B D ` xB ν u ` | v ` y { |B Ω , (41)where we used that v ´ “ v ` “ v in B D .We apply (41) to u “ v “ S D f , with f P L pB D q . Taking into accountthat B ν u ` “ B Ω , we obtain } S D f } H D “ xB ν S D f |´ ´ B ν S D f |` | S D f y { , B D . This and the jump condition (38) entail } S D f } H D “ x f | S D f y { , B D “ p f | S D f q L pB D q . (42)Here p¨|¨q L pB D q is the usual scalar product on L pB D q .In other words, we proved that S D : L pB D q Ñ L pB D q is strictly positiveoperator. Define, as in [20], H ´ D to be the completion of L pB D q with re-spect to the norm }? S D f } L pB D q “ p f | S D f q L pB D q . Let H ` D “ R p? S D q ,the range of ? S D , that can be regarded as the domain of the unboundedoperator ? S D ´ . We observe that H ` D is complete for the norm inducedby the form p? S D ´ f, ? S D ´ f q L pB D q . Therefore S D can be extendedby continuity as an isomorphism, still denoted by S D , from H ´ D onto H ` D and the pairing p? S D f | g q L pB D q “ p f |? S D g q L pB D q defines a duality pairing between H ` D and H ´ D , with respect to the pivotspace L pB D q .Recall that the following closed subspace of H D was introduced in Section5. H D “ t u P H p Ω q ; ∆u “ D p Ω zB D q and B ν u |B Ω “ u . Note that we have seen before that B ν u |B Ω is an element of H ´ { pB Ω q .In that case (41) takes the form p u | v q H D “ xB ν u ´ ´ B ν u ` | v y { , B D , u P H D , v P H D . (43) Lemma 3.
For any g P H { pB D q , there exists a unique v “ E D g P H D so that v |B D “ g and } E D g } H p Ω q ď C } g } H { pB D q , (44) where the constant C only depends on Ω and D .Proof. Take v P H p Ω q so that v | D “ v ´ and v Ω z D “ v ` , where v ´ and v ` are the respective variational solutions of the BVP’s " ∆v ´ “ D,v ´ “ g on B D, and $&% ∆v ` “ Ω z D,v ` “ g on B D, B ν v ` “ B Ω. Clearly, v P H D and (44) holds. [\ nverse conductivity problem 21 Lemma 4.
There holds H ` D “ H { pB D q .Proof. Let g P H { pB D q and f P L pB D q . Apply then (43), with u “ S D f and v “ E D g in order to obtain p f | g q L pB D q “ p S D f | E D g q H D . Whence ˇˇ p f | g q L pB D q ˇˇ ď C } E D g } H D } S D f } H D ď C } g } H { pB D q }? S D f } L pB D q . In other words, the linear form f P L pB D q ÞÑ p f | g q L pB D q is bounded forthe norm }? S D f } L pB D q . Therefore, according to Riesz’s representationtheorem, there exists k P L pB D q so that p f | g q L pB D q “ p f |? S D k q . Since f is arbitrary, we deduce that g “ ? S D k and hence g P H ` D . Thatis H { pB D q Ă H ` D .Conversely, as f P H ` D has the form f “ S D g for some g P H ´ D , thecoupling p S D g | g q : “ p? S D g |? S D g q L pB D q is well defined. Let p g k q be asequence in L pB D q converging to g in the topology of H ´ D . We have } S D g k ´ S D g ℓ } H D “ p S D g k ´ S D g ℓ | g k ´ g ℓ q L pB D q . Hence p S D g k q is a Cauchy sequence in H D which is complete with re-spect to the norm } ¨ } H D . The limit u of the sequence p S D g k q sat-isfies u |B D “ S D g “ f by the continuity of the trace map. Whence f P H { pB D q . [\ As byproduct of the preceding proof we see that S D is extended as abounded operator from H ´ { pB D q onto H D by setting S D f “ lim k S D f k , f P H ´ { pB D q , where p f k q is an arbitrary sequence in L pB D q converging to f in H ´ { pB D q .Moreover, we have u “ S D S ´ D p u |B D q , u P H D . Introduce the double layer type operator D D f p x q “ ż B D B ν p y q N p x, y q f p y q dσ p y q , f P L pB D q , x P Ω zB D. From [3, Theorem 2.4, page 16], we easily obtain, where f P L p D q , D D f |˘ “ p¯ { ` K D q f a.e. on B D. (45)As in [20, Lemma 2, page 154], our spectral analysis is based on thePlemelj’s symmetrization principle. We have Lemma 5.
For S D , K D : L pB D q Ñ L pB D q , the following identityholds K D S D “ S D K ˚ D . Proof.
For f P D pB D q and x P Ω zB D , we have D D S D f p x q “ ż B D B ν p y q N p x, y q ˆż B D N p y, z q f p z q dσ p z q ˙ dσ p y q“ ż B D ˆż B D B ν p y q N p x, y q N p y, z q dσ p y q ˙ f p z q dσ p z q . As N p x, ¨q “ N p¨ , x q and N p¨ , z q are harmonic in D , Green’s formulayields D D S D f p x q “ ż B D ˆż B D N p x, y qB ν p y q N p y, z q dσ p y q ˙ f p z q dσ p z q“ ż B D N p x, y qB ν p y q ˆż B D N p y, z q f p z q dσ p z q ˙ dσ p y q“ S D rB ν p y q S D f |´ s . In light of the jump conditions in (38) and (45) we get D D S D f |` “ p´ { ` K D q S D f “ S D rB ν p y q S D f |´ s |` “ S D p´ { ` K ˚ D q f. This and the fact that D pB D q is dense in L pB D q yield the expectedinequality. [\ We recall the definition of the Shatten class, sometimes called also Shatten-von Neumann class. To this end, we consider a complex Hilbert space H .If T : H Ñ H is a compact operator and if T ˚ : H Ñ H denotes its ad-joint, then | T | : “ ? T ˚ T : H Ñ H is positive and compact and thereforeit is diagonalizable. The non-negative sequence p s k p T qq k ě of eigenvaluesof | T | are usually called the singular values of T . For 1 ď p ă 8 , if ÿ k “ r s k p T qs p ă 8 we say that T belongs to the Shatten class S p p H q . It is worth mentioningthat S p p H q is an ideal of B p H q , S p H q is known as the trace class and S p H q corresponds to the Hilbert-Schmidt class. Lemma 6.
We have K D P S p p L pB D qq , for any p ě p n ´ q{ α .Proof. We first note that ℓ , the kernel of K D , satisfies ℓ p x, y q “ O ´ | x ´ y | ´p n ´ q` α ¯ . (46)Let ω be an open subset of R n ´ and ρ continuous on ω ˆ ω ztp x, x q ; x P ω u and ρ p x, y q “ O ´ | x ´ y | ´p n ´ q` α ¯ , x, y P ω .Let 1 ă q ď α {p n ´ ´ α q . Then elementary computations show thatsup x P ω ż ω | ρ p x, y q| q dx ă 8 . From this we get, in light of (46) and using local cards and partition ofunity, that ℓ P L x pB D ; L qy pB D qq . Consequently, according to [21, Theo-rem 1], K D belogns to S p p L pB D qq , where p “ q {p q ´ q is the conjugateexponent of q . We complete the proof by noting that the condition on q yields p ě p n ´ q{ α . [\ nverse conductivity problem 23We remark that [20, Theorem 4.2] is obtained as an immediate conse-quence of the abstract theorem [20, Theorem 3.1]. Since all the assump-tions of [20, Theorem 3.1] hold in our case, we get, similarly to [20,Theorem 4.2], Theorem 6.In Theorem 6 we anticipated the regularity of the extremal functions g ˘ D,j . This result can be proved by following the same arguments as in[20, page 164].Following [20], λ ˘ j p D q , j ě
1, are called the eigenvalues of the spectralvariational Poincaré problem: find those λ ’s for which there exists u P H D , u ‰
0, so that ż Ω z D ∇ u ¨ ∇ vdx ´ ż D ∇ u ¨ ∇ vdx “ λ ż Ω ∇ u ¨ ∇ vdx, v P H D . As we already mentioned in Section 5, this variational eigenvalue problemis correlated to the eigenvalue problem for the NP operator K D . Tosee this, we observe that as straightforward consequence of (39), (40)together with the jump condition (35), we have ż Ω z D ∇ S D g ¨ ∇ S D hdx ´ ż D ∇ S D g ¨ ∇ S D hdx “ ´ p K ˚ D S D g | S D h q L pB D q “ ´ p S D g | K D S D h q L pB D q . These identities are first established for g, h P L pB D q and then extendedby density to g, h P H ´ { pB D q . Thus, we have in particular } ∇ S D g } L p Ω z D q ´ } ∇ S D g } L p D q } ∇ S D g } L p Ω q “ ´ p S D g | K D S D g q L pB D q p S D g | g q L pB D q , with g P H ´ { pB D q .These comments enable us to deduce Corollary 5 from Theorem 6. Acknowledgments
CJ is supported by NSFC (key projects no.11331004, no.11421110002)and the Programme of Introducing Talents of Discipline to Universities(number B08018). MC is supported by the grant ANR-17-CE40-0029of the French National Research Agency ANR (project MultiOnde). LSis supported by NSFC (No.91730304), Shanghai Municipal EducationCommission (No.16SG01) and Special Funds for Major State Basic Re-search Projects of China (2015CB856003). This work started during thestay of MC at Fudan University on February 2017. He warmly thanksFudan University for hospitality.