Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary

In this paper, we establish conditional stability estimates for two inverse problems of determining metrics in two dimensional Laplace-Beltrami operators. As data, in the first inverse problem we adopt spectral data on an arbitrarily fixed subboundary, while in the second, we choose the Dirichlet-to-Neumann map limited on an arbitrarily fixed subboundary. The conditional stability estimates for the two inverse problems are stated as follows. If the difference between spectral data or Dirichlet-to-Neumann maps related to two metrics \begin{document}$ {\bf{g}}_1 $\end{document} and \begin{document}$ {\bf{g}}_2 $\end{document} is small, then \begin{document}$ {\bf{g}}_1 $\end{document} and \begin{document}$ {\bf{g}}_2 $\end{document} are close in \begin{document}$ L^2(\Omega) $\end{document} modulo a suitable diffeomorphism within a priori bounds of \begin{document}$ {\bf{g}}_1 $\end{document} and \begin{document}$ {\bf{g}}_2 $\end{document} . Both stability estimates are of the same double logarithmic rate.

Throughout this paper, {g jk } := {g jk } 1≤j,k≤2 means a 2×2 matrix and {g jk (x)} denotes the inverse to g(x) = {g jk (x)}. Moreover let ∆ g denote the Laplace-Beltrami operator associated to the Riemannian metric g = {g jk }: Henceforth we write ∆ g = ∆ if g is the 2 × 2 identity matrix. Let λ k be the eigenvalues of the Laplace operator with the zero Dirichlet boundary condition, where we number λ k repeatedly according to their multiplicities. Let e k be the corresponding eigenfunctions: (Ω) = 1 ∀k, j ∈ N, ρ 0 = det g.
Here and henceforth F is the Jacobian matrix of the mapping F and • denotes the composition of mappings. Let λ k (j) and e k (j) be the eigenvalues and the eigenvectors of the Laplace-Beltrami operator associated to the Riemannian metric g j and satisfying (3) and (4).
Then there exist a diffeomorphism F : Ω → Ω, F ∈ C 3+α (Ω) in general depending on g 1 , g 2 , satisfying (6) and a constant C(M 1 , M 2 ) > 0 such that where the function G : R + → R + is defined by In Theorem 1.1, we notice that in general Λ(g 1 , g 2 ) < ∞ does not necessarily hold for g 1 , g 2 satisfying (8), and if Λ(g 1 , g 2 ) = ∞, then the conclusion of the theorem holds trivially. In other words, we cannot expect the reverse inequality for (9). We emphasize that the conclusion in Theorem 1.1 is a conditional stability estimate under condition (8), and we do not know whether a weaker distance than Λ(g 1 , g 2 ) can yield the same stability as (9) or not. The situation is the same also for Theorem 1.2 stated below under condition (14) .
In order to formulate our second inverse problem we introduce the operator The second inverse problem is Inverse boundary value problem.
Then determine β and g by Λ β,g . We see that Λ β,g is a bounded linear operator from L 2 ( Γ) to H −1 ( Γ). By L L(X,Y ) we denote the norm of an operator L from a Banach space X to a Banach space Y .

Remark 1.
It is well-known that the inverse problem (12) - (13) admits the following non-uniqueness: pairs (β, g) and (β, βg) where β satisfies the conditions of Theorem 1.2, generate the same Dirichlet-to-Neumann map. Therefore one can not avoid the appearance of the function β on the left-hand side of inequality (15).
As for stability and related results, we further refer to Bellassoued and Dos Santos Ferreira [7], Blåsten, Imanuvilov and Yamamoto [8], Mandache [21], Montalto [22], Novikov and Santacesaria [25], [26], Santacesaria [29]. Theorems 1.1 and 1.2 are first results for the respective related inverse problems, and for the technical feasibility, we assume the class in C 2+α (Ω) as admissible set of Riemannian metrics under consideration, and we do not pursue admissible sets in less regular function spaces.
The paper is composed of five sections. The proof of Theorem 1.1 is divided into three steps according to the main ideas used for the proof: Sections 2-4 are devoted to these three steps respectively. Section 2: the proofs of the existence of a diffeomorphism and an estimate of F * g 1 − g 2 on Γ (Propositions 1 and 2). Section 3: the establishment of L p -Carleman estimates for the Laplacian (Propositions 6 and 7). Section 4: the completion of the proof of Theorem 1.1: after the construction of a conformal diffeomorphism, we reduce the proof to the stability result in [8] for an inverse boundary value problem for Schrödinger equations.
In Section 5, we complete the proof of Theorem 1.2, which is based on the stability result in [8].

First
Step of the proof of Theorem 1.1: boundary estimate. We define an elliptic operator by L g j (x, D, s) = −∆ g j + s, j = 1, 2 with s ≥ 0.
Let u j = u j (x, s), j = 1, 2, be the solutions to the boundary value problems where f ∈ C ∞ 0 ( Γ). In terms of the eigenfunctions e k (j) and the eigenvalues λ k (j), one can write the solution to equation (16) in the form of the infinite series which is convergent in L 2 (Ω) as follows: Then the normal derivative of the function u j can be represented in the form of infinite series converging in W Indeed, for N ∈ N, we set We need the estimate of the difference of the normal derivatives ∂uj ∂ν g j . The following is known (see e.g., [13]) that there exists a constant C 1 independent of k and j such that In order to prove (18), we write equation (3) as Let P (x, D) = 2 p=1 a p (x)∂ xp be a first order differential operator with coefficients a p ∈ C 2+α (Ω) such that a p (x) = ν p (x) on ∂Ω. Taking the scalar product of P (x, D)e k (j) and equation (19) in L 2 (Ω) after integration by parts, we have Here we set g −1 j := {g m j } for j = 1, 2. This equality and (8) imply (18). Let , 1 ∈ (0, 1) be small parameters. Consider a sequence of functions ρ such that Using (17), we obtain: We estimate the terms I j separately. For the first term I 1 , using (18) we have For I 2 and I 3 , using (18) we obtain and Combining estimates (21) - (23), from (20) we have Let Ω 1 be a bounded domain in R 2 and F ∈ C 3+α (Ω 1 , Ω) be a diffeomorphism of the domain Ω 1 on Ω such that Let u j be some solution to problem (16). Then the function u j = u j • F −1 is a solution to the following boundary value problem where ν is the outward unit normal vector to Ω 1 .
In order to prove (27) we observe Let a function υ(x) satisfy the following conditions The function γ is independent of the metrics g j and, provided that diffeomorphism F satisfies (25), there exist positive constants C 8 , C 9 such that Since the eigenvalues of the operators ∆ g and ∆ F * g are exactly the same, from (28) there exist constants C 10 , C 11 > 0 such that Hence, since the domain Ω is simply connected, after possible change of variables, we can assume that , Ω is symmetric with respect to the x 2 -axis.

and
(33) Proof. We observe that in order to prove the statement of the proposition, it suffices to show the following: For any metrics g 1 , g 2 satisfying (8), there exist constants C 14 , C 15 and a diffeomorphisms F (j) ∈ C 3+α (Ω) such that F (j)| Γ = I and (34) Indeed, if (34) and (35) are both proved, then we set F = F (1) • F −1 (2). This mapping is a diffeomorphism of the domain Ω onto Ω such that F | Γ = (F (1) • F −1 (2))| Γ1 = (I • I)| Γ1 = I. Estimate (32) follows from (34): First we prove the proposition for the case Γ ⊂ Γ 2 ⊂ {x ∈ R 2 ; x 1 = 1}, Ω ⊂ R × R + and (37) g 12 j = g 21 j = 0 on Γ, g 22 j = 1 on Γ, j ∈ {1, 2}. In this case, we have Let w j ∈ C 3+α (Ω) be the solution to the boundary value problem Next we prove some regularity results for the function w j . Let ρ ∈ C 2 (R × R + ) be identically equal to one in some neighborhood of Γ and identically equal to zero on the set {x; dist (x, Γ) ≥ 0 } with some small positive 0 . We set w j = ρw j . Then L g j (x, D, s) w j = r j on R × R + and w j | ∂Ω = f. From the standard estimates in the Schauder spaces and the Sobolev spaces for elliptic equations, there exists a constant C 16 > 0, independent of s, such that Indeed, by taking into consideration that g j ∈ C 2+α (Ω), the standard regularity property for the solution to the boundary value problem (e.g., Chapter 2 in [20]) yields w j H Therefore, applying the regularity property (e.g., [20]) again, we have Therefore the regularity (e.g., [20]) and the Sobolev embedding theorem imply for any β ∈ (0, 1). Then we apply the Schauder internal regularity estimates (e.g., Chapter 6 in [11]) to obtain the above estimate for r j .
Henceforth we understand the double signs correspond. Next we consider a solution to boundary value problem (38) for highly oscillatory boundary data f. Our goal is to construct an asymptotic behavior of the normal derivative of the solution w j on Γ. In order to do that, we employ the standard techniques based on the factorization of elliptic operator into a product of two pseudodifferential operators.

Now we introduce two symbols
Then we can write where the operators K j (x, D, s) satisfy the estimate Here the constant C 23 is independent of s and j ∈ {1, 2}.
where the constant C 24 is independent of s and j ∈ {1, 2}. From (37), (44) and (43), we have Consider the function f = e isx1 g where g ∈ C ∞ 0 ( Γ) and g = 1 on Γ 1 . Denote by q(x 1 , ξ 1 , s) the principal symbol of the pseudodifferential operator We can represent Applying the stationary phase argument (e.g., [10]), we obtain where the term M satisfies Observe that Lemma A.2 of [14] yields Therefore by (45) and (49) we have Using (47) and (48), in order to estimate the left-hand side of (50) from below, we see that there exists a constant C 30 > 0, independent of s, such that Taking s = 1/Λ 1 2 (g 1 , g 2 ) from the above inequality, we obtain (34) under assumptions (37). Now we will remove the restriction (37). Let K > 0 be some parameter which we choose later. First we construct a diffeomorphism R = (R 1 , R 2 ) : For all |x 1 | ≤ K, direct computations yield Here we recall that R is the Jacobian matrix. The formula (7) implies As such an R, we take This equality implies (R * g) 12 The Jacobi matrix of the mapping R is given by We can choose > 0 such that Then (53) yields From (52) and (54), there exists a constant C 31 > 0, which is independent of g, such that (55) sup Let us show that the constructed mapping is injective. Indeed let R(x) = R(y) for some x, y ∈ R × R + . This immediately implies x 2 = y 2 . Consider the function r(t) = t + η( y2 )P (t, y 2 ). Since r (t) > 0 for t > 0, we see that y 1 = x 1 .
Next we prove that the mapping F is surjective. For y = ( y 1 , y 2 ) ∈ R × R + , we have to solve the equation R(x) = y. Obviously x 2 = y 2 . Hence we need to solve the equation R 1 (x 1 , y 2 ) = r(x 1 ) = y 1 . Thanks to (53) this equation has a unique solution, which means that P is surjective.
In the next step, we construct a diffeomorphism G = (G 1 , G 2 ) : Using the diffeomorphism constructed above and making the change of coordinates in R × R + , we can assume We extend the function g 22 By (57), there exist constants δ 0 > 0 and β 1 > 0 such that and so (56). Moreover The Jacobi matrix G of the mapping G is given by .
Taking > 0 sufficiently small, we see that there exists a constant β > 0, independent of Let us show that the constructed mapping is injective. Indeed if G(x) = G(y), then we immediately have x 1 = y 1 . Consider the function r(t) = G 2 (x 1 , t). The derivative of this function is positive for t > 0, so that x 2 = y 2 .
Next we prove that the equation G(x) = y has a solution for an arbitrary y = ( y 1 , y 2 ) ∈ R × R + . We set x 1 = y 1 . By (57) we see G( y 1 , 0) = 0 and lim x2→+∞ G 2 ( y 1 , x 2 ) = +∞. Therefore, for fixed y 1 , the function x 2 → G 2 ( y 1 , x 2 ) takes all the values from R + . Outside of a ball with sufficiently large radius, the mapping G is linear. Then from (58) and (59), we obtain that there exists a constant M > 0, which is independent of g, such that Let Ψ be a conformal diffeomorphism of Ω on D. We recall that D = {x ∈ R 2 ; |x| < 1} = {z ∈ C; |z| < 1}. Without loss of generality we can assume that (0, Obviously this mapping is injective and surjective. Outside of the intersection of any neighborhood of the point (0, −1) with the unit disc D centered at 0, the mapping belongs to the Hölder space C 3+α . On the other hand, for some sufficiently large r 0 , we see that R = G = I: the identity where the mappings G and R are constructed for the corresponding metrics g j , j ∈ {1, 2}, we obtain (37) and, as we have already established, we see that F (j) * g j satisfies (34). From (60) and (55), we obtain (35).
Using Proposition 1, we construct a diffeomorphism F from the domain Ω into itself satisfying (31), (32) and (33). Observe that for the operators ∆ g 1 and ∆ F * g 1 , the eigenvalues are the same. Moreover by e k (1) = e k (1) • F −1 we denote the corresponding eigenvectors of the operator ∆ F * g 1 . By F = I on Γ, we have (see e.g., [18]) This implies that Λ(g 1 , g 2 ) = Λ(F * g 1 , g 2 ). Hence, without loss of generality, we can assume that there exists a constant C 33 independent of g 1 , g 2 such that Let Γ 1 be fixed. We shrink the set Γ up to Γ 1 and later, in order to simplify the notations, we use Γ instead of Γ 1 .
Next we construct a diffeomorphism F of domain Ω on itself such that there exists a smooth positive function µ 2 such that F * g 2 = µ 2 I, where I is the 2 × 2 unit matrix and the mapping F and the function µ depend on metric g continuously. We follow the approach developed in [2] and [31]. More precisely we have the following proposition.
Proposition 2. For any metric g satisfying (8), there exist a diffeomorphism F = (F 1 , F 2 ) : Ω → Ω, F ∈ C 3+α (Ω) and µ ∈ C 2+α (Ω) continuously depending on g, constants β, C 35 , C 36 > 0, depending on the metric g, such that then there exists a continuous function C 36 such that Proof. Without loss of generality, we can assume that 0 ∈ Ω. Using the conformal mapping if necessary, we can assume that Ω = D. Since the domain Ω is simply connected, there exists a conformal diffeomorphism P of Ω in D := {z ∈ C 1 ; |z| < 1} such that P (0) = 0. It is known that for any z ∈ C there exists a conformal diffeomorphism A of D into itself such that A(0) = 0 and ∂ z A(0) = z. After an appropriate choice of z, we can find the above mentioned conformal diffeomorphism, which we denote as A g , such that (A g P ) * g(0) = CI for some positive constant C. Therefore, without loss of generality, we can assume that g(0) = CI. We choose C 37 , C 38 > 0, independent of the metric g, such that Consider the Beltrami equation where (66) µ g = g 11 − g 22 + 2ig 12 Since by (8) Following ( [2], p.264), we consider a solution f g to the Beltarmi equation (65) in the form The function ω g is uniquely determined by the integral equation Since µ g (0) = 0 by g(0) = CI, we see that µ g z ∈ L ∞ (D). By (67), there exist s > 2 and a constant 0 < β 4 < 1 such that the operator norm of µ g S D satisfies Hence, equation (70) has a solution µ g which is uniformly bounded in L s (D). It is shown in [2] (pp. 264-265) that the mapping given by (68) and (69) is a homeomorphism of D onto itself. Thanks to condition (67), the operator ∂ ∂z − µ g ∂ ∂z is an elliptic operator. By the theory of elliptic operators in any domain Ω ⊂⊂ D, there exists a constant C 42 (Ω ) such that Next we obtain the estimates in Hölder spaces near the boundary. Let Ψ : Denoting the real and the imaginary parts of the function f g by f 1 and f 2 respectively, we write the above equation as the system of two equations: The matrix ) and the corresponding eigenvectors e ± = (1, ±i) respectively. Finally we observe that f 2 = 0 on {x 2 = 0}. Therefore the Lopatinskii condition for the system (73) is satisfied and in any bounded domain G ⊂ R × R + we have Combining the estimates (71) and (74), we obtain In order to obtain the corresponding estimate (62) for F −1 (g)(·), we first claim that there exists a constant β 2 > 0 independent of g such that Assume that (77) fails. Then, for some metric g 0 and some point x 0 ∈ D, one should have ∂ z f g 0 (x 0 ) = 0. Let g g 0 be another solution to the Beltrami equation in D such that ∂ z g g 0 (x 0 ) = 0. By the Stoilow factorization (see e.g., [2] p.179), there exists a holomorphic function Φ in D such that The homeomorphism f g can be extended to a homeomorphism: D → D. Hence Φ g 0 ∈ C(D) and Φ g 0 : ∂D → ∂D. Let g g 0 be a homeomorphism of D → D and Ψ : We reach a contradiction. Thus we verify (77).
Since A g P is a conformal diffeomorphism of Ω onto D, we have (A g P ) * (µI) = µ * I with some function µ * ∈ C 2+α (D). Hence we rewrite the inequality (80) as Let the function µ (A g P ) * g be defined by (66). By (81) and (82), we see This inequality and (82) imply Then The inequality (64) is proved.
where the constants M 3 and β are independent of g 1 , g 2 . Then we make the same change of variables in the operator L g 1 (x, D, s). Observe that inequality (29) holds true. Then, by (61) we have Hence, without loss of generality, we can assume that Inverse Problems and Imaging Volume 13, No. 6 (2019), 1213-1258 Henceforth we set x ± = (±1, 1). By Proposition 2, there exists a diffeomorphism F 1 = (F 1,1 , F 1,2 ) of the domain Ω onto itself such that and the operator L g 1 (x, D, s) is transformed to with a smooth positive function µ 1 ∈ C 2+α (Ω) with some α ∈ (0, 1) satisfying Moreover, provided that Λ(g 1 , g 2 ) is sufficiently small, the inequalities (64) and (86) imply We set w j = w j (x, 1) and L g j (x, D) = L g j (x, D, 1). Then Observe that the Dirichlet-to-Neumann maps are given as follows: We construct a solution to the boundary value problem Q µ1I (y, D, 1)w = 0 in Ω, w| Γ0 = 0 of the form: Here P (z) is a smooth holomorphic function constructed in the following proposition and the convergence in (95) is justified after the proof of Proposition 3. Henceforth let Γ x,y denote the arc between points x and y which is clockwise oriented under consideration.
3. Second step of the proof of Theorem 1.1: Carleman estimates. In this section we prove two Carleman estimates (Propositions 6 and 7) for the Laplace operator. Our weight is harmonic in the domain Ω but allows to have the isolated singularities at some points on boundary.
We start with the construction of the weight for the Carleman estimates. Let ϕ * (x, g 1 ) = ϕ * (x) be a harmonic function in Ω such that Let ψ * be a conjugate function to ϕ * and Φ * = ϕ * + iψ * . The function ϕ * is of the class C 3+α at any points of Ω possibly except x ± and in order to estimate the derivatives of the function Φ * near possible singularity points, we provide more details of the construction of this function.
Denote by I = {x ∈ Ω; ∇ϕ * (x) = 0} the set of the critical points of the function ϕ * on Ω. Next we investigate the set of critical points of the function ϕ * in the domain Ω.
Proposition 5. Let a function ϕ * (x, g 1 ) satisfy (102) and metrics g 1 , g 2 satisfy (2). Then there exist positive constants and β independent of g 1 , g 2 such that if Λ(g 1 , g 2 ) ≤ , then and if x 1 , x 2 ∈ I, then there exists a constant C 2 such that Proof. In order to prove (108) and (109) we consider three cases.
Next we prove an L p -Carleman estimate for the Schrödinger equation ∆ + q where a harmonic function ϕ * is taken as the weight function. Typically the proof of L p -Carleman estimate with p = 2 involves some harmonic analysis technique, but such a usual technique can not be directly applied to our case, due to possible singularities of the weight function on the boundary. In our case we developed a rather simple proof using the factorization ∆ = 4∂ z ∂ z of the Laplace operator and the fact that our weight function is a harmonic function. Here we note that ∂ z = 1 2 (∂ x1 + i∂ x2 ) and ∂ z = 1 2 (∂ x1 − i∂ x2 ). More precisely, we have Proposition 6. Let p ∈ (2, +∞), q ∈ L ∞ (Ω) and ϕ * be the harmonic function defined by (102). Let u ∈ H 2 (Ω) satisfy Then there exist constants τ 0 > 0 and C 3 > 0 which is independent of τ , such that Proof. We set w = (∂ z u)e τ Φ * and f τ = (f − qu)e τ Φ * , where the holomorphic function Φ * is given by (104). Proposition 4 yields w ∈ H 1 (Ω). Then where the constant C 4 is independent of τ. By the classical properties of the operator ∂ −1 z (see e.g., [32]), we have Taking the scalar product in L 2 (Ω) of equation (116) with the function W , we obtain By (114), we have . In terms of (119), (118) and (117), we have (120) (∂ z u)e τ Φ * L p (Ω) ≤ C 8 ( f τ L p (Ω) + ∇ue τ ϕ * L p ( Γ) ). Since u is a real-valued function, the above inequality implies (121) (∇u)e τ ϕ * L p (Ω) ≤ C 9 ( f τ L p (Ω) + ∇ue τ ϕ * L p ( Γ) ).
Next we estimate the L p norm of the function ue −τ ϕ * . Let e j ∈ C ∞ 0 (Ω), j = 1, 2 and let supp e j be located in a small neighborhood of the point x j ( x j are the critical points of the function ϕ * ), and e j | B( xj ,δ3) = 1 on some ball B( x j , δ 3 ). The function . On the other hand, by (149), (110), (109) and Morrey's inequality (see e.g., [10]), we obtain The classical estimate implies Estimating the remaining term in (134), using Morrey's inequality and Young's convolution inequality, we have Estimates (150) -(152) and equality (134) imply the L p (Ω)-estimate for the function ue −τ ϕ * in (131). The proof of the proposition is complete.
4. Third step of the proof of Theorem 1.1: construction of conformal diffeomorphism. Thanks to (86) one can apply the Carleman estimate (115) to equation (92). Hence, there exist constants τ 0 and C 1 independent of τ such that

This inequality and (153) yield
Let w be a solution to the initial value problem: Since w 2 is a real-valued function, by Proposition 7, one can find a real-valued solution w satisfying: there exist constants κ > 0, α 0 > 0, τ 1 , and C 7 which is independent of τ , such that for all τ ≥ τ 1 and functiong is given by (128). Here, in order to obtain the last inequality, we used (109). Applying (154) to the right-hand side of inequality (155), we obtain The harmonic function v = w 2 − w verifies Since the domain Ω is simply connected, we can construct a function which is conjugate to v ∈ C α0 (Ω). Denote such a function by ψ 0 (τ, x). We set φ 0 (x) = v(x) − v(0) and Ψ = φ 0 + iψ 0 . Then, thanks to the Cauchy-Riemann equations, we have Next we compute i ∂w2 ∂ν : Estimating the norm of the difference between the Dirichlet-to-Neumann maps defined by (93) and (94), by (86) and (8) we have . Inequality (159) yields Then, by (95), (96) and (101), we have where the function F 1 is given by (102), with mapping F 1 is determined by (87)-(90), and there exist a constant C 16 independent of τ such that function I 2 (τ, ·) satisfies the estimate We set By (87) and (91) we have ≤ C 17 τ Υ(g 1 , g 2 ).
On the other hand, by the argument principle the function Ψ should have at least one zero in the area bounded by the curve γ. We reach a contradiction. Let ψ 2 ∈ C 3+α (Ω) be a harmonic function in Ω such that ψ 2 | ∂Ω = Im P • F 1 and ϕ 2 be the complex conjugate. By the construction of the holomorphic function P , we have By (157), (166), (167) and the maximum principle for the Laplace operator, we see that The Cauchy-Riemann equations and (170) yield We claim that there exist κ > 0 and β 0 > 0 such that (172) |∂ z Φ 2 (z, g 1 )| > β 0 for each z ∈ Ω and g 1 satisfying (8) and Υ(g 1 , g 2 ) ≤ κ.
On the other hand, y is a point of absolute maximum of the function Im P on Γ. Consequently ∂ x2 Im P ( y) = 0. Observe that Re ∂ z Ψ 0 ( x) = ∂ x1 Re Ψ 0 ( x) = ∂ x1 Re P ( y)∂ x1 F 1,1 ( x) = ∂ x2 Im P ( y)∂ x1 F 1,1 ( x) = 0. Therefore the function Ψ 0 (z) = 0 does not have critical points on Γ. Suppose that a critical point z of the function Ψ 0 belongs to Γ 0 . Then this is a critical point of the harmonic function ImΨ 0 . Since z is a point of the global minimum of the function ImΨ 0 on Ω, the Hopf lemma yields that ∂ImΨ0( z) ∂ν = 0. Therefore we reach a contradiction. Suppose that the function Ψ 0 (z) has a critical point z ∈ Ω. Since there are no critical points of the function Ψ 0 on the boundary, one can apply the argument principle. Using the Cauchy-Riemann equations, we compute ∂zΨ0(x±) |∂zΨ0(x±)| = −1. On the segment [x − , x + ] there exists a unique pair ( x + , x − ) of points in Γ 0 such that ∂zΨ0( x±) |∂zΨ0( x±)| = ±1. On the arc between the points x + and x − , there exists a point x + such that ∂zΨ0( x+) |∂zΨ0( x+)| = −i. On the arc between the points x − and x − , there exists a point x + such that ∂zΨ0( x+) |∂zΨ0( x+)| = i. The vector ∂zΨ0(z) |∂zΨ0(z)| makes the full circle around the origin, when the parameter z is moving clockwise from x + to x − . On the other hand, moving clockwise on segment [x − , x + ], the vector ∂zΨ0(z) |∂zΨ0(z)| rotating in the opposite direction, and so arg ∂ z Ψ 0 = 0. We reach a contradiction.
Let R g 1 (x) be the conjugate function to G g 1 (x) in G g 1 (Ω 1 ). Consider the holomorphic function Ψ g 1 (x) = G g 1 (x) + iR g 1 (x) + ln z. Since the boundary ∂G g 1 (Ω 1 ) is smooth, it is known that the mapping e Ψ g 1 is a diffeomorphism of the domain G g 1 (Ω 1 ) in D (see e.g., [12], p. 251, problem 73). Let G g 2 solve the boundary value problem Let R g 2 (x) be the conjugate function to G g 2 (x) in Ω 1 .
Consider the holomorphic function Ψ g 2 (x) = G g 2 (x) + iR g 2 (x) + ln z, z = x 1 + ix 2 . Since the boundary ∂Ω 1 is smooth, it is known that the mapping e Ψ g 2 is a diffeomorphism of the domain Ω 1 onto D (see e.g., [12], p. 251, problem 73). By (87) there exists a constant C 26 , independent of g 1 , g, 2 such that Let 0 < α 1 < α. By (180) and (177), there exists κ = κ(α 1 ) such that We claim that the following estimate is true: In order to prove (182), we construct a diffeomorphism J of the domain G g 1 (Ω 1 ) onto Ω 1 which is close to the unit mapping.
We set By the construction of the mappings P, e Ψ g 2 , e Ψ g 1 and G g 1 we see that the mapping Ξ is a conformal diffeomorphism of the domain Ω on itself. Inequalities (180) and (181) imply (174). From (176) it follows that G g 1 (Γ * 0 ) • P(Γ 0 ) = Γ * 0 .
This complete the proof of the proposition.
Observe that g 2 = µ 2 I. We introduce the function G by formula Thanks to (199) one can apply stability result of [8] to the Schrödinger operators with potentials µ 1 and µ 2 .
Thus the proof of Theorem 1.1 is complete.
The proof of Theorem 1.2 is complete.