PARTIAL CAUCHY DATA FOR GENERAL SECOND-ORDER ELLIPTIC OPERATORS IN TWO DIMENSIONS

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1 PARTIAL CAUCHY DATA FOR GENERAL SECOND-ORDER ELLIPTIC OPERATORS IN TWO DIMENSIONS OLEG YU. IMANUVILOV, GUNTHER UHLMANN, AND MASAHIRO YAMAMOTO Abstract. We consider the inverse problem of determining the coefficients of a general second-order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can determine the coefficients of the operator up to natural obstructions such as conformal invariance, gauge transformations and diffeomorphism invariance. We use the main result to prove that the curl of the magnetic field and the electric potential are uniquely determined by measuring partial Cauchy data associated to the magnetic Schrödinger equation measured on an arbitrary open subset of the boundary. We also show that any two of the three coefficients of a second order elliptic operator whose principal part is the Laplacian, are uniquely determined by their partial Cauchy data.. Introduction Let R 2 be a bounded domain with smooth boundary = N k= γ k, where γ k, k N, are smooth closed contours, and γ N is the external contour. Let Γ be an arbitrarily fixed non-empty relatively open subset of and let Γ 0 = \ Γ. Let ν be the unit outward normal vector to and let u = u ν. ν Henceforth we set i =, x, x 2 R, z = x + ix 2, z denotes the complex conjugate of z C, and we identify x = x, x 2 R 2 with z = x + ix 2 C. We also denote 2 x + i x 2, 2 x i x 2. Let u H be a solution to the following boundary value problem. Lx, Du = g u + 2A u u + 2B + qu = 0, u Γ 0 = 0, u eγ = f. Here g denotes the Laplace-Beltrami operator associated to the Riemannian metric g. We assume that g is a positive definite symmetric matrix in and.2 g = 2 detg g jk, detg x k x j j,k= where {g jk } denotes the inverse of g = {g jk }. From now on we assume that g C 7+α, A, B, q, A j, B j, q j C 5+α C 5+α C 4+α, j =, 2 for some α 0, are complex-valued functions. Henceforth α denotes a constant such that 0 < α <. We set L j x, D = gj + 2A j + 2B j + q j, j =, 2. First author partly supported by NSF grant DMS Second author partly supported by NSF and a Walker Family Endowed Professorship.

2 2 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO We define the partial Cauchy data by.3 { C g,a,b,q = u eγ, u eγ g + 2A ν g + 2B } + qu = 0 in, u H, u Γ0 = 0, where j,k= gjk ν k x j. The goal of this paper is to determine the coefficients of the operator L. In the general case this is impossible. As for the invariance of the Cauchy data, there are of the following three types. i The partial Cauchy data for the operators e η Lx, De η and Lx, D are the same provided ν g = detg 2 that η C 6+α is a complex-valued function and η eγ = η ν e Γ = 0. ii Let β C 7+α be a positive function on. The partial Cauchy data for the operators Lx, D and Lx, D = β βg + 2A + 2B + q are exactly the same. β iii Let F C 8+α : be a diffeomorphism such that F eγ = Id. For any metric g and complex valued functions A, B, q, we introduce a metric F g and functions A F, B F, q F by.4 F g = DF g DF T F, A F = {A + B F x i F 2 x + ib A F x 2 i F 2 x 2 } F det DF, B F = {A + B F + i F 2 + ib A F + i F 2 } F det DF, x x x 2 x 2 q = det DF q F, where DF denotes the differential of F, DF T its transpose and denotes matrix composition. Then the operator Kx, D = F g + 2A F + 2B F + q F and the operator Lx, D have the same partial Cauchy data. We show the converse and state our main result below. Assume that for some α 0, and α > 0 g jk C 7+α, g jk = g kj k, j {, 2}, 2 g jk ξ k ξ j α ξ 2, ξ R 2. j,k= Consider the following set of functions.5 η C 6+α, We have η ν e Γ = 0, η eγ = 0.

3 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 3 Theorem.. Suppose that for some α 0,, there exists a positive function β C 7+α such that g βg 2 eγ = g βg e 2 ν eγ = 0. Then C g,a,b,q = C g2,a 2,B 2,q 2 if and only if there exist a diffeomorphism F C 8+α, F : satisfying F eγ = Id, a positive function β C 7+α and a complex valued function η satisfying.5 such that L 2 x, D = e η Kx, De η, where Kx, D = βf g + 2 β A,F + 2 β B,F + β q,f. and the functions F g, A,F, B,F, q,f are defined for g, A, B, q by.4. We point out that we can prove that that we can assume g βg 2 eγ = 0. However we can not determine the normal derivatives as pointed out in [23] for the case of the operator g. Next we discuss the case of an anisotropic conductivity problem which is an independent interest. In this case the conductivity depends on direction and is represented by a positive definite symmetric matrix σ = {σ jk }. The conductivity equation with voltage potential f on is given by 2 jk u σ = 0 in, x j x k j,k= u = f. We define the partial Cauchy data by { 2.6 V σ = f eγ, σ jk u eγ 2 ν j x k in, j,k= j,k= jk u σ = 0 x j x k u H, u = f, supp f Γ It has been known for a long time that even in the case of Γ =, that the full Cauchy data V σ does not determine σ uniquely in the anisotropic case [20]. Let F : be a diffeomorphism such that F x = x for x on Γ. Then V det DF F σ = V σ. In the case of full Cauchy data i.e., Γ =, the question whether one can determine the conductivity up to the above obstruction has been solved in two dimensions for C 2 conductivities in [24], Lipschitz conductivities in [30] and merely L conductivities in [3]. The method of proof in all these papers is the reduction to the isotropic case using isothermal coordinates []. We have Theorem.2. Let σ, σ 2 C 7+α with some α 0, be positive definite symmetric matrices on. If V σ = V σ2, then there exists a diffeomorphism F : satisfying F eγ = Id and F C 8+α such that det DF F σ = σ 2. }.

4 4 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO For the isotropic case, the corresponding result is proved in [6]. The proof of Theorem.2 is given in section 6. Now we take the matrix g to be the identity matrix. We consider the problem of determining a complex-valued potential q and complex-valued coefficients A and B in a bounded two dimensional domain from the Cauchy data measured on an arbitrary open subset of the boundary for the associated second-order elliptic operator +2A +2B +q. Specific cases of interest are the magnetic Schrödinger operator and the Laplacian with convection terms. We remark that general second order elliptic operators can be reduced to this form by using isothermal coordinates e.g., []. The case of the conductivity equation and the Schrödinger have been considered in [6]. For global uniqueness results in the two dimensional case for the conductivity equation with full data measurements under different regularity assumptions see [2], [6], [24]. Such a problem originates in [9]. Next we will consider the case where the principal part of L j is the Laplacian i.e., g = I; the identity matrix. Then our next result is the following: Theorem.3. Assume that C I,A,B,q = C I,A2,B 2,q 2. Then.7 A = A 2, B = B 2 on Γ, A A 2 B B 2 A A A 2 B 2 + q q 2 = 0 in, 2 B B 2 A A 2 B B B 2 A 2 + q q 2 = 0 in. Remark. In the case that A = A 2 and B = B 2 in, Theorem.3 yields that q = q 2, which is the main result in [6]. The latter result was extended to Riemann surfaces in [3]. The case of full data in two dimensions was settled in [7]. This case is closely related to the inverse conductivity problem, or Calderón s problem. See the articles [24], [6], [2] in two dimensions. Theorem.3 yields Corollary.. The relation C I,A,B,q = C I,A2,B 2,q 2 holds true if and only if there exists a function η C 6+α, η eγ = η ν e Γ = 0 such that.0 L x, D = e η L 2 x, De η. Proof of Corollary.. We only prove the necessity since the sufficiency of the condition is easy to check. By.8 and.9, we have A A 2 = B B 2. This equality is equivalent to  B x = i B +  x 2 where Â, B = A A 2, B B 2. Applying Lemma. p.33 of [3], we obtain that there exists a function η with domain 0 which satisfies. η = η 0 + h, η C 5+α, h = 0 in 0, [ ] h [h] Σk are constants, ν k Σk = h ν γ N = 0 k {,..., N }

5 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 5 and i B + Â, Â B = η. Here 0 = \ Σ is simply connected where Σ = N k= Σ k, Σ j Σ k = for j k, Σ k are smooth curves which do not self-intersect and are orthogonal to. We choose a normal vector ν k = ν k x, k N to Σ k at x contained in the interior Σ 0 k of the closed curve Σ k. Then, for x Σ 0 k, we set [h]x = lim y x, xy,ν k >0 hy lim y x, xy,νk <0 hy where, denotes the scalar product in R 2. Setting 2η = i η, we have Therefore by.8 B + Â, i B Â = 2 η..2 q = q 2 + η + 4 η η + 2η A η B 2. The operator L x, D given by.0 has the Laplace operator as the principal part, the coefficients of x is A 2 + B η x 2, the coefficient of x 2 is ib 2 A η x, and the coefficient of the zero order term is given by the right-hand side of.2. By.7 we have that η ν Γ e = 0 and η eγ = C where the function Cx is equal to constant on each connected component of Γ. Let us show that the function η is continuous. Our proof is by contradiction. Suppose that η is discontinuous say along the curve Σ j. Let the function u 2 H be a solution to the following boundary value problem.3 L 2 x, Du 2 = 0 in, u 2 Γ0 = 0. Assume in addition that u 2 is not identically equal to zero on Σ j. Let Γ be one connected component of the set Γ and C eγ = Ĉ. Without loss of generality we may assume that Ĉ = 0. Indeed if Ĉ 0, then we replace η by the function η Ĉ. Since the partial Cauchy data of the operators L x, D and L 2 x, D are the same, there exists a solution u to the following boundary value problem.4 L x, Du = 0 in, u = u 2 on, Then the function v = e η u 2 verifies u ν = u 2 ν on Γ. L x, Dv = 0 in 0, v Γ0 = 0. Since on η = η = 0 on Γ ν we have that v u. However u H and v is discontinuous along one part of Σ j. Thus we arrive at a contradiction. Let us show that C 0. Suppose that there exists another connected component of Γ 2 of the set Γ such that C eγ2 0. Suppose that the functions u, u 2 satisfy.3 and.4 and u not identically zero. Γ2 Then the function v = e η u 2 verifies L x, Dv = 0 in, v Γ0 = 0.

6 6 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO Moreover, since on η = η = 0 on Γ ν, we have that v v = u, ν = u on ν Γ. By the uniqueness of the Cauchy problem for second-order elliptic equations, we have v u. In particular v = u on Γ 2. Since u = u 2 on, this implies that e η eγ2 =. We arrived at a contradiction. The proof of the corollary is completed. We now apply our result to the case of the magnetic Schrödinger operator. Denote à = Ã, Ã2, where Ãj are real-valued, à = à iã2, rot à = A e 2 x A e x 2, D k = i x k. Consider the magnetic Schrödinger operator 2.5 L ea,eq x, D = D k + Ãk 2 + q. Let us define the following set of partial Cauchy data { C ea,eq = u eγ, u } ν Γ e L ea,eq x, Du = 0 in, u Γ0 = 0, u H. k= For the case of full data in two dimensions, it is known that there is a gauge invariance in this problem and we can recover at best the curl of the magnetic field [29]. The same is valid for the three dimensional case with partial Cauchy data [2]. We prove here that the converse holds in two dimensions. Corollary.2. Let real-valued vector fields Ã, Ã2 C 5+α and complex-valued potentials q, q 2 C 4+α with some α 0, satisfy C ea,eq = C ea 2,eq 2. Then q = q 2, rot à = rot Ã2 in and à = Ã2 on Γ. Proof. A straightforward calculation gives.6 L ea,eq x, D = + 2 i à + 2 x i Ã2 + x à i = + 2 i à + 2 i à + 2 i à + Ã2 + q x i x 2 à rot à + à 2 + q. Then the operator L ea,eq x, D is a particular case of. with the metric g = {δ ij } A = i Ã, B = Ã, q = 2 A e rot i i Ã+ à 2 + q. Suppose that Schrödinger operators with the vector fields Ã, à 2 and the potentials q, q 2 have the same partial Cauchy data. Then.8 gives rot à rot Ã2 + q 2 q 0 and.9 gives.7 Using the identity 2 i 2 à 2 Ã2 2 à + 2 Ã2 + rot i i i i à rot Ã2 + q 2 q 0. A 2 A i = 2rot Ã, we transform.7 to the form rot à rot Ã2 + q 2 q 0.

7 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 7 The proof of the corollary is completed. There is another way to define partial Cauchy data for the Schrödinger operator. { } u Ĉ ea,eq = u eγ, + iã, νu eγ L ea,eq x, Du = 0 in, u Γ0 = 0, u H. ν Corollary.3. Let real-valued vector fields Ã, Ã2 C 5+α and complex-valued potentials q, q 2 C 4+α with some α 0, satisfy Ĉ e A,eq = Ĉ e A 2,eq 2. Then q = q 2, and rot à = rot Ã2 in. Proof. Suppose that there exist two vector fields and potentials Ãj, q j such that Ĉ e A,eq = Ĉ ea 2,eq 2. Consider a complex valued function η C 6+α, η eγ = 0 such that iν, à Ã2 = i η ν on Γ. Then C ea,eq = C ea 2 +i η,eq 2. Applying Corollary.2, we finish the proof. Corollaries.2 and.3 were new in Nov when the second author posed a preliminary version of this manuscript with the proof of Theorem.3, i.e. the metric g is the Euclidean metric. Since then proofs for the magnetic Schrödinger equation and full data in the Euclidean setting was given in [?] and in the Riemann surface case in [?]. In two dimensions, Sun proved in [29] that for measurements on the whole boundary, the uniqueness holds assuming that both the magnetic potential and the electric potential are small. Kang and Uhlmann proved global uniqueness for the case of measurements on the whole boundary for a special case of the magnetic Schrödinger equation, namely the Pauli Hamiltonian [8]. In dimensions n 3, global uniqueness was shown in [25] for the case of full data. The regularity assumptions in the result were improved in [26] and [27]. The case of partial data was considered in [2], based on the methods of [9] and [8], with an improvement on the regularity of the coefficients in [2]. Our main theorem implies that the partial Cauchy data can uniquely determine any two of A, B, q. First we can prove that A and B are uniquely determined if q is known. Consider the operator.8 Lx, Du = u + ax u + bx u + qxu. x x 2 Here a, b, q are complex-valued functions. Let us define the following set of partial Cauchy data { C a,b = u eγ, u ν Γ e u + ax u + bx u } + qxu = 0 in, u Γ0 = 0, u H. x x 2 We have Corollary.4. Let α 0, and two pairs of complex-valued coefficients a, b C 5+α C 5+α and a 2, b 2 C 5+α C 5+α satisfy C a,b = C a 2,b2. Then a, b a 2, b 2. Proof. Taking into account that operator.8 in the form x = + and x 2 Lx, Du = u + ax + ibx u = i, we can rewrite the + ax ibxu + qxu.

8 8 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO For the pairs a, b and a 2, b 2, let the corresponding operators defined by.8 have the same partial Cauchy data. Denote 2A k x = a k x + ib k x and 2B k x = a k x ib k x. By.8, we have A A 2 B B 2 A A A 2 B 2 = 0 in, 2 B B 2 A A 2 B B B 2 A 2 = 0 in. Applying to equation.9 the operator 2 and to equation.20 the operator 2, we have.2.22 By.7 A A 2 2 B B 2 A + A A 2 B 2 = 0 in, B B 2 2 A A 2 B + B B 2 A 2 = 0 in. A A 2 eγ = B B 2 eγ = 0. Using these identities and equations.9 and.20, we obtain A A 2 eγ = B B 2 eγ = 0. ν ν The uniqueness of the Cauchy problem for the system can be proved in the standard way by using a Carleman estimate e.g., [5]. Therefore we have A = A 2 and B = B 2 in. We remark that Corollary.4 generalizes the result of [] uniqueness is obtained assuming that the measurements are made on the whole boundary. In dimensions n 3, global uniqueness was shown in [0] for the case of full data. Similarly to Corollary.4, we can prove that the partial Cauchy data can uniquely determine a potential q and one of the coefficients A and B in.. Corollary.5. For j =, 2, let A j, B j, q j C 5+α C 5+α C 4+α for some α 0, and be complex-valued. We assume either A = A 2 or B = B 2 in. Then C I,A,B,q = C I,A2,B 2,q 2 implies A, B, q = A 2, B 2, q 2. Corollaries.4 and.5 mean that the partial Cauchy data on Γ uniquely determine any two coefficients of the three coefficients of a second-order elliptic operator whose principal part is the Laplacian. The proof of Theorem. uses isothermal coordinates, the Carleman estimate obtained in section 2, and Theorem.3. In this case we need to prove a new Carleman estimate with degenerate harmonic weights to construct appropriate complex geometrical optics solutions. These solutions are different than the case of zero magnetic potential. The new form of these solutions considerably complicate the arguments, especially the asymptotic expansions needed to analyze the behavior of the solutions. In Section 2 we prove the Carleman estimate which we need. In Section 3 we state the estimates and asymptotics which we will use in the construction of the complex geometrical optics solutions. This construction is done in Section 4. The proof of Theorem.3 is completed in Section 5. In section 7 and 8 we discuss some technical lemmas needed in the previous sections.

9 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 9 2. Carleman estimate Notations. We use throughout the paper the following notations. i =, x, x 2, ξ, ξ 2 R, z = x + ix 2, ζ = ξ + iξ 2, z denotes the complex conjugate of z C, D k = i x k, β = β, β 2 where β j N +. We identify x = x, x 2 R 2 with z = x + ix 2 C. We set z = = 2 x i x 2, z = = 2 x +i x 2, O ɛ = {x distx, ɛ}. Throughout the paper we use both notations z and, etc. and for example we denote 2 z = 2. We 2 say that a function ax is antiholomorphic in if z ax 0. The tangential derivative on the boundary is given by = ν 2 x ν x 2, with ν = ν, ν 2 the unit outer normal to. The ball of radius δ centered at x is denoted by B x, δ = {x R 2 x x < δ}. The corresponding sphere is denoted by S x, δ = {x R 2 x x = δ}. If f : R 2 R is a function, then f is the Hessian matrix with entries 2 f x i x j. Let 2 = H k, 2 + H k 2k 2 L 2 be the norm in the standard semiclassical Sobolev space with inner product given by, H k, =, H k + 2k, L 2. For any positive function d we introduce the space L 2 d = {vx v L 2 d = d v 2 dx 2 < }. LX, Y denotes the Banach space of all bounded linear operators from a Banach space X to another Banach space Y. By o X we denote a function f, such that f, κ X = o as +. Finally for κ any x we introduce the left and right tangential derivatives as follows: fls f x D + xf = lim s +0 s where l0 = x, ls is a parametrization of near x, s is the length of the curve, and we are moving clockwise as s increases; f ls f x D xf = lim s 0 s where l0 = x, ls is the parametrization of near x, s is the length of the curve, and we are moving counterclockwise as s increases. For some α 0, we consider a function Φz = ϕx, x 2 + iψx, x 2 C 6+α with real-valued ϕ and ψ such that 2. z = 0 in, Im Φ Γ = 0 0 where Γ 0 is an open set on such that Γ 0 Γ 0. Denote by H the set of all the critical points of the function Φ H = {z z = 0}. Assume that Φ has no critical points on Γ, and that all critical points are nondegenerate: 2 Φ 2.2 H Γ 0, z 0, z H. 2 Then Φ has only a finite number of critical points and we can set: 2.3 H \ Γ 0 = { x,..., x l }, H Γ 0 = { x l+,..., x l+l }. The following proposition was proved in [6].

10 0 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO Proposition 2.. Let x be an arbitrary point in. There exists a sequence of functions {Φ ɛ } ɛ 0, satisfying 2. such that all the critical points of Φ ɛ are nondegenerate and there exists a sequence { x ɛ }, ɛ 0, such that x ɛ H ɛ = {z ɛ z = 0}, x ɛ x as ɛ +0. Moreover for any j from {,..., N } we have H ɛ γ j = if γ j Γ, H ɛ γ j Γ 0 if γ j Γ =, Im Φ ɛ x ɛ / {Im Φ ɛ x x H ɛ \ { x ɛ }} and Im Φ ɛ x ɛ 0. In order to prove.7 we need the following proposition. Proposition 2.2. Let ˆΓ Γ be an arc with left endpoint x and right endpoint x + oriented clockwise. For any x Int Γ there exists a function Φz which satisfies 2., 2.2, Im Φ \Γ = 0 and 2.4 x G = {x ˆΓ Im Φ x = 0}, card G <, Im Φx 0 x G \ {x, x + }, Moreover 2.6 Im Φ x Im Φx x G \ { x} and Im Φ x D + x 6 Im Φ 0, D x + 6 Im Φ 0. Proof. Denote ˆΓ 0 = \ ˆΓ. Let x, x + be points such that the arc [ x, x + ] x, x + and x x, x + be an arbitrary point and x 0 be another fixed point from the interval x, x +. We claim that there exists a pair ϕ, ψ C 6 C 6 which solves the system of Cauchy-Riemann equations in such that A A ψ ˆΓ 0 = 0, ϕ γ j \ˆΓ > 0 if γ j ˆΓ, ψ x = 0, 2 ψ x 0, D + x 6 ψx 0, D x + 6 ψx 0, B The restriction of the function ψ to the arc [ x, x + ] is a Morse function, C ψ > 0 on x, x ], ψ < 0 on [ x +, x +, D ψ x / {ψx x \ { x}, ψ x = 0}, E if γ j ˆΓ =, then the restriction of the function ϕ on γ j has only two nondegenerate critical points.

11 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR Such a pair of functions may be constructed in the following way. Let γ ˆΓ and γ j ˆΓ = for all j {2,... N }. First, by Corollary 7. in the Appendix, for some α 0,, there exists a solution ϕ, ψ C 6+α C 6+α to the Cauchy-Riemann equations with the following boundary data ψ \[x0,bx + ] = ψ, ϕ γ 0 \[x 0,bx + ] < β < 0 and such that if γ j ˆΓ = the function ϕ has only two nondegenerate critical points located on the contour γ j. The function ψ has the following properties: ψ ˆΓ 0 = 0, ψ > ψ 0 on x, x ], < 0 on [ x +, x +. The function ψ on the set [ x, x 0 ] has only one critical point x and ψ x 0. On the set x 0, x + the Cauchy data is not fixed. The restriction of the function ψ on [x 0, x + ] can be approximated in the space C 6+α [x 0, x + ] by a sequence of Morse functions {g ɛ } ɛ 0, such that and k ψx = k g ɛ x x { x +, x 0 }, k {0,,..., 6}, ψ x / {g ɛ x g ɛx = 0}. Let us consider some arc J x, x. On this arc we have e ψ 2.8 ψ > β > 0 on J for some positive β. > 0, say, Let ϕ ɛ, ψ ɛ C 6+α C 6+α be a solution to the Cauchy-Riemann equations with boundary data ψ ɛ = 0 on \ J [x 0, x + ] and ψ ɛ = g ɛ ψ on [x 0, x + ] and on J the Cauchy data is chosen in such a way that 2.9 ψ ɛ C 6+α + ϕ ɛ C 6+α 0 as g ɛ ψ C 6+α [x 0,bx + ] 0. By 2.8, 2.9 for all small positive ɛ, the restriction of the function ψ + ψ ɛ to satisfies ψ + ψ ɛ Γ 0 = 0, ψ + ψ ɛ γ0 \[x ν 0,bx + ] < 0, ψ + ψ ɛ > 0 on [x, x ], ψ + ψ ɛ < 0 on [ x +, x + ], ψ + ψ ɛ [x0,bx + ] = g ɛ, ψ + ψ ɛ [bx,x 0 ] = ψ. If j 2 then the restriction of the function ϕ ɛ + ϕ on γ j has only two critical points located on the contour γ j ˆΓ 0. These critical points are nondegenerate if ɛ is sufficiently small. Therefore the restriction of the function ψ + ψ ɛ on ˆΓ has a finite number of a critical points. Some of these points may be the critical points of ψ + ψ ɛ considered as a function on. We change slightly the function ψ + ψ ɛ such that all of its critical points are in. Suppose that function ψ + ψ ɛ has critical points on ˆΓ. Then these critical points should be among the set of critical points of the function g ɛ, otherwise it would be the point x. We

12 2 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO denote these points by x,..., x m. Let ϕ, ψ C 6+α C 6+α be a solution to the Cauchy-Riemann problem 7. with the following boundary data ψ Γ 0 = 0, ψ x =, ψ G\{bx} = 0, ψ ν γ 0 \J > 0. For all small positive ɛ the function ψ + ψ ɛ + ɛ ψ does not have a critical point on and the restriction of this function on Γ has a finite number of nondegenerate critical points. Therefore we take ϕ + ϕ ɛ + ɛ ϕ, ψ + ψ ɛ + ɛ ψ as the pairs of functions satisfying A - E. The function ϕ+iψ with pair ϕ, ψ satisfying conditions A-E satisfies all the hypotheses of Proposition 2.2 except that some of its critical points might possibly be degenerate. In order to fix this problem we consider a perturbation of the function ϕ + iψ which is constructed in the following way. By Proposition 7.2, there exists a holomorphic function w in such that 2.0 Im w Γ 0 = 0, w H0 = w 2 w H 0 = 0, 2 H 0 0. Denote Φ δ = ϕ + iψ + δw. For all sufficiently small positive δ, we have H 0 H δ {x Φ δx = 0}. We now show that for all sufficiently small positive δ, all critical points of the function Φ δ are nondegenerate. Let x be a critical point of the function ϕ + iψ. If x is a nondegenerate critical point, by the implicit function theorem, there exists a ball B x, δ such that the function Φ δ in this ball has only one nondegenerate critical point for all small δ. Let x be a degenerate critical point of ϕ + iψ. Without loss of generality we may assume that x = 0. In some neighborhood of 0, we have δ = k= c kz k+bk δ k= b kz k for some natural number k and some c 0. Moreover 2.0 implies b 0. Let x,δ, x 2,δ H δ and z δ = x,δ + ix 2,δ 0. Then either 2. z δ = 0 or z b k δ = δb /c + oδ as δ 0. Therefore 2 Φ δ 2 z δ 0 for all sufficiently small δ. Let α 0, and A, B C 6+α be two complex-valued solutions to the boundary value problem A = A in, Im A Γ 0 = 0, 2 B = B in, Im B Γ 0 = 0. Consider the following boundary value problem a 2.3 = 0 in, d = 0 in, aea + de B Γ0 = β. The existence of such functions az and dz is given by the following proposition. Proposition 2.3. Let α 0,, A and B be as in 2.2. If β C 5+α Γ has at least one solution a, d C 5+α C 5+α such that 2.4 a, d C 5+α C 5+α C β C 5+α Γ 0.

13 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 3 If β H 2 Γ 0, then 2.3 has at least one solution a, d H H such that 2.5 a, d H H C 2 β H 2 Γ0. Proof. Let be a domain in R 2 with smooth boundary such that and there exits an open subdomain Γ 0 satisfying Γ 0 Γ 0. Denote Γ = \ Γ 0. We extend A, B to Γ 0 keeping the regularity and we extend β to Γ 0 in such a way that β H 2 e Γ0 C 3 β H 2 Γ0 or β C 5+α e Γ 0 C 3 β C 5+α Γ 0 where the constant C 3 is independent of β. By the trace theorem there exist a constant C 4 independent of β, and a pair r, r such that re A + re B eγ0 = β and if β H 2 Γ 0 then r, r H H and r, r H H C 4 β H 2 Γ0. Similarly if β C 5+α Γ 0 then r, r C 5+α C 5+α and r, r C 5+α C 5+α C 5 β C 5+α Γ 0. Let f = r and f = er. For any ɛ from 0, consider the extremal problem J ɛ p, p = p, p 2 L 2 e + ɛ p f 2 L 2 e + p ɛ f 2 L 2 e inf, p, p K, where K = {h, h 2 L 2 L 2 h e A + h 2 e B eγ0 = 0}. Denote the solution to this extremal problem as p ɛ, p ɛ. Then Hence 2.6 p ɛ, p ɛ, δ, δ L 2 e + ɛ p ɛ Denote P ɛ = ɛ pɛ 2.7 P ɛ = p ɛ, f, P ɛ = ɛ epɛ J ɛp ɛ, p ɛ δ, δ = 0 δ, δ K. δ f, L 2 e + p ɛ δ f, ɛ L 2 e = 0 δ, δ K. f. From 2.6 we obtain P ɛ = p ɛ, P ɛ Γ = P ɛ Γ = 0, ν + iν 2 P ɛ e B ν iν 2 P ɛ e A eγ0 = 0. We claim that there exists a constant C 6 independent of ɛ such that 2.8 P ɛ, P ɛ H e C 6 p ɛ, p ɛ L 2 e + P ɛ, P ɛ L 2 e. It clearly suffices to prove the estimate 2.8 locally assuming that supp p ɛ, p ɛ is in a small neighborhood of zero and the vector 0, is orthogonal to on the intersection of this neighborhood with the boundary. Using a conformal transformation we may assume that supp P ɛ, supp P ɛ {x = 0}. In order to prove 2.8 we consider the system of equations 2.9 u x 2 + B u x = F, supp u B0, δ {x x 2 0}.

14 4 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO Here u = u, u 2, u 3, u 4 = Re P ɛ, Im P ɛ, Re P ɛ, Im P ɛ, F = 2Re p ɛ, Im p ɛ, Re p ɛ, Im p ɛ, B = The matrix B has two eigenvalues ±i and four linearly independent eigenvectors: q 3 = 0, 0,, i, q 4 =, i, 0, 0 corresponding to the eigenvalue i, q =, i, 0, 0, q 2 = 0, 0,, i corresponding to the eigenvalue i. We set r = ν e B, ν 2 e B, ν e A, ν 2 e A, r 2 = ν 2 e B, ν e B, ν 2 e A, ν e A. Consider the matrix D = {d jl } where d jl = r j q l. We have ν iν D = 2 e B ν iν 2 e A ν 2 + iν e B ν 2 + iν e A. Since the Lopatinski determinant det D 0 we obtain 2.8 see e.g. [33]. Next we need to get rid of the second term in the right hand side of 2.8. Suppose that for any C one can find ɛ such that the estimate P ɛ, P ɛ H e C p ɛ, p ɛ L 2 e fails. That is, there exist a sequence ɛ k + and a sequence {C ɛk } such that lim ɛk +0 C ɛk = + and p ɛk, p ɛk P ɛk, P <. ɛk H e C ɛk L 2 e We set Q ɛk, Q ɛk = P ɛk, P ɛk / P ɛk, P ɛk H e and q ɛ k, q ɛk = p ɛk, p ɛk / P ɛk, P ɛk H e. Then q ɛk, q ɛk L 2 e 0 as ɛ k 0. Passing to the limit in 2.7 we have Q = 0 in, Q = 0 in, Q Γ = Q Γ = 0. By the uniqueness for the Cauchy problem for the operator z we conclude Q Q 0. On the other hand, since Q ɛk, Q ɛk H e =, we can extract a subsequence, denoted the same, which is convergent in L 2. Therefore the sequence {Q ɛk, Q ɛk } converges to zero in L 2 L 2. By 2.8, we have /C 7 q ɛk, q ɛk L 2 e + Q ɛ k, Q ɛk L 2 e. Therefore lim inf ɛk 0 Q ɛk, Q ɛk L 2 e 0, and this is a contradiction. Hence P ɛk, P ɛk H e C 8 p ɛk, p ɛk L 2 e, ɛ > 0. Let us plug in 2.6 the function p ɛk, p ɛk instead of δ, δ. Then, by the above inequality, in view of the definitions of P ɛk and P ɛk, we have p ɛk, p ɛk 2 L 2 e C 9f, f, P ɛk, P ɛk L 2 e C 0 f, f L 2 e P ɛ k, P ɛk L 2 e C f, f L 2 e p ɛ k, p ɛk L 2 e.

15 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 5 This inequality implies that the sequence p ɛk, p ɛk is bounded in L 2 and p ɛ k, p ɛ k f, f in L 2 L 2. Then we construct a solution to 2.3 such that 2.20 p, p L 2 e C 2 f, f L 2 e. Observe that we can write the boundary value problem p = f in, p = f in, pe A + pe B eγ0 = 0 in the form of system 2.9 with u = Re p, Im p, Re p, Im p, F = 2Re f, Im f, Re f, Im f. We set r = e A, e A, e B, e B, r 2 = e A, e A, e B, e B. Consider the matrix D = {d jl } where d jl = r j q l. We have e D = B e A. e B Since the Lopatinski determinant det D 0 the estimate 2.20 imply 2.4 and 2.5 see e.g., [33] Theorem This completes the proof of the proposition. The following proposition was proven in [6]: Proposition 2.4. Let Φ satisfy 2.,2.2 and the function C = C +ic 2 belongs to C. Let f L 2, and ṽ H be a solution to ṽ ṽ + Cṽ = f in or ṽ be a solution to e A ṽ ṽ + Cṽ = f in. In the case 2.2 we have ṽ x iim Cṽ 2 L In the case 2.22 we have ṽ x iim z Cṽ L ϕ, ν ν C + ν 2 C 2 ṽ 2 dσ C + C 2 ṽ 2 dx x x 2 +Re i ν 2 ν ṽ ṽdσ x x 2 + ṽ Re i x 2 Cṽ 2 L 2 = f 2 L 2. ϕ, ν ν C ν 2 C 2 ṽ 2 dσ C C 2 ṽ 2 dx x x 2 +Re i ν 2 + ν ṽ ṽdσ x x 2 + i ṽ x 2 Re z Cṽ 2 L 2 = f 2 L 2.

16 6 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO Consider the boundary value problem Kx, Du = 4 + 2A + 2B u = f u = 0. in, For this problem we have the following Carleman estimate with boundary terms. Proposition 2.5. Suppose that Φ satisfies 2., 2.2, u H 0 and A L + B L K. Then there exist 0 = 0 K, Φ and C 3 = C 3 K, Φ independent of u and such that for all > ue ϕ 2 L 2 + ueϕ 2 H + u ν eϕ 2 L 2 Γ ueϕ 2 L 2 C 3 Kx, Due ϕ 2 L 2 + u eγ ν 2 e 2ϕ dσ. Proof. Denote ṽ = ue ϕ, Kx, Du = f. Observe that ϕx, x 2 = Φz + Φz. Therefore e ϕ e ϕ ṽ = 2 2 ṽ = 2 2 ṽ = f = f 2B u 2Au eϕ. Assume now that u is a real valued function. Denote w = 2 ṽ. Thanks to the zero Dirichlet boundary condition for u we have w = 2 ṽ = ν + iν 2 ṽ ν. Let C be some smooth function in such that 2 C = Cx = C x + ic 2 x in, Im C = 0 on Γ 0, where C = C, C 2 is the smooth function in such that div C = in, ν, C = on Γ 0. By Proposition 2.4 we have the following integral equalities: wenc iim x z + NC wenc L 2 ϕ, ν + Nν C + ν 2 C 2 ṽ ν enc 2 dσ +N we NC dx + Re iν 2 ν we NC we x x NC dσ i we NC x 2 Re + NC wenc 2 L 2 = fe ϕ+nc 2 L 2.

17 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 7 We now simplify the integral Re i ν 2 x ν x 2 we NC w e NC dσ. We recall that ṽ = ue ϕ and w = ν + iν 2 ev = ν ν + iν 2 u ν eϕ. Denote R + ip = ν + iν 2 e NImC. Therefore 2.28 Re iν 2 ν we NC we x x NC dσ = 2 Re iν 2 ν [R + ip u x x 2 ν eϕ+nre C ]R ip u ν eϕ+nre C dσ = Re i[ν 2 ν R + ip ] ṽen C 2 R ip dσ. x x 2 ν Using the above formula we obtain wenc iim x z + NC wenc L 2 ϕ, ν + Nν C + ν 2 C 2 ṽ ν enc 2 dσ +N we NC 2 dx + Re i[ν 2 ν R + ip ] ṽenre C 2 R ip dσ x x 2 ν + we NC 2.29 Re i x 2 + NC wenc 2 L 2 = fe ϕ+nc 2 L 2 Taking the parameter N sufficiently large positive and taking into account that the function R + ip is independent of N on Γ 0 we conclude from 2.29 ϕ, ν + N 2 ν C + ν 2 C 2 ṽ 2.30 ν enc 2 dσ + N we NC 2 dx fe ϕ e NC 2 L 2 + CN ṽ Γ ν enc 2 dσ A simple computation gives 4 ṽenre C 2 L z ṽenre C 2 C L 2 = 2ṽeNRe z ṽenre C 2 L 2 = e NRe C 2 ṽ z + 2NRe C ṽ 2 L z we NC 2 L 2 + CN ueϕ 2 L 2. Since by assumption 2.2, the function Φ has zeros of at most second order, there exists a constant C 4 > 0 independent of such that 2.32 ṽe NRe C 2 L 2 C 4 ṽe NRe C 2 H + 2 ṽenre C 2 L 2. Therefore by there exists N 0 > 0 such that for any N > N 0 there exists 0 N that ϕ, ν + N 2 ν C + ν 2 C 2 ṽ ν enc 2 dσ + N we NC 2 dx 2 ṽe NRe C 2 L 2 + ṽenre C 2 H + 2 ṽenre C 2 L fe ϕ e NC 2 L 2 + C 5N Γ ṽ ν enc 2 dσ

18 8 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO In order to drop the assumption that u is the real valued function we obtain the 2.33 separately for the real and imaginary parts of u and combine them. This concludes the proof of the proposition. As a corollary we derive a Carleman inequality for the function u which satisfies the integral equality 2.34 u, Kx, D w L 2 + f, w H, + ge ϕ, e ϕ w H 2, e Γ = 0 for all w X = {w H w Γ0 = 0, Kx, D w L 2 }. We have Corollary 2.. Suppose that Φ satisfies 2., 2.2, f H, g H 2 Γ, u L 2 and the coefficients A, B {C C C C K}. Then there exist 0 = 0 K, Φ and C 6 = C 6 K, Φ, independent of u and, such that for solutions of 2.34: 2.35 ue ϕ 2 L 2 C 6 fe ϕ 2 H, + geϕ 2 H 2, e Γ 0. Proof. Let ɛ be some positive number and dx be a smooth positive function of Γ which blow up like for any y Γ. Consider the extremal problem xy J ɛ w = 2 weϕ 2 L 2 + 2ɛ Kx, D w ue 2ϕ 2 L weϕ 2 L 2 d e Γ inf, 2.37 w X = {w H 2 Kx, D w L 2, w Γ0 = 0}. There exists a unique solution to 2.36, 2.37 which we denote by ŵ ɛ. By Fermat s theorem J ɛŵ ɛ [δ] = 0 δ X. Using the notation p ɛ = ɛ Kx, D ŵ ɛ ue 2ϕ this implies 2.38 Kx, Dp ɛ + ŵ ɛ e 2ϕ = 0 in, p ɛ = 0, By Proposition 2.5 we have 2.39 p ɛ ν e Γ = d ŵɛ e2ϕ. p ɛ e ϕ 2 L 2 + p ɛe ϕ 2 H + p ɛ ν eϕ 2 L 2 Γ p ɛe ϕ 2 L 2 C ŵ ɛ e ϕ 2 L 2 + ŵ ɛ 2 e 2ϕ dσ 2C 7 J ɛ ŵ ɛ. eγ Taking the scalar product of equation 2.38 with ŵ ɛ we obtain 2J ɛ ŵ ɛ + ue 2ϕ, p ɛ L 2 = 0. Applying to the second term of the above equality estimate 2.39 we have J ɛ ŵ ɛ C 8 ue ϕ 2 L 2. Using this estimate we pass to the limit in 2.38 as ɛ goes to zero. We obtain 2.40 Kx, Dp + ŵe 2ϕ = 0 in, p = 0, 2.4 Kx, D ŵ ue 2ϕ = 0 in, ŵ Γ0 = 0, p ν e Γ = d ŵ e2ϕ,

19 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 9 and 2.42 ŵe ϕ 2 L 2 + ŵeϕ 2 L 2 e Γ C 9 ue ϕ 2 L 2. Since ŵ L 2 we have p H 2 and by the trace theorem p ν H 2. The relation 2.40 implies ŵ H 2 Γ. But since ŵ L 2 d Γ and ŵ Γ0 = 0 we have ŵ H 2. By we get 2.43 ŵe ϕ H 2, C 20 2 ue ϕ L 2. Taking the scalar product of 2.4 with ŵe 2ϕ and using the estimates 2.43, 2.42 we get 2.44 ŵeϕ 2 L 2 + ŵeϕ 2 L 2 + ŵeϕ 2 C 2 ue ϕ 2 H 2, Γ e L 2. From this estimate and a standard duality argument, the statement of Corollary 2. follows immediately. Consider the following problem 2.45 Lx, Du = fe ϕ in, u Γ0 = ge ϕ. We have Proposition 2.6. Let A, B C 5+α, q L and ɛ, α be a small positive numbers. There exists 0 > 0 such that for all > 0 there exists a solution to the boundary value problem 2.45 such that 2.46 ue ϕ L 2 + ue ϕ L 2 C 22 f L 2 + g H 2, Γ 0. Let ɛ be a sufficiently small positive number. If suppf G ɛ = {x distx, H > ɛ} and g = 0 then there exists 0 > 0 such that for all > 0 there exists a solution to the boundary value problem 2.45 such that 2.47 ue ϕ L 2 + ue ϕ L 2 C 23 ɛ f L 2. Proof. First we reduce the problem 2.45 to the case g = 0. Let rz be a holomorphic function and rz be an antiholomorphic function such that e A r + e B r Γ0 = g where A, B C 6+α are defined as in 2.2. The existence of such functions r, r follows from Proposition 2.3, and these functions can be chosen in such a way that We look for a solution u in the form where r H + r H C 24 g H 2 Γ0. u = e A+Φ r + e B+Φ r + ũ, 2.48 Lx, Dũ = fe ϕ in, ũ Γ0 = 0 and f = f q 2 A ABeA re iψ q 2 B ABeB re iψ.

20 20 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO In order to prove 2.46 we consider the following extremal problem: 2.49 Ĩ ɛ u = 2 ueϕ 2 H, + 2ɛ Lx, Du fe ϕ 2 L ueϕ 2 H 2, e Γ inf, 2.50 u Y = {w H w Γ0 = 0, Lx, Dw L 2 }. There exists a unique solution to problem 2.49, 2.50 which we denote as û ɛ. By Fermat s theorem 2.5 Ĩ ɛû ɛ [δ] = 0 δ Y. Let p ɛ = ɛ Lx, Dû ɛ fe ϕ. Applying Corollary 2. we obtain from p ɛe ϕ 2 L 2 C 25 û ɛ e ϕ 2 H, + û ɛe ϕ 2 H 2, e Γ 2C 25Ĩɛû ɛ. Substituting in 2.5 with δ = û ɛ we get 2Ĩɛû ɛ + fe ϕ, p ɛ L 2 = 0. Applying to this equality estimate 2.52 we have Ĩ ɛ û ɛ C 26 f 2 L 2. Using this estimate we pass to the limit as ɛ +0. We obtain 2.53 Lx, Du fe ϕ = 0 in, u Γ0 = 0, and 2.54 ue ϕ 2 H, + ueϕ 2 L 2 e Γ C 27 f 2 L 2. Since f L 2 C 28 f L 2 + g H 2 Γ0, inequality 2.54 implies In order to prove 2.47 we consider the following extremal problem 2.55 Jɛ u = 2 ueϕ 2 L 2 + 2ɛ Lx, Du feϕ 2 L ueϕ 2 L 2 d e Γ inf, 2.56 u X = {w H 2 Lx, Dw L 2, w Γ0 = 0}. Here dx is a smooth positive function of Γ which blow up like for any y Γ. xy 8 There exists a unique solution to problem 2.55, 2.56 which we denote as û ɛ. By Fermat s theorem J ɛû ɛ [δ] = 0 δ X. This equality implies 2.57 Lx, D p ɛ + û ɛ e 2ϕ = 0 in, p ɛ = 0, By Proposition 2.5 p ɛ ν e Γ = d ûɛ e2ϕ. p ɛe ϕ 2 H, + p ɛ ν eϕ 2 L 2 Γ p ɛe ϕ 2 L 2

21 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR C 29 û ɛ e ϕ 2 L 2 + û ɛ 2 e 2ϕ dσ C 30 Jɛ û ɛ. eγ Taking the scalar product of equation 2.57 with û ɛ we obtain 2 J ɛ û ɛ + fe ϕ, p ɛ L 2 = 0. Applying to this equality estimate 2.58 we have Jɛ û ɛ C 3 f 2 L 2. Using this estimate we pass to the limit in We conclude that 2.60 Lx, D p + ue 2ϕ = 0 in, p = 0, p ν Γ e = u e2ϕ, 2.6 Lx, Du fe ϕ = 0 in, u Γ0 = 0. Moreover 2.59 implies ue ϕ 2 L 2 + ueϕ 2 L 2 Γ e C 32 f 2 L 2. This finishes the proof of the proposition. 3. Estimates and Asymptotics In this section we prove some estimates and obtain asymptotic expansions needed in the construction of the complex geometrical optics solutions in Section 4. Consider the operator L x, D = 4 + 2A + 2B + q = 2 + B 2 + A + q 2 A A B = 2 + A 2 + B + q 2 B 3. A B. Let A, B C 6+α, with some α 0,, satisfy A = A in, Im A Γ0 = 0, 2 B = B in, Im B Γ0 = 0 and let A 2, B 2 C 6+α be defined similarly. Observe that 2 + A e A = 0 in, 2 + B e B = 0 in. Therefore if az, Φz are holomorphic functions and bz is an antiholomorphic function, we have L x, De A ae Φ = q 2 A A B e A ae Φ, Let us introduce the operators: g = 2πi L x, De B be Φ = q 2 B A B e B be Φ. gξ, ξ 2 ζ z dζ dζ = gξ, ξ 2 π ζ z dξ dξ 2,

22 22 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO g = gξ, ξ 2 2πi ζ z dζ dζ = π We have e.g., p.47, 56, 72 in [32]: gξ, ξ 2 ζ z dξ dξ 2. Proposition 3.. A Let m 0 be an integer number and α 0,. Then z LC m+α, C m+α+. B Let p 2 and < γ < 2p 2p CLet < p <. Then,. Then z, LL p, L γ. LL p, Wp. Assume that A, B, A, B satisfy 2.2. Setting T B g = e B e B g and P A g = e A e A g, for any g C α we have We define two other operators: 2 + BT Bg = g in, 2 + AP Ag = g in. 3.3 R,A g = 2 ea e ΦΦ ge A e ΦΦ, R,B g = 2 eb e ΦΦ ge B e ΦΦ for A, B, A, B satisfying 2.2. The following proposition follows from straightforward calculations. Proposition 3.2. Let g C α for some positive α. The function R,A g is a solution to R,Ag 2 R,Ag + AR,A g = g in. The function R,B g solves R,B g + 2 R,B g + B R,B g = g in., z Using the stationary phase argument e.g., Bleistein and Handelsman [4], we will show Proposition 3.3. Let g L and a function Φ satisfy 2.,2.2. Then lim + ge ΦzΦz dx = 0. Proof. Let {g k } k= C 0 be a sequence of functions such that g k g in L. Let ɛ > 0 be arbitrary. Suppose that ĵ is large enough such that g g b j L ɛ. Then 2 ge ΦzΦz dx g g b j eφzφz dx + g b j eφzφz dx. The first term on the right-hand side of this inequality is less then ɛ/2 and the second goes to zero as approaches to infinity by the stationary phase argument see e.g. [4]. We have Proposition 3.4. Let g C 2, g Oɛ = 0 and g H = 0. Then for any p < 3.6 R,Ag + g 2 z Φ + R,B g g L p 2 z Φ = o as +. L p

23 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 23 Proof. We give a proof of the asymptotic formula for R,B g. The proof for R,A g is similar. Let gζ, ζ = ge B. Then 2e B R,B g = eφφ gζ, ζ π ζ z eφζφζ dξ dξ 2 = eφφ gζ, ζ lim π δ +0 ζ z eφζφζ dξ dξ 2. \Bz,δ Let z = x + ix 2 and x = x, x 2 be not a critical point of the function Φ. Then 2e B R,B g = eφφ gζ, ζ ζ e ΦζΦζ 3.7 lim dξ dξ 2 π δ +0 \Bx,δ ζ z ζ Φζ = eφφ gζ, ζ lim e ΦζΦζ dξ dξ 2 π δ +0 \Bx,δ ζ z ζ ζ Φζ e ΦΦ π lim δ +0 Sx,δ Since g H = 0, we have 3.8 gζ, ζ C ζ ζ Φζ gζ, ζ ζ z l k= Hence, passing to the limit in 3.7 we get 2e B R,B g = eφφ π ζ z ζ Denote G x = ζz ζ egζ,ζ ζ Φζ ν i ν 2 2 ζ Φζ eφζφζ dξ dξ 2. g C ζ x k Lp p, 2. gζ, ζ ζ Φζ e ΦζΦζ dξ dξ 2 gz, z z Φz. e ΦζΦζ dξ dξ 2. By Proposition 3.3, we see that 3.9 G x 0 as + x. Denote ζ gζ, ζ T ξ, ξ 2 = ζ Φζ χ, where χ is the characteristic function of. Clearly T ξ, ξ G x dξ dξ 2 a.e. in. z ζ By 3.8 T belongs to L p R 2 for any p, 2. For any f L p R 2, we set I r fz = z ζ 2 r fζ, ζdξ dξ 2. R 2 Then, by the Hardy-Littlewood-Sobolev inequality, if r > and = r p q p < q <, then I r f L q R 2 C p,q f L p R 2. for <

24 24 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO Set r = 2. Then we have to choose =, that is, we can arbitrarily choose p > 2 close p q 2 to 2, so that q is arbitrarily large. Hence T dξ zζ dξ 2 belongs to L q with positive q. By 3.9, 3.0 and the dominated convergence theorem The proof of the proposition is finished. G 0 in L q q,. We now consider the contribution from the critical points. Proposition 3.5. Let Φ satisfy 2. and 2.2. Let g C 4+α for some α > 0, g Oɛ = 0 and g H = 0. Then there exist constants p k such that 3. ge ΦzΦz dx = l p 2 k e 2iψex k + o as +. 2 k= Proof. Let δ > 0 be a sufficiently small number and ẽ k C0 B x k, δ, ẽ k Bexk,δ/2. By the stationary phase argument l I = ge ΦΦ dx = ẽ k ge ΦΦ dx + o = 2 l k= e 2iψex k Bex k,δ k= Bex k,δ ẽ k ge ΦΦ2iψex k dx + o 2 as +. Since all the critical points of Φ are nondegenerate, in some neighborhood of x k one can take local coordinates such that Φ Φ 2iψ x k = z 2 z 2. Therefore l I = e 2iψex k q k e z2 z 2 dx + o as +, 2 k= B0,δ where q k C0B0, 4 δ and q k 0 = 0. Hence there exist functions r,k, r 2,k C0B0, 3 δ such that q k = 2zr,k + 2zr 2,k. Integrating by parts, one can decompose I as I = l e 2iψex k r,k r 2,k z2 ez2 dx + o = 2 k= l k= l k= e 2iψex k e 2iψex k B0,δ B0,δ B0,δ r,k r 2,k z2 0χxez2 dx q k e z2 z 2 dx + o 2 as +, where χ, q k C0B0, 2 δ, χ B0,δ /2 and q k 0 = 0. Hence there exist functions r,k, r 2,k C0B0, δ such that q k = 2z r,k + 2z r 2,k. Integrating by parts and applying Proposition 3.4 we obtain lim + B0,δ q k e z2 z 2 dx = l k= e 2iψex k lim + B0,δ r,k r 2,k ez2 z2 dx = 0.

25 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 25 Therefore 3. follows from a standard application of stationary phase. The proof of the proposition is completed. Proposition 3.6. Let 0 < ɛ < ɛ, a function Φ satisfy 2., 2.2 and O ɛ H \ Γ 0 =. Suppose that g C α H for some α 0,, g Oɛ = 0 and g H = 0. Then 3.2 R,B g L O ɛ + R,B g L O ɛ C ɛ, α g C α H. Moreover 3.3 R,B g L R,B g L 2 + R,B g L 2 C 2 ɛ, α g C α H. Proof. Denote g = ge B. Let x = x, x 2 be an arbitrary point from O ɛ and z = x + ix 2. Then π e ΦΦ g ge ΦΦ l = ζ z dξ ge ΦΦ dξ 2 = lim δ +0 ζ z dξ dξ 2. k= \Bex k,δ Integrating by parts and taking δ sufficiently small we have π e ΦΦ g = eg lim ζ δ lim δ lim δ +0 \ l k= Bex k,δ \ l k= Bex k,δ l k= Sex k,δ ζ z ζ e ΦΦ dξ dξ 2 g 2 Φ ζ 2 ζ z ζ 2 eφφ dξ dξ 2 ν i ν 2 g ζ z ζ e ΦΦ dσ. Since g H = 0 we have that g C 0 Sex k,δ δ α g C α. Using this inequality and the fact that all the critical points of Φ are nondegenerate we obtain 2 lim δ +0 l k= Sex k,δ Since eg 2 Φ ζ 2 ζ, ζ C 3 g l C ζ 2 α k= 3.5 ν i ν 2 ξex k 2α π e ΦΦ g = + g ζ z ζ e ΦΦ dσ = 0. we see that eg 2 Φ ζ 2 ζ, ζ L and ζ 2 eg ζ ζ z ζ e ΦΦ dξ dξ 2 g 2 Φ ζ 2 ζ z ζ 2 eφφ dξ dξ 2. From this equality, Proposition 3.3 and definition 3.3 of the operator R,B, the estimate 3.2 follows immediately. To prove 3.3 we observe R,B g = B R,B g + R,B { g B g} + 2π eφφ+b ζ ζ z ζ z ge ΦΦ dξ dξ 2.

26 26 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO By Proposition 3. B R,B g + R,B { g B g} L 2 C 4 g H. Using arguments similar to 3.4, 3.5 we obtain Hence 2π ζ ζ z ζ z ge ΦΦ dξ dξ 2 L 2 C 5 g C α H. R,B g L 2 C 6 g C α H. Combining this estimate with 3.2 we conclude R,B g L 2 C 7 g C α H. Using this estimate and equation 3.4 we have finishing the proof of the proposition. R,B g L 2 C 8 g C α H, Let e, e 2 C be functions such that 3.6 e + e 2 = in, e 2 vanishes in some neighborhood of H \ Γ 0 and e vanishes in a neighborhood of. Proposition 3.7. Let for some α 0, A, B C 5+α, and the functions A, B C 6+α satisfy 2.2. Let e, e 2 be defined as in 3.6. Let g L p for some p > 2, supp g supp e and distγ 0, supp g > 0. We define u by u = R,B e P A g Me A + e 2P A g Me A, 2 z Φ where M = Mz is a polynomial such that k k P A g Me A H = 0 for any k from {0,..., 6}. Then we have 3.7 Px, Due Φ 2 + A2 + BueΦ = ge Φ + eϕ h as +, where h L C 9 p g L p and for some sufficiently small positive ɛ we have: 3.8 u L u L 2 + u H, O ɛ C 0 g L p. 2

27 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 27 Proof. By Proposition 3. P A g belongs to Wp. Since p > 2, by the Sobolev embedding theorem there exists α > 0 such that P A g C α. By properties of elliptic operators and the fact that supp e 2 supp g = { } we have that P A g C 5 supp e 2. The estimate 3.8 follows from Proposition 3.6. Short calculations give 3.9 Px, Due Φ = ge Φ + eφ Px, D This formula implies 3.7 with h = e iψ Px, D e 2 P A g Me A 2 z Φ /sign. e2 P A g f Me A 2 zφ The following proposition will play a critically important role in the construction of the complex geometric optic solutions. Proposition 3.8. Let f L p for some p > 2, distγ 0, supp f > 0, q H 2 Γ 0, ɛ be a small positive number such that O ɛ H \ Γ 0 =. Then there exists 0 such that for all > 0 there exists a solution to the boundary value problem 3.20 Lx, Dw = fe Φ in, w Γ0 = qe ϕ / such that we ϕ L 2 + we ϕ L 2 + we ϕ H, O ɛ C f L p + q H 2 Γ0. Proof. Let χ C 0 be equal to one in some neighborhood of the set H\Γ 0. By Proposition 2.6 there exists a solution to the problem 3.20 with inhomogeneous term χf and boundary data q/ such that 3.2 w e ϕ H, C 2 f L 2 + q H 2 Γ0. Denote w 2 = R,B e P A χf Me A + e 2P A χfme f A 2 zφ where M = Mz is a polynomial such that k P k A χf Me A H = 0 for any k from {0,..., 6}. Let q be the restriction of w 2 to Γ 0. By 3.2 there exists a constant C 3 independent of such that q H 2 Γ0 C 3 f L p. By Proposition 3.7 there exists a constant C 4 independent of such that 3.23 w2 e ϕ L 2 + w 2 e ϕ L 2 + w 2 e ϕ H, O ɛ C 4 f L p. Let ã, b H be holomorphic and antiholomorphic functions, respectively, such that ã e A + b e B Γ0 = q. By 3.22 and Proposition 2.3 there exist constants C 5, C 6 independent of such that 3.24 ã H + b H C 5 q C f L p H 2 Γ0 6. The function W = w 2 + ã e A e Φ + b e B+Φ satisfies Lx, DW = χfe Φ + e h ϕ in, W Γ0 = 0,.

28 28 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO where 3.25 h L 2 C 7 f L 2 with some constant C 7 independent of. By 3.23, W e ϕ L 2 + W e ϕ L 2 + W e ϕ H, O ɛ C 8 f L p. Let W be a solution to problem 2.45 with inhomogeneous term and boundary data f = e h, g 0 respectively given by Proposition 2.6. The estimate 2.46 has the form 3.27 W e ϕ H, C 9 h L 2 C 20 f L 2. Then the function w + W + W solves The estimate 6. follows form 3.2, 3.26 and The proof of the proposition is completed. 4. Complex Geometrical Optics Solutions For a complex-valued vector field A, B and complex-valued potential q we will construct solutions to the boundary value problem 4. L x, Du = 0 in, u Γ0 = 0 of the form 4.2 u x = a ze A +Φ + d ze B +Φ + u e ϕ + u 2 e ϕ. Here A and B are defined by 3.2 respectively for A and B, a z = az+ a z + a 2, z d z = dz + d z + d 2, z 2, 4.3 a, d C 5+α, a = 0 in, d 4.4 ae A + de B Γ0 = 0. = 0 in, Let x be some fixed point from H \. Suppose in addition that 4.5 k a k H = k d k H = 0 k {0,..., 5}, a H\{ex} = d H\{ex} = 0, a x 0, d x 0. Such functions exists by Proposition 7.2. Denote g = T B q 2 B A B de B M 2 ze B, g 2 = P A q 2 A A B ae A M ze A, where M z and M 2 z are polynomials such that 4.6 k g k H = k g 2 k H = 0 k {0,..., 6}, g = g 2 = 0 on H \ { x}. Thanks to our assumptions on the regularity of A, B and q, g, g 2 belong to C 6+α. 2,

29 PARTIAL CAUCHY DATA FOR GENERAL OPERATOR 29 Note that by 4.6, k+j g k j H = k+j g 2 k j H = 0 if k + j 6. The function a z is holomorphic in and d z is antiholomorphic in and a e A + d e B = g 2 z Φ + g 2 2 z Φ on Γ 0. The existence of such functions is given again by Proposition 2.3. Observe that by 4.7 the functions e 2g, e 2g 2 zφ C4 zφ. Let ĝ = T B q 2 B A B d e B M 2 ze B, ĝ 2 = P A q 2 A A B a e A M ze A, where M z and M 2 z are polynomials such that 4.8 k ĝ k H = k ĝ 2 k H = 0 k {0,..., 3}. Henceforth we recall 3.3. The function u is given by u = e iψ R,A {e g + ĝ /} e iψ e 2g + bg 2 z Φ + eiψ 4 2 z Φ L e2 g x, D z Φ 4.9 : e iψ R,B {e g 2 + ĝ 2 /} e iψ e 2g 2 + bg 2 2 z Φ + eiψ 4 2 z Φ L x, D e2 g 2 z Φ Now let us determine the functions u 2, a 2 z and d 2 z. First we can obtain the following asymptotic formulae for any point on the boundary of 4.0 R,A {e g } = 4. R,B {e g 2 } = where e A +2iψ 2 2 det ψ x 2 e B 2iψ 2 2 det ψ x σ x = z g x 2 zφ x, m x = 2. e 2iψex σ x + e2iψex m x + W z z 2,, z z e 2iψex σ x + e2iψex m x + W,2, z z 2 z z z g x 2 zφ x Φ x 3 Φ x + 2 z g x 2 Φ x 2 z g x 2 Φ x σ x = z g 2 x Φ x, m x = z g 2 x Φ x Φ x 2 2Φ x 2 z g 2 x Φ x + 2 z g 2 x, 2 Φ x 2, g = e A g, g 2 = e B g 2 and W,, W,2 H 2 Γ 0 satisfy 4.4 W, H 2 Γ0 + W,2 H 2 Γ0 = o 2 The proof of 4.0 and 4. is given in Section 8. as +.

30 30 O. IMANUVILOV, G. UHLMANN, AND M. YAMAMOTO Denote p + x = e A x σ x z z + m x, p 2 x = e B σ x x z z z z + m x. 2 z z Thanks to Proposition 2.3 we can define functions a 2,± z C 2 and d 2,± z C 2 satisfying 4.5 a 2,± e A + d 2,± e B = p ± on Γ 0. Straightforward computations give 4.6 L x, Da + a e A +Φ + d + d e B +Φ + e ϕ u = q 2 A A B e Φ R,B {e g 2 + ĝ 2 /} e 2g 2 +bg 2 / 2 zφ +q 2 B A B e Φ R,A {e g + ĝ /} e 2g +bg / 2 zφ + eφ 2 L x, D L 4 zφ x, D e 2 g 2 zφ + eφ L 2 x, D L 4 zφ x, D e 2 g zφ. Using Proposition 3.4 we transform the right-hand side of 4.6 as follows. 4.7 L x, Da + a e A+Φ + d + d e B+Φ + u e ϕ = q 2 A A B e Φ g 2 zφ q 2 B A B e Φ g 2 2 zφ + χ O ɛ O L χ \Oɛ o L 4 2 as +. We are looking for u 2 in the form u 2 = u 0 + u. The function u is given by 4.8 where 4.9 u = eiψ R,B {e g 5 } + eiψ R,A {e g 6 } + e 2g 5 e iψ 2 2 zφ + e 2g 6 e iψ 2 2 zφ, g 5 = P A q 2 A A B g M 5 ze A, g 6 = T B q 2 B A B g 2 M 6 ze B. 2 z Φ 2 z Φ Here M 5 z, M 6 z are polynomials such that g 5 H = g 6 H = g 5 H = g 6 H = 0. Using Proposition 2.3 we introduce functions a 2,0, d 2,0 C 2 holomorphic and antiholomorphic respectively such that 4.20 Next we claim that a 2,0 e A + d 2,0 e B = g 5 2 z Φ + g 6 2 z Φ on Γ R,A {e g 6 } Γ0 = o as +, R,B {e g 5 } Γ0 = o as +.