On uniqueness in the inverse conductivity problem with local data Victor Isakov June 21, 2006
1
Introduction
The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain Ω ⊂ Rn , n = 2, 3 from the Neumann data given for all Dirichlet data (Dirichlet-to-Neumann map). Calderon [5] proposed the idea of using complex exponential solutions to demonstrate uniqueness in the linearized inverse condictivity problem. Complex exponential solutions of elliptic equations have been introduced by Faddeev [7] for needs of inverse scattering theory. Sylvester and Uhlmann in their fundamental paper [19] attracted ideas from geometrical optics, constructed almost complex exponential solutions for the Schr¨odinger operator, and proved global uniqueness of a ( and of potential c in the Schr¨odinger equation) in the three-dimensional case. In the two-dimensional case the inverse conductivity problem is less overdetermined, and the Sylvester and Uhlmann method is not applicable, but one enjoys advantages of the methods of inverse scattering and of theory of complex variables. Using these methods Nachman [17] demonstrated ¯ and Astala and P¨aiv¨arinta [1] showed uniqueness uniqueness of a ∈ C 2 (Ω) of a ∈ L∞ (Ω) which is a final result in the inverse conductivity problem in R2 with many measurements from the whole boundary. There is a known hypothesis (see for example, [11], Problem 5.3, [14], [20]) that the Dirichlet-to-Neumann map given at any (nonvoid open) part Γ of the boundary also uniquely determines conductivity coefficient or potential in the Schr¨odinger equation. This local boundary measurements model important 1
applications, for example to geophysics or to semiconductors when collecting data from the whole boundary is either not possible or extremely expensive. Despite extended long term efforts this hypothesis remains not proven, altough there is some progress. Kohn and Vogelius [15] showed uniqueness of the boundary reconstruction (of all existing partial derivatives of a) and hence uniqueness of piecewise analytic a. When coefficients are known in a neighborhood of the boundary, then Runge type approximation argument reduces the partial Dirichlet-to-Neumann map to complete map [9], [11], Exercise 5.7.4, [15], and hence the hypothesis follows. In smooth case Bukhgeim and Uhlmann [4] made use of Carleman estimates (with linear phase function) to show that the Neumann data on a sufficiently large part Γ of the boundary given for all Dirichlet data on the whole boundary uniquely determine potential c in the three-dimensional Schr¨odinger equation. The most advanced and recent result is due to Kenig, Sj¨ostrand and Uhlmann [14]. They modified the scheme of [19], [4] by using quadratic phase function and demonstrated uniqueness of c from Neumann data on Γ for all Dirichlet data on a complementary part Γ1 . While Γ in [14] can be arbitrarely small one can not assume zero Dirichlet data on ∂Ω \ Γ (although they can be zero on ∂Ω \ Γ1 ) and they need Γ and Γ1 to have nonvoid intersection. Only result concering zero boundary data on part Γ0 of the boundary is due to H¨ahner [8] who by explicit calculations proved completeness of products of harmonic functions which are zero on a spherical Γ0 . An inverse scattering in half-space was considered by Karamyan [13]. In this paper we give a complete proof of this hypothesis when the Dirichlet-to-Neumann map is given on arbitrary part Γ of ∂Ω while on the remaining part Γ0 one has homogeneous Dirichlet or Neumann data. Our restrictive assumption is that Γ0 is a part of a plane or of a sphere. In some applications this assumption is natural, but available uniqueness results [6], [16] require that coefficients of the differential equation are known near ∂Ω. An exception is the paper [13] where one is given scattering data in half-space. This assumption enables to reflect almost complex exponential solutions across Γ0 and to avoid use of a special fundamental solution (Green’s function) for the Schr¨odinger equation and of corresponding exponential weighted estimates with a large parameter in Ω. Currently, such fundamental solution and estimates are available only when homogeneous boundary data are given at a part of the boundary (but not at the whole boundary) [4], [14]. It is not likely that such fundamenatl solutions and 2
estimates can be found when homogeneous Dirichlet (or Neumann) are prescribed at the whole boundary. We add new ingredients to the SylvesterUhlmann method. A crucial observation is that contributions of products of almost complex exponential solutions and of their reflections converge to zero when large parameter τ goes to ∞.
2
Main results
Let Ω be a domain in R3 with Lipschitz boundary. We consider the Schr¨odinger equation −∆u + cu = 0 in Ω (2.1) with the Dirichlet boundary data u = g0 on ∂Ω
(2.2)
∂ν u = g1 on ∂Ω.
(2.3)
or the Neumann data Let B0 be some ball. We will assume that the (complex valued) potential c ∈ L∞ (Ω), c = −k 2 on Ω \ B0 . Let Γ0 be an open bounded part of ∂Ω and Γ = ∂Ω \ Γ0 . We define the local Dirichlet-to-Neumann map Λc (D, Γ) as 1
Λc (D, Γ)g0 = ∂ν u on Γ, g0 ∈ H 2 (∂Ω), g0 = 0 on Γ0 and the local Neumann-to-Dirichlet map as 1
Λc (N, Γ)g1 = u on Γ, g1 ∈ H − 2 (∂Ω), g1 = 0 on Γ0
(2.4)
provided the Dirichlet or Neumann problems are uniquely solvable. In Theorems 2.1 and 2.2 we consider two cases a) Ω is a bounded subdomain of {x : x3 < 0}, Γ = ∂Ω ∩ {x3 < 0}; and ¯ \ B. b) 0 < k, Ω = {x : x3 < 0} and Γ ⊂ Ω
(2.5)
It is well known that in case (2.5), a), the boundary value problems (2.1), (2.2) or (2.3) have unique solutions u ∈ H 1 (Ω) for any boundary data 1 1 g0 ∈ H 2 (∂Ω), g1 ∈ H − 2 (∂Ω) provided there is uniqueness of a solution. 3
Uniqueness is guaranteed by maximum principles or energy integrals when =c = 0, 0 ≤ c on Ω or when =c 6= 0 on a nonempty open subset of Ω. In case (2.5), b), we will assume that =c ≤ 0 on Ω and we will augment the equation and the boundary condition by the Sommerfeld radiation condition lim r(σ · ∇u − iku)(x) = 0, σ = r−1 x, as r = |x| → ∞.
(2.6)
By using integral equations or the Lax-Phillips method one can demonstrate uniqueness and existence of a solution u ∈ H 1 (Ω ∩ B) for any ball B to the scattering boundary value problem (2.1), (2.2) or (2.3), and (2.6) with 1 1 compactly supported g0 ∈ H 2 (∂Ω), g1 ∈ H − 2 (∂Ω) [11], [16]. Theorem 2.1 If Λc1 (D, Γ) = Λc2 (D, Γ)
(2.7)
Λc1 (N, Γ) = Λc2 (N, Γ),
(2.8)
or then c1 = c2 This result has immediate corollary for the conductivity equation −div(a∇u) − k 2 u = 0 in Ω.
(2.9)
¯ a = 1 on Ω \ B0 , a > 0 on Ω, ¯ and 0 < k We will assume that a ∈ C 2 (Ω), in case (2.5), b). As for the Schr¨odinger equation, in case (2.5), a), the elliptic boundary value problems (2.9), (2.2) or (2.3) are uniquely solvable in H 1 (Ω) for all k except discrete set of eigenvalues. When k = 0, the Dirichlet problem is uniquely solvable. In case (2.5), b) the boundary value scattering scattering problems (2.9), (2.2) or (2.3), (2.6) are uniquely solvable in the same functional spaces as for the Schr¨odinger equation. We will assume the unique solvability condition and we define local Dirichlet-to-Neumann and Neumann-to-Dirichlet maps for the conductivity equation similarly to the Schr¨odinger equation and we will denote them by Λ(a; D, Γ), Λ(a; N, Γ). Theorem 2.2 If Λ(a1 ; D, Γ) = Λ(a2 ; D, Γ)
(2.10)
Λ(a1 ; N, Γ) = Λ(a2 ; N, Γ)
(2.11)
or
4
with the additional assumption ∂3 a1 = ∂3 a2 = 0 on Γ0 ,
(2.12)
then a1 = a2 on Ω. These results imply similar results for bounded domains Ω when Γ0 is a part of a sphere. In Theorems 2.3, 2.4 we use the following notation and assumptions. Let Ω be a subdomain of B0 . Let Γ0 = ∂B0 ∩ ∂Ω and Γ = ∂Ω \ Γ0 . We will assume that Γ0 6= ∂B0 . Theorem 2.3 The equality (2.7) implies that c1 = c2 Theorem 2.4 The equality (2.10) implies that a1 = a2 Finally we give available results the plane case and assume that Ω is a bounded simply connected domain in R2 with Lipschitz boundary. According to established theory of elliptic boundary value problems if a ∈ L∞ (Ω), k = 0, then the Dirichlet problem (2.9), (2.2) has a unique solution u ∈ H 1 (Ω) 1 for any g0 ∈ H 2 (∂Ω), and the Neumann problem (2.9), (2.3) has a unique solution in the same space with normalization condition Z
u=0
∂Ω
provided Z ∂Ω
g1 = 0. 1
We remind that the (conormal) derivative a∂ν u ∈ H − 2 (∂Ω) is defined as Z ∂Ω
(a∂ν u)ϕ =
Z
1
a∇u · ∇ϕ, ϕ ∈ H 2 (∂Ω),
(2.13)
Ω
where the integral on the left side is understood as dual pairing between 1 1 H − 2 (∂Ω) and H 2 (∂Ω). After these reminders we can define the partial Dirichlet-to-Neumann and Neumann-to-Dirichlet maps as above. Theorem 2.5 Let a1 , a2 ∈ L∞ (Ω). Let Γ be any nonvoid open arc of ∂Ω. Then equalities (2.10) and (2.11) imply that a1 = a2 in Ω. Theorem 2.5 was proven by Astala, Lassas, and P¨aiv¨arinta [2], Theorem 2.3. Theorems 2.1-2.5 can be immediately generalized to semilinear Schr¨odinder equations and quasilinear conductivity equations as in [12], [18]. 5
3
Proofs for half-space
In this section we consider the case of Ω ⊂ {x : x3 < 0} when unobservable part Γ0 of the boundary is contained in the plane {x3 = 0}. We give proofs for Neumann boundary condition, because of its applied importance, and we will indicate how to modify them for Dirichlet condition. We start with a standard orthogonality relation. Lemma 3.1 Under condition (2.8) Z Ω
(c1 − c2 )v1 v2 = 0
(3.14)
for all functions v1 , v2 ∈ H 1 (Ω ∩ B), for any ball B, satisfying −∆v1 + c1 v1 = 0 in Ω, ∂ν v1 = 0 on Γ0
(3.15)
−∆v2 + c2 v2 = 0 in Ω, ∂ν v2 = 0 on Γ0 .
(3.16)
and
Proof: First we consider case a) of bounded domain Ω. Let v1 be any solution to (3.15). Let u2 be the solution to the Schr¨odinger equation with c = c2 and with the Neumann data ∂ν u2 = ∂ν v1 on ∂Ω. Subtracting the equations −∆u2 + c2 u2 = 0 and −∆v1 + c1 v1 = 0 and letting v = u2 − v1 we yield −∆v + c2 v = (c1 − c2 )v1 on Ω.
(3.17)
We have v = 0 on Γ by condition (2.8) and (2.4) and ∂ν v = 0 (by definition of v) on ∂Ω. Multiplying equation (3.17) by solution v2 to (3.16) and integrating by parts we yield Z Ω
(c1 − c2 )v1 v2 =
Z Ω
(−∆v + c2 v)v2 =
Z Ω
v(−∆v2 + c2 v2 ) = 0
where we used the boundary conditions ∂ν v = 0 = ∂ν v2 on Γ0 and the equality v = ∂ν v = 0 on Γ. Summing up we have the orthogonality relation (3.14). 6
The case of Ω = {x3 < 0} needs in addition a Runge type approximation argument Let u1 be any solution to (3.15). Let u2 be the solution to the Schr¨odinger equation with c = c2 and with the Neumann data ∂ν u2 = ∂ν u1 on ∂Ω. Subtracting the equations −∆u2 + c2 u2 = 0 and −∆u1 + c1 u1 = 0 and letting v = u2 − u1 we yield −∆v + c2 v = (c1 − c2 )u1 on Ω.
(3.18)
Since v = 0 by condition (2.8) and ∂ν v = 0 (by definition of v) on Γ we have v = 0 on Ω \ B0 due to uniqueness in the Cauchy problem for the Laplace equation. Multiplying equation (3.18) by v2 and integrating by parts we yield Z Ω∩B0
(c1 − c2 )u1 v2 =
Z Ω∩B0
(−∆v + c2 v)v2 =
Z Ω∩B0
v(−∆v2 + c2 v2 ) = 0
where we used the boundary conditions ∂ν v = 0 = ∂ν v2 on Γ0 and the equality v = 0 on Ω \ B0 . Summing up we have the orthogonality relation (3.14) for v1 = u1 . To complete the proof we will L2 (Ω ∩ B0 )-approximate arbitrary v1 by u1 . Let us assume the opposite: the subspace {u1 } is not dense in {v1 }. Then by Hahn-Banach theorem there is f ∈ L2 (Ω), f = 0 outside B0 , such that Z
f u1 = 0
(3.19)
f v1 6= 0 for some v1 .
(3.20)
Ω
for all u1 , but
Z Ω
Let u∗1 (f ) solve the Neumann problem −∆u∗1 + c1 u∗1 = f in Ω, ∂ν u∗1 = 0 on ∂Ω.
(3.21)
From (3.19) and (3.21) we have 0=
Z Ω
(−∆u∗1 + c1 u∗1 )u1 =
Z Γ
u∗1 ∂ν u1
where we used the Green’s formula and boundary conditions for u1 , u∗1 , Since ∂ν u1 can be arbitrary smooth function on Γ we conclude that u∗1 is zero on Γ. Now u∗1 solves the elliptic equation −∆u∗1 = 0 on Ω \ B0 and has zero 7
Cauchy data on Γ, so by uniqueness in the Cauchy problem u∗1 = 0 on Ω \ B0 . Applying again the Green’s formula in Ω ∩ B0 we yield Z Ω
f v1 =
Z Ω∩B0
(−∆u∗1 + c1 u∗1 )v1 = 0
which contradicts (3.20). The proof is complete. Lemma 3.2 Under condition (2.7) Z Ω
(c1 − c2 )v1 v2 = 0
for all functions v1 , v2 ∈ H 1 (Ω ∩ B), for any ball B, satisfying −∆v1 + c1 v1 = 0 in Ω, v1 = 0 on Γ0 and −∆v2 + c2 v2 = 0 in Ω, v2 = 0 on Γ0 Proof is similar to the proof of Lemma 3.1. Proof of Theorem 2.1 First we remind known results about existence of special almost complex exponential solutions to the Schr¨odinger equation in R3 . Let ξ = (ξ1 , ξ2 , ξ3 ), ξ ∗ = (ξ1 , ξ2 , −ξ3 ). We introduce 1
e(1) = (ξ12 + ξ22 )− 2 (ξ1 , ξ2 , 0), e(3) = (0, 0, 1), and the unit vector e(2) to get orthonormal basis e(1), e(2), e(3) in R3 . We denote the coordinates of x in this basis by (x1e , x2e , x3e )e . Observe that 1
ξ = (ξ1e , 0, ξ3 )e , ξ1e = (ξ12 + ξ22 ) 2 and that in general
x · y = x1 y1 + x2 y2 + x3 y3 = x1e y1e + x2e y2e + x3e y3e . 8
We define ζ(1) = (
1 ξ3 ξ1e 1 − τ ξ3 , i|ξ|( + τ 2 ) 2 , + τ ξ1e )e , 2 4 2
1 ξ1e 1 ξ3 − τ ξ3 , i|ξ|( + τ 2 ) 2 , − − τ ξ1e )e , 2 4 2 1 ξ3 ξ1e 1 ζ(2) = ( + τ ξ3 , −i|ξ|( + τ 2 ) 2 , − τ ξ1e )e , 2 4 2 1 1 ξ3 ξ1e + τ ξ3 , −i|ξ|( + τ 2 ) 2 , − + τ ξ1e )e , (3.22) ζ ∗ (2) = ( 2 4 2 where τ is a positive real number. By direct calculations we can see that
ζ ∗ (1) = (
ζ(1) · ζ(1) = ζ ∗ (1) · ζ ∗ (1) = ζ(2) · ζ(2) = ζ ∗ (2) · ζ ∗ (2) = 0.
(3.23)
Let us extend c1 , c2 onto R3 as even functions of x3 . Since (3.23) holds, it is known [11], section 5.3, [19], that there are almost exponential solutions eiζ(1)·x (1 + w1 ), eiζ(2)·x (1 + w2 ) to the equations −∆u1 + c1 u1 = 0, −∆u2 + c2 u2 = 0 in R3
(3.24)
kw1 k2 (B0 ) + kw2 k2 (B0 ) → 0 as τ → ∞,
(3.25)
with where k k2 (B) is the standard norm in L2 (B). We define f ∗ (x1 , x2 , x3 ) = f (x1 , x2 , −x3 ) and we let u1 (x) = eiζ(1)·x (1 + w1 ) + eiζ
∗ (1)·x
(1 + w1∗ ),
u2 (x) = eiζ(2)·x (1 + w2 ) + eiζ
∗ (2)·x
(1 + w2∗ ).
(3.26)
It is obvious that u1 , u2 ∈ H 2 (Ω ∩ B) for any B, solve the partial differential equations (3.24) and that ∂ν u1 = ∂ν u2 = 0 on Γ0 .
(3.27)
c = c1 − c2 .
(3.28)
Let
9
By (3.24), (3.27), (3.28), and Lemma 3.1 0=
Z Ω
Z
cu1 u2 =
c(x)(ei(ζ(1)+ζ(2))·x (1 + w1 (x))(1 + w2 (x))+
Ω
ei(ζ
∗ (1)+ζ(2))·x
(1 + w1∗ (x))(1 + w2 (x))+
∗ (2))·x
(1 + w1 (x))(1 + w2∗ (x))+
ei(ζ(1)+ζ ei(ζ
∗ (1)+ζ ∗ (2))·x
(1 + w1∗ (x))(1 + w2∗ (x)))dx
due to (3.26). Using (3.22) we conclude that Z Ω
c(x)(eiξ·x (1 + w1 (x))(1 + w2 (x))+
ei(ξ1e x1e −2τ ξ1e x3 ) (1 + w1∗ (x))(1 + w2 (x))+ ei(ξ1e x1e +2τ ξ1e x3 ) (1 + w1 (x))(1 + w2∗ (x))+ eiξ
∗ ·x
(1 + w1∗ (x))(1 + w2∗ (x)))dx = 0.
(3.29)
Now we let τ → ∞. Observe that moduli of all exponents are bounded by 1. So due to (3.25) limits of all terms containing factors wj , wj∗ are zero. By the Riemann-Lebesgue Lemma limits of Z
c(x)ei(ξ1e x1e −2τ ξ1e x3 ) ,
Z
c(x)ei(ξ1e x1 +2τ ξ1e x3 ) dx
Ω
Ω
as τ → ∞ are also zero provided ξ1e 6= 0. Therefore from (3.29) we derive that Z ∗ c(x)(eiξ·x + eiξ ·x )dx = 0 (3.30) Ω
for any ξ, ξ1e 6= 0. Since cj and hence c (given by (3.28) )are compactly supported, the right side in (3.30) is analytic with respect to ξ, so we have (3.30) for all ξ ∈ R3 . Since c is an even function of x3 , Z
c(x)(eiξ·x + eiξ
∗ ·x
)dx =
Ω
Z R3
c(x)eiξ·x dx.
Hence from (3.30) Z R3
c(x)eiξ·x dx = 0 10
for any ξ ∈ R3 . By uniqueness of the inverse Fourier transformation c = 0, and hence c = 0 and c1 = c2 . This completes the proof under condition (2.8). Now we will show how to adjust it to the case of Dirichlet boundary conditions. The argument until (3.25) is the same. Then we let u1 (x) = eiζ(1)·x (1 + w1 (x)) − eiζ
∗ (1)·x
(1 + w1∗ (x)),
u2 (x) = eiζ(2)·x (1 + w2 (x)) − eiζ
∗ (2)·x
(1 + w2∗ (x)).
(3.31)
It is obvious that u1 , u2 ∈ H 2 (Ω ∩ B0 ), solve the partial differential equations (3.24) and that u1 = u2 = 0 on Γ0 . (3.32) By (3.24), (3.32) and Lemma 3.2 0=
Z Ω
Z Ω
cu1 u2 =
c(x)(ei(ζ(1)+ζ(2))·x (1 + w1 (x))(1 + w2 (x))− ei(ζ
∗ (1)+ζ(2))·x
(1 + w1∗ (x))(1 + w2 (x))−
∗ (2))·x
(1 + w1 (x))(1 + w2∗ (x))+
ei(ζ(1)+ζ ei(ζ
∗ (1)+ζ ∗ (2))·x
(1 + w1∗ (x))(1 + w2∗ (x)))dx
due to (3.31). Using (3.22) we conclude that Z Ω
c(x)(eiξ·x (1 + w1 (x))(1 + w2 (x))−
ei(ξ1e x1e −2τ ξ1e x3 ) (1 + w1∗ (x))(1 + w2 (x))− ei(ξ1e x1e +2τ ξ1e x3 ) (1 + w1 (x))(1 + w2∗ (x))+ eiξ
∗ ·x
(1 + w1∗ (x))(1 + w2∗ (x)))dx = 0.
As above we let τ → ∞ and repeating the argument after (3.29) we conclude that Z Z ∗ 0 = c(x)(eiξ·x + eiξ ·x )dx = c(x)eiξ·x dx. R3
Ω
11
for any ξ ∈ R3 . By uniqueness of the inverse Fourier transformation c = 0, and hence c1 = c2 . The proof is complete. Proof of Theorem 2.2 The well known substitution 1
u = a− 2 v
(3.33)
transforms the conductivity equation (2.9) into the Schr¨odinger equation (2.1) with 1 1 c = a− 2 ∆a 2 . (3.34) From (3.33) it follows that homogeneous boundary Dirichlet condition for u on Γ0 implies the same condition for v and since ∂3 a = 0 on Γ0 due to (2.12) the same is true for the Neumann condition. As known ([11]), section 5.2, ([15]) the local Dirichlet-to-Neumann (Neumann-to-Dirichlet) map on Γ uniquely determines a, ∂3 a on Γ. Hence again (3.33) implies that partial Dirichlet-to-Neumann map Λ(a; D, Γ) for the conductivity equation uniquely determines partial Dirichlet-to-Neumann map Λc (D, Γ) for the Schr¨odinger equations. The same holds for the Neumann-to-Dirichlet maps. By Theorem 2.1 the potential c given by (3.34) is unique. (3.28) can be viewed as an linear 1 1 1 elliptic equation −∆a 2 + ca 2 = 0 in Ω with respect to a 2 . As mentioned Λ(a; D, Γ) or Λ(a; D, Γ) uniquely determine c on Ω and a, ∂3 a on Γ. So due to uniqueness in the Cauchy problem for elliptic equations ([11], section 3.3) a is uniquely determined on Ω as well. The proof is complete. We observe that Theorem 2.2 holds for complex valid a with
0 which are of importance in applications [11], p.7. The above proof is valid with the substitution (3.33) where a 1 is the principal branch which is well 2 defined on the half-plane 0.
4
Proofs for subsets of balls: use of the Kelvin transform
We remind the definition of the Kelvin transform of a function in R3 . Let X(x) = |x|−2 x, x(X) = |X|−2 X 12
(4.35)
and U (X) = |X|−1 u(x(X))
(4.36)
∆X U (X) = |X|5 ∆x u(x(X))
(4.37)
It is known [10] that
Proof of Theorem 2.3 We can assume that B is the ball of radius 21 centered at x0 = (0, 0, 21 ) and that the origin in R3 does not belong to Ω. Let us apply the Kelvin transform to the equation (2.1). Using (4.37) and (4.36) we yield −∆X U + CU = 0, where C(X) = |X|6 c(x(X)) The inversion (4.35) transforms the sphere {|x − x0 | = 12 } into the plane {X3 = 1}. Hence the domain Ω in X-variables is a subdomain of the halfspace {X : 1 < X3 }, and parts Γ0 , Γ of its boundary are correspondingly parts of the plane {X3 = 1} and of the halfspace {1 < X3 }. Due to (4.36) homogeneous Dirichlet data on Γ0 are transformed into homogeneous Dirichlet data in new variables. Obviously, the Dirichlet-to-Neumann map in x-variables uniquely determines the Dirichlet-to-Neumann map on Γ in new variables. Applying Theorem 2.1 to the inverse problem in X variables we conclude that C(X) = |X|6 c(x(X)) is uniquely determined. Hence c is uniquely determined on Ω. The proof is complete.
5
Conclusion
The main remaining open question is of course how obtain uniqueness from local Dirichlet-to-Neumann map when Γ0 is an arbitrary surface. When this map is given at all k uniqueness follows from the results on inverse hyperbolic problems obtained by the methods of boundary control [3]. In particular, it is important to relax topological assumptions on Ω and to consider case of several connected components of ∂Ω, especially when Γ0 is an inner connected component of the boundary. For spherical Γ0 one can most likely to use methods of this paper. In case of half-space by more careful study of bahavior at inifinity one can relax the assumtpion that c is compactly supported. We formulated results assuming that Dirichlet or Neumann problems are uniquely 13
solvable only to follow traditions. Arguments in the three-dimensional case will not change if we drop these assumptions and instead require equalities of Cauchy pairs (u, a∂ν u) on Γ (as in [20], section 3), under homogeneous Dirichlet or Neumann conditions on Γ0 . It is very interesting and probably more difficult to recover simulteneously a, Γ0 and coefficient b of the boundary condition a∂ν u + bu = 0 on Γ0 . Again such results are available for hyperbolic equations. It is still not clear if it is possible to construct semilocalized (in weighted spaces) almost complex exponential solutions to the Schr¨odinger equation and to use them in uniqueness proofs and constructive methods of solution of the inverse problem with local data or with unknown multiple inclusions. Localization would most likely remove all geometrical assumptions and to obtain most general results. Due to substantial overdeterminancy of this inverse problem it is feasible that a completely different approach will work and resolve remaining questions.
Aknowledgement: This research was in part supported by the NSF grant DMS 04-05976.
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