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# Determination of second-order elliptic operators in two dimensions from partial Cauchy data

Edited* by Victor Guillemin, Massachusetts Institute of Technology, Cambridge, MA, and approved October 27, 2010 (received for review August 5, 2010)

## Abstract

We consider the inverse boundary value problem in two dimensions of determining the coefficients of a general second-order elliptic operator from the Cauchy data measured on a nonempty arbitrary relatively open subset of the boundary. We give a complete characterization of the set of coefficients yielding the same partial Cauchy data. As a corollary we prove several uniqueness results in determining coefficients from partial Cauchy data for the isotropic conductivity equation, the Schrödinger equation, the convection–diffusion equation, the anisotropic conductivity equation modulo a group of diffeomorphisms that are the identity at the boundary, and the magnetic Schrödinger equations modulo gauge transformations. The key step is the construction of novel complex geometrical optics solutions using Carleman estimates.

## Main Result

Let Ω⊂**R**^{2} be a bounded domain with smooth boundary , where *γ*_{k}, , are smooth closed contours, and is the external contour. Let be an arbitrarily fixed nonempty relatively open subset of ∂Ω. Let ν be the unit outward normal vector to ∂Ω, and let . We set and identify *x* = (*x*_{1}, *x*_{2})∈**R**^{2} with *z* = *x*_{1} + *ix*_{2}∈**C**, and by we denote the complex conjugate of *z*∈**C**.

We consider a second-order elliptic operator: [1]

Here, *g* = *g*(*x*) = {*g*_{jk}}_{1≤j,k≤2} is a positive definite symmetric matrix in Ω, and Δ_{g} is the Laplace–Beltrami operator associated to the Riemannian metric *g*: where we set {*g*^{jk}} = *g*^{-1}. Throughout this paper, we assume that , and (*A*,*B*,*q*), , *j* = 1, 2 for some *α*∈(0,1), are complex-valued functions. We set

We define the set of partial Cauchy data by where is the conormal derivative with respect to the metric *g*.

The goal of this paper is to determine the metric *g* and coefficients *A*, *B*, *q* from the partial Cauchy data . In the general case, this is impossible. There are the following main invariants of the Cauchy data in the problem.

*Conformal Invariance*. Let be a strictly positive function. Then, [2]This follows because the Laplace–Beltrami operator is conformal invariant in two dimensions:

*Gauge Transformations*. It is easy to see that the set of partial Cauchy data of the operators*e*^{-η}*L*(*x*,*D*)*e*^{η}and*L*(*x*,*D*) are the same provided that η is a smooth complex-valued function such that [3]*Diffeomorphism Invariance*. Let be a diffeomorphism such that . The pull back of a Riemannian metric*g*is given as composition of matrices by [4]where*DF*denotes the differential of*F*, (*DF*)^{T}denotes its transpose, and ∘ denotes matrix composition. Moreover, we introduce the functions: . Then, [5]

We show the converse; namely, a complete list of invariants of the problem. We have

Suppose that for some *α*∈(0,1), there exists a positive function such that . Then, if and only if there exist a diffeomorphism , satisfying , a positive function , and a complex-valued function η satisfying **3** such that where

## Calderón’s Problem and Other Applications

### Calderón’s Problem.

The question proposed by Calderón (1) is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary.

In the anisotropic case, the conductivity depends on direction and is represented by a positive definite symmetric matrix {*σ*^{jk}}. The conductivity equation with voltage potential *f* on ∂Ω is given by

We define the partial Cauchy data by [6]

It has been known for a long time that does not determine σ uniquely in the anisotropic case (2). Let be a diffeomorphism such that *F*(*x*) = *x* for *x* on . Then, where *F*^{∗}*σ* is given by **4**.

In the case of full Cauchy data (i.e., ), the question of whether one can determine the conductivity up to the above obstruction has been solved in two dimensions for *C*^{2} conductivities in ref. 3, Lipschitz conductivities in ref. 4, and merely *L*^{∞} conductivities in ref. 5. See also ref. 6. The method of proof in all these papers is based on the reduction to the isotropic case using isothermal coordinates (7).

We can prove the uniqueness for Calderón’s problem with partial Cauchy data:

Let *σ*_{1}, with some *α*∈(0,1) be positive definite symmetric matrices on . If , then there exists a diffeomorphism satisfying and such that

For the isotropic case, this result was proven in ref. 8 and in fact follows from Theorem 1 in the case where *g* = *I* and *A* = *B* = 0. We mention that ref. 9 has proven a similar result for general Riemann surfaces in the case where *g* is not the identity but fixed.

### Case Where the Principal Part Is the Laplacian.

In the rest of this section, we assume that the principal parts of second-order elliptic operators under consideration are the Laplacian: *g* = *I* ≡ {*δ*_{jk}}.

For the case when *A*_{1} = *A*_{2}, *B*_{1} = *B*_{2}, and full data, this result was proven by Bukhgeim (10).

If , then [7]and in the domain Ω we have [8][9]

The relation holds true if and only if there exists a function satisfying such that [10]

We only prove the necessity because the sufficiency of the condition is easy to check. By **8** and **9**, we have . This equality is equivalent to

Applying Lemma 1.1 (p. 313) of ref. 11, we obtain that there exists a function in the domain Ω^{0} that satisfies [11][*h*]|_{Σk} are constants, and

Here, Ω^{0} = Ω∖Σ is simply connected where , Σ_{j}∩Σ_{k} = ∅ for *j* ≠ *k*, Σ_{k} are smooth curves that do not self-intersect and are orthogonal to ∂Ω. We choose a normal vector *ν*_{k} = *ν*_{k}(*x*), to Σ_{k} at *x* contained in the interior of the closed curve Σ_{k}. Then, for , we set , where (·,·) denotes the scalar product in **R**^{2}. Setting , we have

Therefore, by **8**, [12]

The operator *L*_{1}(*x*,*D*) given by the right-hand side of **10** has the Laplace operator as the principal part, the coefficient of is , the coefficient of is , and the coefficient of the zero-order term is given by the right-hand side of **12**. By **7**, we have that and , where the function is equal to a constant on each connected component of . 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*^{1}(Ω) be a solution to the following boundary value problem: [13]

Assume, in addition, that *u*_{2} is not identically equal to zero on Σ_{j}. Let be one connected component of the set and . Without loss of generality, we may assume that . Indeed, if , we replace η by the function . Because the partial Cauchy data generated by the operators *L*_{1}(*x*,*D*) and *L*_{2}(*x*,*D*) are the same, there exists a solution *u*_{1} to the following boundary value problem: [14]

Then, the function *v* = *e*^{-η}*u*_{2} verifies

Because on , we have that *v* ≡ *u*_{1}. On the other hand, *u*_{1}∈*H*^{1}(Ω) and *v* are discontinuous along one part of Σ_{j}, and we arrive at a contradiction.

Let us show that . Suppose that there exists another connected component of of the set such that . Assume that *u*_{1}, *u*_{2} satisfy **13**, **14**.

Then, the function *v* = *e*^{-η}*u*_{2} verifies

Moreover, because on , we have that

The uniqueness of the Cauchy problem for the second-order elliptic equation yields *v* ≡ *u*_{1}. In particular, *v* = *u*_{1} on . Because *u*_{1} = *u*_{2} on ∂Ω, this implies that . We arrived at a contradiction. The proof of the corollary is completed.

Next, we apply Theorem 3 to several cases and state our results on the unique identifiability, modulo the natural obstructions, of some important inverse boundary value problems with partial Cauchy data arising in mathematical physics.

### The Magnetic Schrödinger Equation.

We consider the case of the magnetic Schrödinger operator.

Denote , where are real-valued, , rot . The magnetic Schrödinger operator is defined by

Let us define the following set of partial Cauchy data:

Theorem 3 implies the following corollary.

Let real-valued vector fields , and complex-valued potentials , with some *α*∈(0,1), satisfy . Then, , on Ω and on .

We mention that this result was not known even in the case of full data. In this case, ref. 12 proved a uniqueness result assuming that both the electric and magnetic potentials are small. Still in the case of full data, ref. 13 proved a uniqueness result for a special case of the magnetic Schrödinger equation, namely the Pauli Hamiltonian.

### Laplace Equation with Convection Terms.

Another application of Theorem 3 is to the Laplace equation with convection terms. For real-valued *a*, *b*, and complex-valued *q*, we define the following set of partial Cauchy data:

Then, we have the following corollary.

Let *α*∈(0,1), , and . If , then (*a*^{(1)},*b*^{(1)}) ≡ (*a*^{(2)},*b*^{(2)}).

This corollary generalizes the result of ref. 14, where the uniqueness was proved assuming that the measurements are made on the whole boundary.

We also mention that Theorem 3 implies that partial Cauchy data on arbitrary uniquely determine any two coefficients of the triple (*A*,*B*,*q*). A particular case is the following corollary.

For *j* = 1, 2, let for some *α*∈(0,1) be complex-valued. We assume either *A*_{1} = *A*_{2} or *B*_{1} = *B*_{2} in Ω. Then, implies (*A*_{1},*B*_{1},*q*_{1}) = (*A*_{2},*B*_{2},*q*_{2}).

## Sketch of Proof of Theorem 3

The key of the proof is the constructions of families of τ-parameterized solutions *u*_{1} = *u*_{1}(*τ*)(*x*) and *v* = *v*(*τ*)(*x*) with *τ*∈**R** satisfying *L*_{1}(*x*,*D*)*u*_{1} = 0, *u*_{1}|_{Γ0} = 0 and *L*_{2}(*x*,*D*)^{∗}*v* = 0, *v*|_{Γ0} = 0. Here, *L*_{2}(*x*,*D*)^{∗} is the adjoint to *L*_{2}(*x*,*D*) and . By *u*_{2} we denote the solution to *L*_{2}(*x*,*D*)*u*_{2} = 0 with *u*_{2}|_{∂Ω} = *u*_{1}|_{∂Ω} (we assume for simplicity that there is a unique solution of the boundary value problem). Then, the coincidence of the partial Cauchy data yields ∇*u*_{1}(*τ*) = ∇*u*_{2}(*τ*) on . Therefore, integration by parts gives [15]

Then, the proof relies on the constructions of suitable *u*_{1} and *v*, which are complex geometrical optics (CGO) solutions.

**CGO Solution.** We look for the geometrical optics solution *u*_{1} of the form: [16]

**The Phase Function.** Let the holomorphic function Φ = *φ* + *iψ* satisfy [17]where . The critical points of the function Φ play an important role in the proof. The following proposition shows that the union of the sets of critical points of the functions satisfying **17** is dense in Ω.

Let be an arbitrary point in Ω. There exists a sequence of functions {Φ_{ϵ}}_{ϵ∈(0,1)} satisfying **17** such that there exists a sequence , *ϵ*∈(0,1), and

Moreover, for any *j* from we have

Let be a tangential vector field to ∂Ω. In order to prove **7**, we use the phase function Φ given by the following lemma.

Let be an arc oriented clockwise with left endpoint *x*_{-} and right endpoint *x*_{+}. For any there exists a function Φ(*z*) that satisfies **17**, ImΦ|_{∂Ω∖Γ∗} = 0 and all the critical points of ImΦ from the set are nondegenerate, and the left or the right derivative of ImΦ of order seven is not equal to zero at *x*_{±}.

The functions , are defined by in Ω, , in Ω, . The amplitudes are of the forms , , where *a* is a holomorphic function and *d* is an antiholomorpic function such that on Γ_{0}. Here and henceforth, if ∂_{z}*a*(*z*) = 0, then we call *a* antiholomorphic.

Let be some fixed point from . In addition, the functions *a* and *d* have the following properties:

We introduce the operators and and the operators

Here, , , , , , *ζ* = *ξ*_{1} + *iξ*_{2}.

Denote , , where *M*_{1}(*z*) and are polynomials such that

Thanks to our assumptions on the regularity of *A*_{1}, *B*_{1}, and *q*, the functions *g*_{1}, *g*_{2} belong to .

The function *a*_{1}(*z*) is holomorphic in Ω and is antiholomorphic in Ω and

**Construction of the Correction Term** *u*_{11}. Let where and are polynomials such that

Let *e*_{1}(*x*), *e*_{2}(*x*) be smooth functions such that *e*_{1} + *e*_{2} ≡ 1, *e*_{2} vanishes in some neighborhood of , and *e*_{1} vanishes in some neighborhood of ∂Ω.

The function *u*_{11} is given by

**Construction of the Correction Terms** *a*_{2,τ}(*z*) **and**

Observe that the following asymptotic formulas hold true for any point on the boundary of Ω: where *σ*_{1}, , *m*_{1}, and are some smooth functions, , , , and , satisfy

We define the functions and satisfying

Let

Here, *M*_{5}(*z*), are polynomials such that . Let a holomorphic function *a*_{2,0} and an antiholomorphic function *d*_{2,0} satisfy

Finally, we set

**Construction of the Correction Term** *u*_{12}. We look for the function *u*_{12} in the form *u*_{12} = *u*_{-1} + *u*_{0}. The function *u*_{-1} is given by

We set *φ* = Re Φ and *O*_{ε} = {*x*∈Ω; dist(*x*,∂Ω) ≤ *ε*}. For the construction of *u*_{0}, first we consider the following boundary value problem: [18]

(a) Let *ε* > 0 be small such that , *f*∈*L*^{p}(Ω) with *p* > 2 and . There exists a solution of **18** satisfying

(b) Let *f*∈*L*^{2}(Ω) and *q* = 0. There exists a solution of **18** satisfying

Here, *C*_{1} > 0 does not depend on the choices of τ, *f*, *q*.

Let . Observe that the function can be represented as a sum of *m*_{j}(*τ*,·) where and

Moreover, . Then, the correction term *u*_{0} can be constructed using Lemma 1.

**Carleman Estimate.** Lemma 1 is derived from the following Carleman estimate with a degenerate weight function.

Suppose that Φ satisfies **17**. Then, there exist *τ*_{0} and *C* independent of *u* and τ such that [19]for all and all |*τ*| > *τ*_{0}.

**CGO for the Adjoint Equation.** The operator *L*_{1}(*x*,*D*)^{∗} has the form of the operator *L*_{1}(*x*,*D*) with different coefficients for the first- and zero-order terms. Similarly to *u*_{1}, we construct the CGO solution

Here, the functions , satisfy , in Ω, , and , . The smooth holomorphic function *b*(*z*) and the antiholomorphic function have zeros of order five on , are not equal to zero at , and satisfy the boundary condition on Γ_{0}.

Using the phase function Φ constructed in Proposition 9, we compute the right-hand side of **15** up to the terms of order :

Here, *F*_{0} and *F*_{1} are independent of τ. This immediately implies **7**. Moreover, the equation *F*_{1} = 0 implies that there exist a holomorphic function and an antiholomorphic function such that [20]and [21]

Computing the asymptotic formula of the right-hand side of **15** with an error up to the order and using **20**, **21**, we have [22]where , , and are some constants independent of τ.

Let η be a smooth function such that η is zero in some neighborhood of ∂Ω and . Observe that the partial Cauchy data of the operator *L*_{2}(*x*,*D*) and the operator *e*^{-sη}*L*_{1}(*x*,*D*)*e*^{sη} are exactly the same. Therefore, we have the analog of **22** for these two operators with and replaced by and . The coefficients *A*_{1}, *B*_{1} should be replaced by , . The functions will not change. The function *q*_{1} should be replaced by . This immediately implies that . By Proposition 8 we construct the set of functions Φ_{ϵ} satisfying **17** such that the union of the sets of the critical points of these functions is dense in Ω. This finishes the proof of **8** and **9**. The proof of the theorem is completed.

## Sketch of Proof of Theorem 1

For simplicity, we restrict ourselves to the case that Ω is simply connected. Suppose that the two operators generate the same partial Cauchy data. Multiplying the metric *g*_{2}, if necessary, by some positive smooth function , we may assume that [23]

Observe that without loss of generality, we may assume that there exists a smooth positive function *μ*_{2} such that *g*_{2} = *μ*_{2}*I*. Indeed, using isothermal coordinates we make a change of variables in the operator *L*_{2}(*x*,*D*) such that *g*_{2} = *μ*_{2}*I*. Then, we make the same changes of variables in the operator *L*_{1}(*x*,*D*). The partial Cauchy data for both operators obtained by this change of variables are the same.

Let ω be a subdomain in such that Ω∩*ω* = ∅, , and the boundary of the domain is smooth. We extend *μ*_{2} in as a smooth positive function and set in ω. By **23**, .

There exists an isothermal mapping *χ*_{1} = (*χ*_{1,1}, *χ*_{1,2}) such that the operator *L*_{1}(*x*,*D*) is transformed to the following form: [24]where *μ*_{1} is a smooth positive function in , and *C*_{1}, *D*_{1}, *r*_{1} are some smooth complex-valued functions. Consider a solution to the problem of the form **16** with the holomorphic weight function Φ_{1}. Then, the function *u*_{1}(*x*) = *w*(*χ*_{1}(*x*)) satisfies

Because the partial Cauchy data for the operators *L*_{1}(*x*,*D*) and *L*_{2}(*x*,*D*) are the same, there exists a function *u*_{2} such that [25]

Using **23** and **25**, we extend *u*_{2} on such that [26]

Let *φ*_{2} be the harmonic function in such that

We claim that [27]

Thanks to the Carleman estimate (**19**) there exists *τ*_{0} = *τ*_{0}(*ϵ*) such that [28]where *C*_{0} = *C*_{0}(*ϵ*) is independent of τ and *δ*_{ϵ} → 0 as *ϵ* → 0. On the other hand, . Here, , , . Then, by **26**, the following holds true: [29]

This equality implies **27** immediately. Indeed, let for some point from ω [30]

Then, there exists a ball such that [31]

Let us fix positive *ϵ*_{1} such that and 2*δ*_{ϵ1} < *α*^{′}. Form **29** by **28** and **31** we have where *τ* > *τ*_{0} if and *τ* < -*τ*_{0} if . The above inequality contradicts **30**.

Let Ξ = *χ*_{1,1} + *iχ*_{1,2}. Using the Cauchy–Riemann equations, we construct a harmonic function *ψ*_{2} such that the function Φ_{2} = *φ*_{2} + *iψ*_{2} is holomorphic in . Moreover, we take the function Φ_{1}, which may be holomorphically extended to some domain such that . Observe that Φ_{2} = Φ_{1}∘Ξ in ω. Then, in ω. The function Ξ may be extended up to a single-valued holomorphic function in such that and .

In Ω, consider the new infinitesimal coordinates for the operator *L*_{1}(*x*,*D*) given by the mapping . In these coordinates, the operator *L*_{1}(*x*,*D*) has the form [32]

Because , the partial Cauchy data for the operators *L*_{2}(*x*,*D*) and are exactly the same. The operators *L*_{2}(*x*,*D*) and are particular cases of the operator **1**. Because , the Cauchy data and are equal. We multiply the operator by the function and denote the resulting operator as . Therefore, by Corollary 4, there exists a function η that satisfies **3** such that . The proof of the theorem is completed.

## Footnotes

^{1}To whom correspondence should be addressed. E-mail: gunther{at}math.washington.edu.Author contributions: O.Y.I., G.U., and M.Y. performed research and wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

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