Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

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334

Chapter 19: Developments in Inverse Problems

19.8

Anisotropic Conductors

In §§19.1-19.5 we considered isotropic conductivities, i.e., the electrical prop-

erties of 11 do not depend of direction. Examples of anisotropic media are

muscle tissue. In this section we consider the inverse conductivity problem

for anisotropic medium. The problem is well understood in two dimensions.

Using isothermal coordinates [2] one can in fact reduce, by a change of vari-

ables, the anisotropic conductivity equation to an isotropic one and therefore

one can apply the two-dimensional results of §19.5. Of course, this is not

available in dimension n > 2. In fact, in this case the problem is equivalent

to the problem of determining a Riemannian metric from the Dirichlet to

Neumann map associated to the Laplace-Beltrami operator [34]. Only the

real-analytic case is understood at present [34].

In this section we will consider the case of a quasilinear anisotropic con-

ductivity. We outline recent results [57] proving identical results to the linear

case. One needs to go further than the linearization procedure of Lemma

19.13 for isotropic non-linear conductivities. In fact, we show that one can

reduce the problem question about the density of product of solutions for the

linear anisotropic conductivities by using a second linearization.

We assume that ry(x, t) E Cl,°(S2 x IR) be a symmetric, positive definite

matrix function satisfying

'y(x, t) > CT',

(x, t) E Ti x [-T,T], T > 0,

(19.163)

where CT > 0 and I denotes the identity matrix.

It is well known (see, e.g., [21]) that, given f E C2'"(SI), there exists a

unique solution of the boundary-value problem

f 0 - (7(x, u) Vu) = 0 in 12,

ulan = f.

(19.164)

We define the Dirichlet to Neumann map A., : C1,11(8Q) --+ Cl,"(00) as the

map given by

A7: f

f)Dul

an, (19.165)

where u is the solution of (19.164) and v denotes the unit outer normal of

852.

Physically, y(t, u) represents the (anisotropic, quasilinear) conductivity

of Sl and Ay(f) the current flux at the boundary induced by the voltage f.

We study the inverse boundary-value problem associated to (19.164): how

much information about the coefficient matrix y can be obtained from knowl-

edge of the Dirichlet to Neumann map A,.?

Uhlmann 335

The uniqueness, however, is false in the case where y is a general matrix

function as it is also in the linear case [33]: if 4i : ci -3 ci is a smooth

diffeomorphism which is the identity map on 811, and if we define

(44'Y)(x,t) _

(D4))TID I)(D4)) o4t-1(x),

(19.166)

then it follows that

A*..y = Ay,

(19.167)

where D4) denotes the Jacobian matrix of 4i and ID4I = det(D4)).

The main results of [57] concern with the converse statement. We have

Theorem 19.41. Let 52 C R2 be a bounded domain with C3,°` boundary,

0 < a < 1. Let y1 and y2 be quasilinear coefficient matrices in C2"a(S2 x R)

such that A.y, = A.y..

Then there exists a C3'° diffeomorphism 4i

S2 -* S2

with 1Iest = identity such that y2 = 4).71

Theorem 19.42. Let SZ C R", n > 3, be a bounded simply connected do-

main with real-analytic boundary. Let y1 and y2 be real-analytic quasilinear

coefficient matrices such that A, = Ate. Assume that either y1 or y2 extends

to a real-analytic quasilinear coefficient matrix on R". Then there exists a

real-analytic diffeomorphism 4):11 - 12 with 4ol., = identity, such that

y2 = Ob.'Y1

Theorems 19.41 and 19.42 generalize all known results for the linear case

[64]. In this case and when n = 2, with a slightly different regularity assump-

tion, Theorem 19.41 follows using a reduction theorem of Sylvester [60], using

isothermal coordinates, and Theorem 19.14 for the isotropic case.

In the linear case and when n > 3, Theorem 19.42 is a consequence of the

work of Lee and Uhlmann [34], in which they discussed the same problem

on real-analytic Riemannian manifolds. The assumption that one of the

coefficient matrices can be extended analytically to R" can be replaced by a

convexity assumption on the Riemannian metrics associated to the coefficient

matrices. Thus Theorem 19.42 can also be stated under this assumption,

which we omit here. We mention that, in the linear case, complex geometrical

optics solutions have not been constructed for the Laplace-Beltrami operator

in dimensions n > 3. The proof of Theorem 19.1 in the linear case follows a

different approach.

A. Linearization.

The proof of linearization Lemma 19.13 is also valid in

the anisotropic case. We shall use yt to denote the function of x obtained by

freezing tin y(x, t).

336

Chapter 19: Developments in Inverse Problems

Under the assumptions of Theorem 19.6, using Lemma 19.13 we have

that

A7i = A7,t,

V t E JR.

(19.168)

Since Theorems 19.41 and 19.42 hold in the linear case, it follows that there

exists a diffeomorphism fit, which is in CQ when n = 2 and is real-analytic

when n > 3, and the identity at the boundary such that

(19.169)

It is proven in [57] that 4?t is uniquely determined by y and thus by yt,

l = 1, 2. We then obtain a function

4?(x,t)=V(x):n xR -+sixR, (19.170)

which is in C3"(SZ) for each fixed t in dimension two and real analytic in

dimension n > 3. It is also shown in [57] that 41 is also smooth in t. More

precisely we have, in every dimension n > 2, that ee E C2,a(S2).

In order to prove Theorems 19.41 and 19.42, we must then show that 4

is independent of t. Without loss of generality, we shall only prove

- I

in S2.

(19.171)

t=o

It is easy to show, using the invariance (19.167) that we may assume that

41(x, 0) - x, that is,

4i° = identity.

(19.172)

Let us fix a solution u E C3,,, (Sj) of

V AVu = 0, ul = f, (19.173)

where we denote A = y° = 72.

For every t E JR and I = 1, 2, we solve the boundary-value problem

(19.173) with 7t replaced by yi. We obtain a solution u1,):

inf

u'(t)I =

1=1,2.

It is easy to check that

(19.174)

u(2)((Dt(x)), x E S2.

Uhlmann

337

Differentiating this last formula in t and evaluating at t = 0 we obtain

autl) -

8u 2i

- X Vu = 0,

x E SZ,

(19.175)

t=o

where

X = (19.176)

It is easy to show that X Vu = 0 for every solution of (19.173) implies

X = 0. So we are reduced to prove

Nl)

au(2)

at at

= 0. (19.177)

two

Using (19.174) we get

V (71 (XI t)Dut(l)) - V ('Y2(x, t)Vu'(2)) = 0.

(19.178)

Differentiating (19.178) in t at t = 0 we conclude

+V.

+ V

'2) = 0.

{(7' _

222

'(1)

t=o

(19.179)

We claim that in order to prove (19.177) it is enough to show that

[(°yl _

72)I

Qul

=o.

at J

t=o J

This is the case since we get from (19.179) and (19.180)

V [Av

(

(1) -

I -

= 0.

(19.180)

The claim now follows since the operator V AV : HH(Sl) n H1(1) -+ L2(Q)

is an isomorphism and therefore

au(l)

_ Ou(2) /

=0.

t=o

B. Second linearization and products of solutions. In order to show

(19.180) we now study the second linearization. We introduce, for every

338

Chapter 19: Developments in Inverse Problems

t E R, the map K7,t : C2,"(8Sl) -a H'2(80) which is defined implicitly as

follows (see [53]): for every pair (fl, f2) E C21"(8f2) X C2,"(8fZ),

JI1KA,t(f2)dS

Dul

Vu dx

(19.181)

with u1,

l = 1, 2, as in (19.174) with f replaced by ft,

l = 1, 2. We have

Proposition 19.43 ([53]). Let y(x, t) be a positive definite symmetric ma-

trix in C2(fZ X R), satisfying (19.163). Then for every f E C2,* (0Q) and

tEIIY,

lim11s

[8'A,(t + sf)

AA,(f)J

-KA,t(f)II 0.

Under the assumptions of Theorems 19.41 and 19.42, using Proposi-

tion 19.43 with t = 0, we obtain

K7,,.(f) = K720(f),

V f E C3."(8fZ).

Thus, by (19.181) we have

VuT

a_Y1

t_o

Due dx = f Dui

two

Vu 22 dx,

with u1, u2 solutions of (19.174). By writing

_ L71

_ 8'Y2

at at

}

t=o

(19.182)

(19.183)

and replacing in (19.182) ul by u, and u2 by (u1 + u2)2 - ui - u2i we obtain

Vu B(x)V(ulu2)dx = 0

(19.184)

with u, ul and u2 solutions of (19.174).

To continue from (19.184), we need the following two lemmas.

Lemma 19.44. Let h(x) E C' (S2) be a vector-valued function. If

h(x)V (ulu2)dx = 0

for arbitrary solutions ul and u2 of (19.174), then h(x) lies in the tangent

space Tx(8f2) for all x E 00.

Lemma 19.45. Let A(x) be a positive definite, symmetric matrix in C2

Define

DA = Spanp(n){uv : u, v E C3," (St), V AVu = V AVv = 0}.

Then the following are valid:

Uhlmann

339

(a) If I E 00 (U) and l1 DA, then l = 0 in S2,

(b) If n = 2, then DA = L2(I).

Now we finish the proof of (19.180) concluding the proofs of Theorems

19.41 and 19.42.

By Lemma 19.44 we have that v B(x)Vu - 0 in 852. Integrating by

parts in (19.184), we obtain

fEy B(x)Vu]u1u2dx

= 0. (19.185)

We now apply Lemma 19.44 to (19.185). If n > 3, we have that y1 and

y2 are real-analytic on S2 x It Thus B E C'(Sl). Since the solutions u solve

an elliptic equation with a real-analytic coefficient matrix, we have that u is

analytic in Q. If u is analytic on fl, we can conclude from Lemma 19.45 that

V (B(x)Vu) = 0, x E SZ. (19.186)

We shall prove that (19.186) holds independent of whether u is analytic

up to 852 or not. This is due to the Runge approximation property of the

equation. Using the assumptions of Theorem 19.42, we extend A analytically

to a slightly larger domain Il D a. For any solution u E C3,°(Sl) and an

open subset 0 with 0 C fl, we can find a sequence of solutions {u,.} C

C"(?), which solves (19.184) on Sl, and u,n l

-+ ul in the L2 sense,

1101

m-+oo 0,

where d1 C 52, O C O1. By the local regularity theorem of elliptic equations

this convergence is valid in H2(0). Since (19.186) holds with u = u,n, letting

m -+ oo yields the desired result for u on O. Thus (19.184) holds. If n = 2,

Lemma 19.44(b) implies that V V. (B(x)Vu) = 0 for any solution u E C3.°(SZ).

The proof of Lemma 19.44 follows an argument of Alessandrini [4], which

relies on the use of solutions with isolated singularities. It turns out that in

our case, only solutions with Green's function type singularities are sufficient

in the case n > 3, while in the case n = 2, solutions with singularities of

higher order must be used. There are additional difficulties since we are

dealing with a vector function h. We refer the readers to [57] for details.

The proof of part (a) of Lemma 19.45 follows the proof of Theorem 1.3

in [3] (which also follows the arguments of [31]).

Namely, one constructs

solutions u of (19.173) in a neighborhood of fl with an isolated singularity

of arbitrary given order at a point outside of Q. We then plug this solution

into the identity

lu2dx = 0.

By letting the singularity of u approach to a point x in Oil, one can show

that any derivative of I must vanish on x and thus by the analyticity of 1,

1 - 0 in St. For more details, see (57].

340

Chapter 19: Developments in Inverse Problems

To prove the part (b) of Lemma 19.45, we first reduce the problem to the

Schrodinger equation.

Using isothermal coordinates (see [2]), there is a conformal diffeomor-

phism F : (S2, g) -3 (V, e), where g is the Riemannian metric determined by

the linear coefficient matrix A with gii = A7'. One checks that F transforms

the operator V AV (on Sl) to an operator V AT (on S2') with A' a scalar

matrix function ,Q(x)I. Therefore the proof of the part (b) is reduced to the

case where A =,8I, with /3(x) E

C2.°(Il). By approximating by smooth so-

lutions, we see that the C3"° smoothness can be replaced by H2 smoothness.

Thus we have reduced the problem to showing that

Da = SpanL2{uv : u, v E H2(S2); V # Vu = V # Vv = 01 = L2(1).

We make one more reduction by transforming, as in §19.3, the equation

V QVu = 0 to the Schrodinger equation

Ov-qv=0

with

u = /3 2v, q =

E C°(S2). (19.187)

This allows us to reduce the proof to showing that

Dq = SpanL2{vlv2 : Vi E H2(fl), Ovi - qvi = 0, i = 1, 2} = L2(SZ)

(19.188)

for potentials q of the form (19.187).

Statement (19.188) was proven by Novikov [46]. In [57) it was shown that

it is enough to use the Proposition below which is valid for any potential

q E L°°(SZ). This result uses some of the techniques of [61,62] similar to the

proof of Theorem 19.20.

Proposition 19.46. Let q E L°°(1l), n = 2. Then Dq has a finite codimen-

sion in L2(0).

It is an interesting open question whether Dq = L2(1l) in the two-

dimensional case.

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