A Mathematical Journal
Vol. 7, No 3, (65 - 73). December 2005.

Conjectures in Inverse Boundary Value
Problems for Quasilinear Elliptic Equations

Ziqi Sun
Department of Mathematics and Statistics Wichita State university

Wichita, KS 67226, USA
ziqi.sun@wichita.edu

ABSTRACT
Inverse boundary value problems originated in early 80’s, from the contri-

bution of A.P. Calderon on the inverse conductivity problem [C], in which one
attempts to recover the electrical conductivity of a body by means of bound-
ary measurements on the voltage and current. Since then, the area of inverse
boundary value problems for linear elliptic equations has undergone a great deal
of development [U]. The theoretical growth of this area contributes to many ar-
eas of applications ranging from medical imaging to various detection techniques
[B-B][Che-Is].

In this paper we discuss several conjectures in the inverse boundary value
problems for quasilinear elliptic equations and their recent progress. These prob-
lems concern anisotropic quasilinear elliptic equations in connection with nonlin-
ear materials and the nonlinear elasticity system.

RESUMEN

Problemas inversos a valores en la frontera se desarrollaron a comienzos de
la década de los 80, a partir de contribuciones de A.P. Calderon en el problema
de conductividad inversa [C], en el cual se intenta recuperar las conductividad
eléctrica de un cuerpo mediante mediciones de voltaje y corriente en la frontera.
Desde entonces, el área de problemas a valores en la forntera inversos para ecua-
ciones lineales eĺıpticas ha sido objeto de mucho desarrollo [U]. El crecimiento de
la teoŕıa en esta área tiene aplicaciones en muchas aplicaciones, las que vaŕıan
desde imagenoloǵıa médica, hasta diversos métodos de detección [BB], [Che-Is].
En este art́ıculo, discutimos varias conjeturas en problemas inversos de valores en



66 Ziqi Sun
7, 3(2005)

la frontera para ecuaciones eĺıpticas quasi-lineales y sus progresos recientes. Es-
tos problemas dicen relación con ecuaciones eĺıpticas quasilineales anisotrópicas
en conexión con materiales nolineales y sistemas de elasticidad no lineal.

Key words and phrases: Inverse boundary value problem. Dirichlet to
Neumann map

Math. Subj. Class.: 35R30

1 Anisotropic Quasilinear Conductivity Equations

Consider the quasilinear elliptic equation

LAu =
n∑

i,j=1

(aij (x, u)uxi )xj = 0, u|Γ = f ∈ C
2,α(Γ) (1)

on a bounded domain Ω ⊂ Rn, n ≥ 2, with smooth boundary Γ. Here A(x, t) =
(aij (x, t))n×n is the quasilinear coefficient matrix which is assumed to be in the C1,α

class with 0 < α < 1. The nonlinear Dirichlet to Neumann map

ΛA : f → ν · A(x, f )∇u|Γ

is an operator from C2,α(Γ) to C1,α(Γ), which carries essentially all information about
the solution u which can be measured on the boundary. Here we denote ν to be the
unit outer normal of Ω. The inverse problem under discussion is to recover information
about the quasilinear coefficient matrix A from the knowledge of ΛA.

This problem was raised by R. Kohn and M. Vogelius [KV] in mid 80’s as a nonlin-
ear analogue of the well known inverse conductivity problem posed by A.P. Calderon
[C]. Physically, the problem is connected to Electrical Impedance Tomography in
nonlinear media.

It has been shown in [Su1] that, in the isotropic case of the problem, i.e., when A
is a scalar matrix, the Dirichlet to Neumann map ΛA gives full information about A.
In other words, ΛA determines A uniquely as a function on Ω × R. This generalizes
to the quasilinear case the well known uniqueness theorems of the linear case (i.e.,
when A is scalar and is indenpendent on t)[SU1,2][SuU2] [N].

In the anisotropic case, however, one only expects to recover A module the group

G = {all C3,α diffeomorphisms Φ: Ω̄ → Ω̄ with Φ
∣∣
∂Ω

= identity}.

In fact, ΛA is invariant under G: For any A and Φ ∈ G, ΛA = ΛHΦA. Here HΦA is
the pull back of A under Φ:

HΦA(x, t) = (|detDΦ|−1(DΦ)T A(x, t)(DΦ)) ◦ Φ−1 (2)



7, 3(2005)
Conjectures in Inverse Boundary Value Problems for ... 67

where DΦ is the Jacobian matrix of Φ. One should observe that (2) holds only when
Φ is independent on t. Thus, the following conjecture is natural:

Conjecture 1: Assume that ΛA1 = ΛA2 . Then there exists a unique diffeomorphism
Φ ∈ G so that A2 = HΦA1.

In [SuU1] we have verified this conjecture in the C2,α category for dimension n = 2
and in the real analytic category for dimension n ≥ 3. These results extend all known
results regarding this conjecture in the case of linear coefficient matrices (i.e. when
A is independent of t), obtained earlier in the works of Sylvester [S], Nachman [N]
and Lee-Uhlmann [LU]. We mention that in the two dimensional case the unique
diffeomorphism Φ in the result belongs to the C3,α class, which is one order smoother
than A1 and A2 and in the case n ≥ 3, Φ is in the real analytic category. Assuming
Holder smoothness for the coefficient seems quite essential to assure that Φ is one
order smoother than the coefficient matrices. As explained in [SuU1], this is closely
related to the elliptic regularity theory.

The proof is based on a well known linearization technique introduced in [I1] and
further developed in [I2][IS][IN][Su1,3] which reduces the nonlinear problem to a linear
one. Let t ∈ R and g ∈ C2,α(Γ). From ΛA one determines two linear operators:

K
(1)
A,t : g → d/dsΛA(t + sg)|s=0

K
(2)
A,t : g → d

2/ds2(s−1ΛA(t + sg))|s=0 (3)

One observes that K(1)A,t = ΛAt , the Dirichlet to Neumann map corresponding to the
linear coefficient matrix At(x) = A(x, t) for a fixed t. So, if ΛA1 = ΛA2 for two
quasilinear coefficient matrices A1 and A2, then ΛAt1 = ΛAt2 , ∀t ∈ R, and since the
conjecture is true in the linear case, one obtains a family of diffeomorphisms Φt ∈ G,
depending on the parameter t, so that

HΦt A
t
1 = A

t
2, ∀t ∈ R. (4)

The mathematical difficulty is to show that Φt is actually independent on t, which
would imply the result. It has been verified in [SuU1] that Φt is smooth in t. For
dimension n ≥ 3, this was achieved by studying a related geometrical problem in
which Φt becomes a family of isometries between two families of Riemannian metrics
|Ati|

1/(n−2)(Ati)
−1 on Ω̄, i = 1, 2. For n = 2, One can transform it to a similar problem

where Φt becomes a family of conformal diffeomorphisms between Riemannian metrics
(Ati)

−1, i = 1, 2. In the latter case, the smoothness is verified via the standard theory
of the Beltrami equation [AB].

So, the task is to show that Φ̇|t=0, where dot means differentiation in t variable.
We only give a very brief description of the proof. One only needs to show

Φ̇0 = Φ̇t|t=0 = 0 (5)



68 Ziqi Sun
7, 3(2005)

since the same argument works for t 6= 0. By a transformation one may assume that
Φ0 = identity map. The proof of (5) is then based on the information obtained from
(3):

K
(2)
A1,t

= K(2)A2,t. (6)

A crucial step of the proof is to show that one can recover from K(2)A,t information
about ∂A/∂t(x, 0). So (6) implies

∂

∂t
A1(x, 0) =

∂

∂t
A2(x, 0), ∀x ∈ Ω. (7)

One views (7) as a certain control over the flows At1 and A
t
2 at t = 0. Actually,

the assumption Φ0 =Id. together with (7) give A01 = A
0
2 and Ȧ

0
1 = Ȧ

0
2. Consider

now the solution flows uti,f for the linear equations LAti (u
t
i,f ) = 0 with u

t
i,f |Γ = f ,

i = 1, 2. One observes that the control over the flows of coefficient matrices translates
to a control over the solution flows. In fact, for every f , u01,f = u

0
2,f and u̇

0
1,f = u̇

0
2,f .

Since the transformation in (4) links ut1,f to u
t
2,f via the relation u̇

t
1,f = u̇

t
2,f ◦ Φ

t, one
differentiates it in t at t = 0 to get Φ̇0 ·∇u01,f = 0 for all boundary value f , from which
(5) follows by an argument based on Runge approximation. See [SuU1] for details.

The above result obtained in [SuU1] covers the two dimensional case and the real
analytic case in dimension three or higher. However, the remaining case in dimension
n ≥ 3 is essentially open even when the equation (1) is linear. An interesting problem
for further study in this direction is whether one can reduce the conjecture in the
quasilinear case directly to the conjecture in the linear case. In other words, one
would like to verify Conjecture 1 under the assumption that Conjecture 1 holds in
the linear case. Such a full reduction has already been obtained in the scalar case
(where A is a scalar matrix) [Su1]. It is possible that the same reduction also hold
in the anisotropic case. One possible approach to attack this problem is to further
study the relation between (6) and (7) in the general case, which is the heart of proof
in [SuU1]. The main issue is how to avoid the use of the property of completeness
of products of solutions which is currently available only in the two dimensional case
and the case of real analytic coefficient matrices.

2 Quasilinear Equations in Connection with Non-
linear Elastic Materials

Consider the quasilinear elliptic equation

∇ · A(x, ∇u) = 0, u|Γ = f ∈ C3,α(Γ), (8)

on a bounded domain Ω ⊂ Rn, n ≥ 2, with smooth boundary Γ. Here A(x, p) =
(a1(x, p), a2(x, p), ..., an(x, p)) is the quasilinear coefficient vector. We assume that
A and Ap (which is assumed to be symmetric) are both in C2,α(Ω̄×R) with 0 < α < 1,
A(x, 0) = 0 and the structure conditions which guarantee the unique solvability in
the C3,α class [HSu].



7, 3(2005)
Conjectures in Inverse Boundary Value Problems for ... 69

The nonlinear Dirichlet to Neumann map

ΛA : f → ν · A(x, ∇u)|Γ, (9)

is an operator from C3,α(Γ) to C2,α(Γ), which carries essentially all information about
the solution u observable on the boundary. One verifies that ΛA is invariant under
the group G: ΛA = ΛHΦA for all Φ ∈ G. Here the transformation HΦ is defined as

HΦA(x, p) = (|detDΦ|−1(DΦ)T A(x, (DΦ)p)) ◦ Φ−1.

The main problem is whether the converse is true.

Conjecture 2: Assume that ΛA1 = ΛA2 . Then there exists a unique diffeomorphism
Φ ∈ G so that A2 = HΦA1.

The equation (8) can be considered as a simple scalar model of the nonlinear
elasticity system, which takes the form

∇{σ(x, E) + (∇u)σ(x, E)} = 0, (10)

where u is the displacement vector function resulting from a deformation of an elastic
body and the matrix function σ is the constitutive relation with the strain tensor

E =
1
2

(∇uT + ∇u + ∇uT ∇u).

In [HSu], we developed a mathematical framework towards proving this conjecture
in the case of two dimensions. In the discussion below, we assume ΛA1 = ΛA2 for two
quasilinear coefficient vectors A1 and A2 in dimension two. By linearizing (9) one
obtains, as in the case of Conjecture 1, a family of diffeomorphisms {Φf }⊂ G which
transforms A1,p(x, ∇u1,f ) to A2,p(x, ∇u2,f ):

A2,p(x, ∇u2,f ) = HΦf A1,p(x, ∇u1,f ),

and the main problem is to show that Φf is independent on f . Here we denote by
ui,f solution of (11) with A replaced by Ai, i = 1, 2.

One notices that {Φf , f ∈ C2,α(Γ)} is an infinite dimensional family rather than
an one dimensional family in the case of Conjecture 1. Also, contrary to (3), any
further linearization on (9) would not provide any new information about Φf . So,
technically, the task in this case is much harder to accomplish.

For a f ∈ C3,α(Γ), let gi,f be the Riemannian metric (on Ω̄) generated by the
metrix A−1i,p (x, ∇ui,f ), i = 1, 2. One verifies that Φf is a family of conformal diffeo-
morphisms sending (Ω̄, g1,f ) to (Ω̄, g2,f ). If one uses Φ∗f g to denote the pullback of a
tensor g under Φf , then (15) can be rewritten as

Φ∗f g2,f = |DΦf |g1,f .

Given f , h ∈ C3,α(Γ), Let’s denote by ġi,f,h the Frechet derivative of gi,f at the
reference point f in the direction h, i = 1, 2. Once again, one can show that Φf



70 Ziqi Sun
7, 3(2005)

is smooth in f (parallel to those in Conjecture 1) and we denote by X = Φ̇f,h the
corresponding derivative of Φf in the direction h (viewed as a vector field). For a
fixed f , we may once again assume that Φf = identity and set g1,f = g2,f =: gf and
u1,f = u2,f =: uf .

In order to prove the conjecture by showing

X = Φ̇f,h = 0, ∀h ∈ C3,α(Γ), (11)

Let us take a deep look at the relation Φ∗f g2,f =| DΦf | g1,f by differentiating it in f
with a direction h ∈ C3,α(Γ). We get

ġ1,f,h − ġ2,f,h =  LX gf − (eσ∇gf · (e
−σX))gf . (12)

where LX gf stands for Lie derivative of gf under the vector field X and σ = log
√

det(g).
Equation (12) implies that X is connected to the inhomogeneous conformal Killing
field equation (with respect to the metric gf ) with the boundary condition X |Γ= 0.
However, this equation has no real consequence if one just considers one direction.
The main observation made in [HSu] is that if one considers a pair of directions, then
one can use the theory of conformal Killing field to obtain useful consequences leading
to (11). Indeed, when one is given a pair of directions h1, h2 ∈ C2,α(Γ), one can show
that the following symmetric relation

ġf,h1 lf,h2 = ġf,h2 lf,h1

holds for ġf,h1 = ġ1,f,h1 or ġ2,f,h1 and lf,h = ∇gf u̇f,h = g
−1
f ∇u̇f,h. This is proven in

[HSu] using the special structure of the linearized coefficient matrix. Combining this
symmetric relation together with (12) one gets

lf,h2c(LX1 gf − (e
σ∇gf · (e

−σX1))gf ) = lf,h1c(LX2 gf − (e
σ∇gf · (e

−σX2))gf ), (13)

where Xi = Φ̇f,hi , i = 1, 2. Equation (13) implies that both Xi, i = 1, 2, satisfy the
inhomogeneous conformal Killing field equation of the type

lc(LX (g) − (eσ∇ · (e−σX))g) = F (14)

with the same inhomogeneous term F , which is a 1-form. The equation (14) is the
crucial equation for the proof. We have proven that if X and l satisfy the equation (14)
with X |Γ= 0, then both inner products

〈
l, X

〉
g

and
〈
l⊥, X

〉
g

are uniquely determined
by F, where l⊥ stands for the unique vector perpendicular to l with

∥∥l⊥∥∥ = ∥∥l∥∥
in the counterclockwise direction under the metric g [Su2], Base on this result, one
concludes from (13) that the vector fields Xi and lf,hi must satisfy the following system
of equations: { 〈X1,lf,h2 〉gf = 〈X2, lf,h1 〉gf〈

X1,l⊥f,h2
〉

gf
=

〈
X2, l

⊥
f,h1

〉
gf

, (15)



7, 3(2005)
Conjectures in Inverse Boundary Value Problems for ... 71

To understand (15) better, consider now a two-parameter family of conformal
diffeomorphisms Φf +η1h1+η2h2 ⊂ G with parameters η1 and η2 in R. For a fixed
point x ∈ Ω, define

ω(η1, η2) = Φf +η1h1+η2h2 (x) : R
2 → Ω̄

as a function from (η1, η2) to the image of x under Φf +η1h1+η2h2 . One checks that

ωη1 = Φ̇f +η1h1+η2h2,h1 (x), ωη2 = Φ̇f +η1h1+η2h2,h2 (x).

By Replacing f by f + η1h1 + η2h2 one can shows from (15) that the function ω
satisfies the following first order system:

{ 〈ωη1 , l2〉g = 〈ωη2 , l1〉g〈
ωη1 , l

⊥
2

〉
g

=
〈
ωη2 , l

⊥
1

〉
g

, (16)

where
lj = lf +η1h1+η2h2,hj ◦ Φf +η1h1+η2h2 , j = 1, 2.

Here the additional term Φf +η1h1+η2h2 is needed once one removes the assumption
Φf = identity.

System (16) can be viewed as a generalized Cauchy-Riemann system under the
vector fields l1 and l2. The proof of (11) with h = h1 and h2 is now reduced to
showing that System (16) admits no bounded nonconstant solution ω. Note that ω is
always bounded. In order to do that, one way is to apply Liouville’s type theorems
to the system (16). However, one must choose the directions h1 and h2 in a way that
the gradients of the solution l1 and l2 are uniformly independent. Once (11) is proven
with two independent directions, one can show that (11) holds for all directions. This
is proven in [HSu] using the geometric argument developed in [Su2].

In [HSu] the above framework has been successfully to two important special
cases: The case in which A(x, p) is independent of x and the case in which Ap(x, p)
is independent of p. In both cases one is allowed to construct the needed independent
directions h1 and h2. See [HSu] for details.

To verify the conjecture completely, the main difficulty is the construction of
special directions. The construction of special directions in the known cases has
been completed by using techniques of exponentially growing solutions, which is not
available in the general case. One possible way to overcome this difficulty is to replace
the two-parameter family of conformal diffeomorphisms Φf +η1h1+η2h2 by ΦF (η1,η2),
where F (η1, η2) is a two dimensional nonlinear variety in C3,α(Γ) passing through f .
The nonlinearity of F (η1, η2) should correspond to the quasilinear nature of A(x, p).
Once one identifies the correct form of F (η1, η2), the rest of the argument can be
modified to cover the general case.

Received: April 2004. Revised: May 2004.



72 Ziqi Sun
7, 3(2005)

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