Localisation Inverse Problem and Dirichlet-to-Neumann Operator for Absorbing Laplacian Transport ()
1. Laplacian Transport and Dirichlet-to-Neumann Operators
The theory of Dirichlet-to-Neumann operators is the basis of many research domains in analysis, particularly, those concerning Laplacian transports. It is also very important in mathematical-physics, geophysics, electrochemistry. Moreover, it is very useful in medical diagnosis, such as electrical impedance tomography:
In 1989, J. Lee and G. Uhlmann have introduced an example on the determination of conductivity matrix field in a bounded open domain, see e.g. [1]. This example is related to measuring the elliptic Dirichlet-toNeumann map for associated conductivity equation, see e.g. [1].
The problem of electrical current flux is an example of so-called diffusive Laplacian transport. Besides the voltage-to-current problem, the motivation to study this kind of transport comes for instance, from the transfer across biological membranes, see e.g. [2,3].
Let some species of concentration
, 
, diffuse stationary in the isotropic bulk 
from a (distant) source localised on the closed boundary 
towards a semipermeable compact interface 
of the cell
, where they disappear at a given rate
. Then the steady field of concentrations (Laplacian transport with a diffusion coefficient
) obeys the set of equations:
(P1) 
Usually, one supposes that
, 
, is a constant concentration of the species inside the cell
.
This example motivates the following abstract stationary diffusive Laplacian transport problem with absorption on the surface
:
(P2) 
This is the Dirichlet-Neumann problem for domain 
with the Robin [4] boundary condition on the absorbing surface
. Varying 
between 
and
, one recovers respectively the Neumann and the Dirichlet boundary conditions.
Now, we can associate with the problem (P2) a Dirihlet-to-Neumann operator
(1)
Domain 
belongs to a certain Sobolev space of functions on the boundary
, which contains
, the solutions of the problem (P2) for given
and for the Robin boundary condition on 
fixed by 
and
.
The advantage of this approach is that as soon as the operator (1) is defined, one can apply it to study the mixed boundary value problem (P2). This gives, in particularly, the value of the particle flux due to Laplacian transport across the membrane
. Moreover, the total current across the boundary 
can be defined (for given
) in term of Dirihlet-to-Neumann operator (1) as follows:
(2)
where 
designed the differential element relative to
.
There are at least two inverse problems derived from problem (P2):
a) geometrical inverse problem: given Dirichlet data 
and the corresponding (measured) Neumann data
, in (1), on the accessible outer boundary
, to reconstruct the shape of the interior boundary
, see [5].
b) localisation inverse problem: concerns to localisate of the domain (cell) 
with a given shape and the fixed parameters 
and
, see [6].
The main question in this context is to find sufficient conditions insuring that the localization inverse problem is uniquely soluble. Indeed:
First, we relate the above problems a) and b) with the Dirichlet-to-Neumann operator (1) by defining explicitly this operator, whose can define the local and total current across the external boundary
, which are useful to resolve a) and b).
Second, we study the localisation inverse problem in the framework of application outlined in the problem (P2), which consist in finding sufficient (Dirichlet-toNeumann) conditions to localise the position of the cell 
from the experimentally measurable macroscopic response parameters.
In Section 2, we introduce the existence and uniqueness for the solution of problem (P2). In Section 3, we introduce our first main result concerning the study of spherical case of problem (P1), whose we give a general method to resolve the type of partial derivative system like (P1), see proposition 3.2. Indeed, we allow an explicit calculations, based on Green-Ostrogradski theorem, for the solution of this problem.
In Section 4, it is our second main result which consist in showing that total current across the external boundary
, involving Dirihlet-to-Neumann operator (1), can resolve the localisation inverse problem in three dimensional case, when the compact
.
2. Uniqueness of the Problem (P2)
We suppose that 
and 
be open bounded domains in 
with 
-smooth disjoint boundaries 
and
, that is 
and 
.
Then the unit outer-normal to the boundary 
vector-field 
is well-defined, and we consider the normal derivative in (P2) as the interior limit:
(3)
The existence of the limit (3) as well as the restriction 
is insured since 
has to be harmonic solution of problem (P2) for 
-smooth boundaries 
[7].
Now, we introduce some indispensable standard notations and definitions, see [8]. Let 
be Hilbert space 
on domain 
and 
denote the corresponding boundary space. We denote by 
the Sobolev space of 
-functions, whose 
derivatives are also in
, and similar, 
is the Sobolev space of 
-functions on the 
-smooth boundary
.
Proposition 2.1. Let 
for 
-smooth boundaries
. Then the Dirichlet-Neumann problem (P2) has a unique (harmonic) solution in domain
.
Proof. For existence we refer to [7]. To prove the uniqueness, we consider the problem (P2) for 
and
. Then by Gauss-Ostrogradsky theorem, one gets that the corresponding solution 
yields:
(4)
The estimate (4) implies that
. Hence by the boundary condition one gets
, and from
we obtain that for
, the harmonic function 
for
. 
The next statement is a key for analysis of inverse localisation problems:
Proposition 2.2. Consider two problems (P2) corresponding to a bounded domain 
with 
- smooth boundary 
and to two subsets 
and 
with the same smoothness of the boundaries
,
. If for solutions
, 
of these problems one has
(5)
then
.
Proof. By virtue of 
and by condition (5), the problem (P2) has two solutions for identical external (on
) and internal (on 
and
) Robin boundary conditions. Then by the standard arguments based on the Holmgren uniqueness theorem [9] for harmonic functions on
, one obtains that
. 
3. Dirichlet-to-Neumann Operators for Absorbing Laplacian Transport
Here, we consider the spherical shell of the problem (P1) so that 
and the absorbing cell is also a ball
, whose we denote by 
the distance between the two centers
.
Hereafter, we denote the previous hypothesis by spherical case.
In the sequel, we resolve the problem (P1) in order to calculate explicitly Dirichlet-to-Neumann operator relative to this case.
Before resolving problem (P1), we need the following theorem which the key of the solution:
Theorem 3.1. (Gauss-Ostrogradski)
Let 
a field vector across the domain
, having as border
.
(6)
whose 
designated the divergence of field vector
. 
and 
designated respectively the differential elements relative to 
and
. 
designated the unit outer-normal vector on 
at arbitrary point
.
Remark 1. Let the orthonormal reference with origin 
and axis
, which is keen on the line 
in the sense of the vector
.
On the other hand, since for all
, spherical harmonic function 
is independent of 
if
, then we note:
Since 
is harmonic function, then it takes the following form, see [10]:
(7)
Therefore, we need to calculate the coefficients of (7) from the condition boundaries. Indeed, since the radius of points of 
are equal to constant
, then the condition boundary on 
implies easily, by identification, the following system:
But, on the boundary
, the radius aren’t equal, and depend of spherical angle
. Then, for this reason, we use Gauss-Ostrogradski theorem’s, whose we show that it is useful to find another relation between the coefficients of (7) like
. Consequently, we get for each
, a system of two equations with two unknowns 
and
, which it is sufficient to calculate 
and
:
Proposition 3.2. The condition boundary on 
implies:
(8)
Proof. Let
, we construct the following vector field 
by:
(9)
whose, 
is a primitive relative to 
for the following function:
Calculate the flux of field 
across the domain 
using Gauss-Ostrogradski theorem 3.1:
(10)
where:
1. Calculate
:
In domain
, we have:
On the other hand, 
can be calculated from (9) by:
Then, we deduce that:
Therefore, from Fubini’s theorem of multiple integrals, we obtain:
(11)
Moreover, condition boundary on 
implies that:
So, by replacing 
by its value 
in (11), we deduce:
(12)
But, we can prove that:
(13)
Indeed: from (7), we have that
Multiplying by 
the previous equation, and integrating it on domain
, we obtain:
(14)
On the other hand, spherical harmonic functions form a basis for the Hilbert space 
following inner product:
Consequently, we deduce, since
that:
(15)
Here, 
designed Dirac function.
So, by inserting (15) in (14), we deduce above equality (13) as follows:
We continue the proof by inserting (13) in (12):
(16)
2. Calculate
:
knowing that,
then:
(17)
2.1 Showing that:
(18)
Indeed: unit outer-normal vector 
relative to domain 
at arbitrary point 
is
. This implies:
2.2 Showing that:
(19)
Indeed: the symmetry of the shape implies that unit outer-normal vector of 
relative to domain 
is below in plan generated by the two vectors 
and 
which are orthogonal to field vector 
directed by
. So, we obtain:
.
Then, by inserting (18) and (19) in (9), we deduce that:
(20)
3. Boundary Equation Finally, by inserting (16) and (20) in (10), we obtain that:
The previous equation ends the proof since it is true for any
. 
Proposition 3.3. If
, then problem (P1) have unique solution with the form (7), whose the coefficients are given by:
• 
• 
• 
• 
.
where, 
is the distance 
between arbitrary point 
on sphere
and the center
.
Proof. It is enough to resolve for any
, the system of two unknowns 
et 
given by the two boundary conditions (8) and
. 
Since the solution of problem (P1) is given from proposition 3.3, then we can deduce its relative Dirichletto-Neumann operator:
Corollary 3.4. The Dirichlet-to-Neumann operator (1) is defined by
(21)
Here, the coefficients 
are given by proposition 3.3.
Proof. Since we have that
, then unit outer-normal vector 
for arbitrary point 
is given by:
and consequently, this implies:
But, we have from the proof of proposition 3.2 that
. Then:
Consequently, it is enough to replace 
in the previous equation by its value given in (7). 
Remark 2. For general properties of Dirichlet-toNeumann operators, mainly existence and uniqueness, we refer to [10], chapter 4.
Remark 3. Notice that definition of Dirichlet-toNeumann operator (21) implies that it has as eigenfunctions the spherical harmonic function
, and a discrete spectrum
, whose
.
Corollary 3.5. Dirichlet-to-Neumann operator (21) is unbounded, non-negative, self-adjoint, first-order elliptic pseudo-differential operator with compact resolvent on the Hilbert space
.
Proof. For the proof, we refer to [10], chapter 4. 
Remark 4. Corollary 3.4 implies using Hille-Yosida’s theorem that Dirichlet-to-Neumann operator (21) can be generate certain semigroup
. Moreover, we can prove using Arzela-Ascoli’s criterion that this semigroup is contractant holomorphic in the both Banach space 
and
.
Proof. For a complete proof, see [10], chapter 4. 
4. Localisation Inverse Problem
We are interested by resolving the localisation inverse problem of (P1) using the explicit formula of
, which will be calculated in terms of measurable Dirichlet-toNeumann boundary hypothesis on external boundary
.
For resolving this problem, we need the following:
i) First, we aim to calculate the total flux 
across external boundary
.
ii) Second, we aim to find an equation involving the distance 
between 
center of cell 
and
center of
.
Proposition 4.1. The total flux 
and 
satisfy the following:
(22)
Proof. Since the differential element at the boundary 
and unit outer-normal vector 
at arbitrary point 
are respectively equal to 
and
, then we deduce from (7) that:
On the other hand, by Gauss-Ostrogradsky theorem, one gets:
where 
is the volume differential element. Therefore, (22) is deduced. 
Since we have, from proposition 4.1, that total flux 
across external boundary 
depended only of one coefficient 
of development (7), whose 
depended of distance
, then an equation of 
can be find easily. Indeed:
Corollary 4.2. The distance 
verifies the following equation:
(23)
Proof. It is enough to insert (22) in the expression of 
given in proposition 3.2 in order to substitute
, after replacing 
by its value in term of
:
Consequently, the fact that 
ends the proof. 
5. Conclusions
(23) is an equation of the only unknown 
involving the parameters 
and
, which are the Dirichlet-to-Neumann hypothesis of problem (P1) on the external boundary, and we can found them from an experimental measures.
To summarize, we have found an equation for 
which is the distance between the center 
of the cell 
and the center 
of
, so it remains to find the position of the center
. In fact:
Let 
and 
be two points at the external boundary 
whose the norm of the local current 
reaches respectively its maximum and minimum values, see Figure 1. Then, from the symmetry of the shape, we deduce that the center 
of the cell 
is localized at the line passed by the points
, 
and
, exactly between 
and 
where the distance 
between 
and 
is given by Equation (23).
By conclusion, we can now answer the question posed in the introduction about the uniqueness of the inverse localisation problem for (P1), and we can conclude that total flux 
(2), involving Dirihlet-to-Neumann operator (1), is sufficient to resolve the localisation inverse problem, in three-dimensional case, if the shape is regular. But, it is not enough in other type of inverse problem like geometrical inverse problem, see [5].
6. Acknowledgements
I want to thank the CNRS Federation “Francilienne de
Figure 1. Position of the cell overline B.
Mecanique, Materiaux Structures et Procedes” CNRS FR2609, in particularly Prof. Didier Clouteau, for financial support in this study