Suppose that a bounded simply connected domain **D** in
with a smooth boundary is occupied by a conducting
material. Also, suppose that **D** is known to contain one single,
smooth and non-self-intersecting curve **S**, or a crack, in its
interior, but that the location and the shape of **S** are unknown. To
determine **S** we carry out an experiment in which we prescribe known
amount of electric current on the boundary and measure the resulting
voltage **u** on a part of . Our interest then is to
determine **S** from the data obtained in this experiment.

Inverse problems of this type have been considered both by engineers
and by mathematicians, because the analysis of these problems is
important in the practice of non-destructive evaluations and because
the formulation of these problems is relatively simple mathematically.
In this section we shall present some known mathematical results
concerning this inverse problem. To begin with we shall formulate
the direct problem in which the geometry of **S** is known. Obviously,
the field **u** is found as the solution to the following boundary value
problem:

where **f** stands for the current (multiplied by a constant)
prescribed on . The function **f** is
assumed to have a vanishing integral on . Also, the
side of **S** into which the normal vector points is denoted by
, while stands for the other side of **S**.
Correspondingly, the limit of a certain quantity on **S** from the
positive or negative side is denoted by a superposed **+** or **-**. The
boundary condition on **S** in (1) indicates that the
crack is insulating across **S**. The solution **u** to
(1) is unique up to an arbitrary additive constant.

In the inverse problem, the geometry of the crack **S** is not known.
We give **N** functions () as the Neumann data **f**
in (1), and then measure the corresponding Dirichlet
data () on a part of denoted by
. From the data thus obtained we determine **S**. We notice
that a similar problem in which one prescribes Dirichlet data on
and measures Neumann data on is also possible, and
these problems are equivalent when . We shall, however,
consider the `give **f**, measure **u**' setting for the purpose of
definiteness unless stated otherwise.

Friedman & Vogelius[10] considered the following uniqueness issue
related to this inverse problem: Let be the solution to the
same problem as that for , except that **S** is replaced by .
Suppose that () holds to within an additive
constant on . The question is if we can then conclude
that . Obviously the answer is no if **N=1**. Indeed, suppose
and take on the stream line of the function
. Then we have . However, Friedman & Vogelius showed
that one can take * two* (**N=2**) functions (**i=1,2**) in a
way that the uniqueness of **S** holds. Bryan & Vogelius[5]
extended the results of Friedman & Vogelius by showing that the
unique determination of cracks is possible with
experiments. This result was further improved by Alessandrini & Diaz
Valenzuela[1] who showed that 2 experiments are enough to
determine any number of cracks uniquely. Kim & Seo[13] also
showed the same conclusion almost at the same time.

The results mentioned so far are for 2D problems. As regards the
corresponding 3D results, Kubo et al[19] stated that 3
measurements are enough to determine 1 crack in a 3D bounded domain.
Their argument might not please mathematicians,
but a rigorous proof of uniqueness by Eller[9] considers
a somewhat impractical case where , or more precisely,
where the Neumann-to-Dirichlet map (a `function' which yields the
Dirichlet data given the Neumann data on ) is known on
the whole exterior boundary. As mentioned in Eller, Alessandrini has
announced that he and his colleagues have shown that 2 experiments
determine a crack uniquely when the boundary condition on the crack is
of the homogeneous Dirichlet type. They are also reported to have
shown the uniqueness of **S** when **S** is known to
lie on a plane and the boundary condition on the crack is of the
homogeneous Neumann type. These results, however, do not seem to have
been published so far in printed forms.

Thu Feb 19 01:36:51 JST 1998