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 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 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 who showed that 2 experiments are enough to determine any number of cracks uniquely. Kim & Seo 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 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 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.