The author considers three inverse problems related to elliptic equations. The first of them consists in identifying the coefficient a appearing in the Dirichlet problem
under the assumption that a is a discontinuous function of the form and b being unknown. Here is an open subset of , is the characteristic function of and the functions and satisfy the relations: i) ; ; iii) b(x)
The author proves the uniqueness of the unknown pair when , under the assumption that the Dirichlet-Neumann map is known for any with Supp , being a neighbourhood if . In the case he obtains a weaker result.
Similar uniqueness results are deduced also in the case where equation (1) is replaced by
where (x) is the delta function with pole at x and is constant outside a bounded domain with , being a convex domain with analytic boundary.
The additional information needed to prove the uniqueness result for is the following: u(x, is assigned on for any , and being two disjoint neighbourhoods in
The third identification problem is related to the anisotropic equation
where the matrix A(x) admits the decomposition
Here and B(x) are known positive definite matrices , while the open set is unknown. In this case the additional information is again the Dirichlet-Neumann map as in the first problem: it ensures the uniqueness of the open set .