The inverse conductivity problem in an unbounded domain

In this paper we present an analytic treatment of the inverse problem of reconstructing the internal electric conductivity of an in nite domain (the lower half space) from electrical measurements performed on its surface. To this end we transform the initial di erential equation into an integral equation, a procedure which has the advantage of permitting a simple discussion of the regularization process. If the domain is unbounded the integral kernel describing the forward problem is no longer compact and this should permit, in principle, the inversion to be achieved without di culty. However this requires the existence of data up to in nite distances, so that we prefer to restrain ourselves to nite cells and use a staightforward regularization by means of model functions. In this way we can write a stable Fredholm integral equation of the second kind. This equation can be solved either numerically or by using its eigenfunctions, which we have computed explicitly.