Complex spherical waves and inverse problems in unbounded domains

Abstract

Abstract. This work is motivated by the inverse conductivity problem of identifying an embedded object in an infinite slab. The novelty of our approach is that we use complex spherical waves rather than classical Calderón type functions. For Calderón type functions, they are growing exponentially on one side of a hyperplane and decaying exponentially on the other side. Without extra modifications, they are inadequate for treating inverse problems in unbounded domains such as the infinite slab. The obvious reason for this is that Calderón type functions are not integrable on hyperplanes. So they can not be used as measurements on infinite boundaries. For complex spherical waves used here, they blow up faster than any given positive polynomial order on the inner side of the unit sphere and decay to zero faster than any given negative polynomial order on the outer side of the unit sphere. We shall construct these special solutions for the conductivity equation in the unbounded domain by a Carleman estimate. Using complex spherical waves, we can treat the inverse problem of determining the object in the infinite slab like the problem in the bounded domain. Most importantly, we can easily localize the boundary measurement, which is of great value in practice. On the other hand, since the probing fronts are spheres, it is possible to detect some concave parts of the object. 1.