Probe method and a Carleman function

A Carleman function is a special fundamental solution with a large parameter for the Laplace operator and gives a formula to calculate the value of the solution of the Cauchy problem in a domain for the Laplace equation. The probe method applied to an inverse boundary value problem for the Laplace equation in a bounded domain is based on the existence of a special sequence of harmonic functions which is called a needle sequence. The needle sequence blows up on a special curve which connects a given point inside the domain with a point on the boundary of the domain and is convergent locally outside the curve. The sequence yields a reconstruction formula of unknown discontinuity, such as cavity, inclusion in a given medium from the Dirichlet-to-Neumann map. In this paper an explicit needle sequence in three dimensions is given in a closed-form. It is an application of a Carleman function introduced by Yarmukhamedov. Furthermore, an explicit needle sequence in the probe method applied to the reduction of inverse obstacle scattering problems with an arbitrary fixed wave number to inverse boundary value problems for the Helmholtz equation is also given. AMS: 35R30, 35R25, 35J05, 33E12, 35C05