Probe method and a Carleman function

Abstract

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.

Revisiting the probe and enclosure methods

“…It is known at the early stage of the probe method [11](see also [16]) that under the additional assumption that k 2 is not a Dirichlet eigenvalue of −∆ in Ω, there exists a needle sequence for an arbitrary (y, σ) by the Runge approximation property of the Helmholtz equation. Besides, if σ is given by a part of a line which is called a straight needle, then one can give an explicit needle sequence without any restriction on k 2 , see [18].…”

Extracting discontinuity using the probe and enclosure methods

“…In [95] the author pointed out an unexpected relationship between the needle sequence for the Laplace and Helmholtz equations and a special family of fundamental solutions of the Laplace equation introduced in [193].…”

“…The method presented here provides us how to use some special Carleman function itself in inverse obstacle problems. It is different from another one [95] as a generator of an explicit needle sequence in the probe method since therein only its regular part is used.…”

Vekua theory for the Helmholtz operator

“…In this particular case, simple explicit integral operators have been defined in the original work of Vekua in any space dimension N ≥ 2 (see [34,35], [36, p. 59], and Fig. 1), but no proofs of their properties are provided, and to the best of our knowledge, these results have been used later on only in very few cases [9,25].…”