Commentary on Calderón’s paper 28, On an Inverse Boundary Value Problem

In the paper [U] we surveyed some of the most important developments motivated by Calderon’s beautiful paper [C1] up to Calderon’s 75th anniversary conference. As we describe in this note, his only article on inverse problems has continued to have a crucial impact in the field. In this section we recall the problem which was considered by Calderon in the 40’s when he was an engineer working for the Argentinian state oil company “Yacimientos Petroĺi feros Fiscales” (YPF). Parenthetically Calderon said in his speech accepting the “Doctor Honoris Causa” of the Universidad Autonoma de Madrid that his work at YPF had been very interesting but he was not well treated there; he would have stayed at YPF otherwise ([C2]). It goes without saying that the bad treatment of Calderon by YPF was very fortunate for Mathematics! Calderon’s motivation was geophysical prospection, in particular oil exploration, using electrical methods. The question is whether one can determine the conductivity of the subsurface of the Earth by making voltage and current measurements at the surface. The problem of determining the electrical properties of a medium by making voltage and current measurements at the boundary has also raised the interest of the medical imaging community and is known as Electrical Impedance Tomography (EIT). One exciting potential application is the early diagnosis of breast cancer. The conductivity of a malignant breast tumor is typically 0.2 mho which is significantly higher than normal tissue which has been typically measured at 0.03 mho. See the book [Ho] and the issue of Physiological Measurement [DIMS] for applications of EIT to medical imaging and other fields. Let Ω ⊆ R, n ≥ 2, be a bounded domain with smooth boundary (many of the results we will describe are valid for domains with Lipschitz boundaries). The electrical conductivity of Ω is represented by a bounded and positive function γ(x). In the absence of sinks or sources of current the potential u ∈ H(Ω) with given boundary voltage potential f ∈ H 1 2 (∂Ω) is a solution of the Dirichlet problem