On uniqueness for an inverse boundary value problem in electrical prospection1

摘要

In this paper, we consider the problem of determining the discontinuous conductivity λ in the equation div(λ grad u)=0 in Ω when for any Dirchlet data g, we are given the Neumann data h; in other words, results of all possible boundary measurements are known. We derive a uniqueness proof of inclusions of different(analytic) conductivities under the minimal assumptions: (i) the boundaries of inclusions are only Lipschitz, (ii) no topological assumptions. After some modification of Alessandrini's construction [J. Diff. Eq. 84 (1990) 252–272], we gave an existence proof of singular solution for Lipschitz piecewise analytic coefficients and used it for our purpose.