Elliptic Equations: Many Boundary Measurements

We consider the Dirichlet problem (4.0.1), (4.0.2). We assume that for any Dirichlet data g we are given the Neumann data h; in other words, we know the results of all possible boundary measurements, or the so-called Dirichlet-to-Neumann operator Λ: H_(1/2)(∂Ω) → H_(−1/2)(∂Ω), whichmapsthe Dirichlet data g into the Neumann data h. From Theorem 4.1 the operator Λ is well-defined and continuous, provided that Ω is a bounded domain with Lipschitz ∂Ω. In Sections 5.1, 5.4, 5.7 we consider scalar a, b = 0, c = 0. The study of this problem was initiated by the paper of Calderon [C], who studied the inverse problem linearized around a constant and suggested a fruitful approach, which was extended by Sylvester and Uhlmann in their fundamental paper [SyU2], where the uniqueness problem was completely solved in the three-dimensional case.