The Calderón Problem with Partial Data in Two Dimensions

We consider the problem of determining a complex-valued potential q in a bounded two-dimensional domain from the Cauchy data measured on an arbitrary open subset of the boundary for the associated Schrödinger equation Δ+q. A motivation comes from the classical inverse problem of electrical impedance tomography. In this inverse problem one attempts to determine the electrical conductivity of a body by measurements of voltage and current on the boundary of the body. This problem was proposed by Calderón [9] and is also known as Calderón’s problem. In dimensions n ≥ 3, the first global uniqueness result for C-conductivities was proven in [28]. In [25], [5] the global uniqueness result was extended to less regular conductivities. Also see [14] for the determination of more singular conormal conductivities. In dimension n ≥ 3 global uniqueness was shown for the Schrödinger equation with bounded potentials in [28]. The case of more singular conormal potentials was studied in [14]. In two dimensions the first global uniqueness result for Calderón’s problem was obtained in [24] for C-conductivities. Later the regularity assumptions were relaxed in [6] and [2]. In particular, the paper [2] proves uniqueness for L∞conductivities. In two dimensions a recent breakthrough result of Bukhgeim [7] gives unique identifiability of the potential from Cauchy data measured on the whole boundary for the associated Schrödinger equation. As for the uniqueness in determining two coefficients, see [10], [18]. In all the above-mentioned articles, the measurements are made on the whole boundary. The purpose of this paper is to show global uniqueness in two dimensions, both for the Schrödinger and conductivity equations, by measuring all the Neumann data on an arbitrary open subset Γ̃ of the boundary produced by inputs of Dirichlet data supported on Γ̃. We formulate this inverse problem more precisely below. Let Ω ⊂ R be a bounded domain with smooth boundary which consists of N smooth closed curves γj , ∂Ω = ⋃N j=1 γj , and let ν be the unit outward normal vector to ∂Ω. We denote ∂u ∂ν = ∇u · ν. A bounded and positive function γ̃(x)