The Calderón Problem with Partial Data in Two Dimensions
نویسنده
چکیده
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)
منابع مشابه
Recent Progress in the Calderón Problem with Partial Data
We survey recent results on Calderón’s inverse problem with partial data, focusing on three and higher dimensions.
متن کاملThe Calderón Problem with Partial Data on Manifolds and Applications
We consider Calderón’s inverse problem with partial data in dimensions n ≥ 3. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flatness condition include parts of cylindrical sets, conical sets, and surfaces of re...
متن کاملParameter determination in a parabolic inverse problem in general dimensions
It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse prob...
متن کاملOn the Linearized Local Calderón Problem
In this article, we investigate a density problem coming from the linearization of Calderón’s problem with partial data. More precisely, we prove that the set of products of harmonic functions on a bounded smooth domain Ω vanishing on any fixed closed proper subset of the boundary are dense in L(Ω) in all dimensions n ≥ 2. This is proved using ideas coming from the proof of Kashiwara’s Watermel...
متن کاملThe Calderón problem with partial data
In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n ≥ 3, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the ...
متن کامل