Uniqueness of a Potential from Boundary Data in Locally Conformally Transversally Anisotropic Geometries

2018/02/07 by Feizmohammadi, Ali
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.02645

Abstract

Let (Ω³,g) be a compact smooth Riemannian manifold with smooth boundary and suppose that U is a an open set in Ω such that g|_U is the Euclidean metric. Let Γ= U ∩ ∂ Ω be connected and suppose that U is the convex hull of Γ. We will study the uniqueness of an unknown potential for the Schrödinger operator -\triangle_g + q from the associated Dirichlet to Neumann map, Λ_q. We will prove that if the potential q is a priori explicitly known in U^c, then one can uniquely reconstruct q over the convex hull of Γ from Λ_q. We will also outline a reconstruction algorithm. More generally we will discuss the cases where Γ is not connected or g|_U is conformally transversally anisotropic and derive the analogous result.

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