Inverse Boundary-Value Problems on an Infinite Slab

Abstract

In this work, we study the stability aspect of two inverse boundary-value problems (IBVPs) on an infinite slab with partial data. The uniqueness aspects of these IBVPs were considered and studied by Li and Uhlmann for the case of the Schrödinger equation as well as by Krupchyk, Lassas and Uhlmann for the case of the magnetic Schrödinger equation. Here we quantify the method of uniqueness proposed by Li and Uhlmann and prove a log-log stability estimate for the IBVPs associated to the Schrödinger equation. The boundary measurements considered in these problems are modelled by partial knowledge of the Dirichlet-to-Neumann map; more precisely, we establish log-log stability estimates for each of the following two IBVPs: in the first inverse problem, the corresponding Dirichlet and Neumann data are known on different boundary hyperplanes of the slab; in the second inverse problem, they are known on the same boundary hyperplane of the slab.