An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data

Sombuddha Bhattacharyya

doi:10.3934/ipi.2018034

Abstract

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schrodinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.

Highlights

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schrodinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary
A Riemannian manifold (M, g) with boundary is simple if ∂M is strictly convex, and for any point x ∈ M the exponential map expx is a diffeomorphism from its domain in TxM onto M
801 c 2018 American Institute of Mathematical Sciences where dB = d + iB∧ : C∞(M ) → Ω1(M ) and d∗B is the formal adjoint of dB

Summary

C3 α u

We prove a Carleman estimate for u ∈ C∞(M ) with u|∂M supported in an open subset of ∂M. We follow very closely the ideas of [6] in the derivation of the Carleman estimate . We consider small open sets in M where the Riemannian metric g is nearly Euclidean after a suitable change of coordinates. We first prove the estimate on those open sets and later we patch it up over M using a partition of unity. We crucially use the fact that the metric g is close to Euclidean on these coordinate patches. Our aim is to prove the following estimate holds for all u ∈ Cc∞(Ω) and for 0 < h < sufficiently small:. We will start with proving the Carleman estimate for the following special case.

The function f is so that

E can be parameterized by a smooth real valued function f so that