Global uniqueness and stability in determining the electric potential coefficient of an inverse problem for Schrödinger equations on Riemannian manifolds
Abstract
In the present paper, we consider the inverse problem of a Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional Riemannian manifold M with metric g. It is subject to a non-homogenous Dirichlet boundary term. We aim at determining the potential coefficient by means of a Neumann boundary measurement on a portion Γ1 of the boundary Γ of Ω. Under sharp conditions on the complementary part Γ0 = Γ∖Γ1, and under weak regularity requirements on the data, we establish two canonical results of the inverse problem: (i) global uniqueness and (ii) global Lipschitz stability. The lower bound inequality corresponding to the upper bound inequality contained in (ii) is also given. Our proofs rely on four main ingredients: (a) sharp Carleman estimate at the H 1-level for Schrödinger equations on Riemannian manifolds [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339–404], (b) related continuous observability inequality at the H 1-level [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339–404], (c) a continuous observability inequality at the L 2-level [J. Inverse Ill-Posed Probl. 12 (2004), 43–123], [Functional Analysis and Evolution Equations, Birkhäuser, Basel (2008), 613–636], (d) optimal regularity theory for Schrödinger equations with Dirichlet boundary data ([Differential Integral Equations 5 (1992), 521–535], [Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia Math. Appl. 2, Cambridge University Press, Cambridge, 2000] as well as the new Theorem 3.6).
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Jul 07, 2015
- Source ID
- 10.1515/jiip-2014-0003
Entities
People
- Roberto Triggiani
- Zhifei Zhang
Organizations
- Air Force Office of Scientific Research
- Huazhong University of Science and Technology
- National Natural Science Foundation of China
- National Science Foundation
- University of Memphis