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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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1569-3945
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Volume 23, Issue 6

Issues

Global uniqueness and stability in determining the electric potential coefficient of an inverse problem for Schrödinger equations on Riemannian manifolds

Roberto Triggiani / Zhifei Zhang
  • Corresponding author
  • Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China
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Published Online: 2015-07-07 | DOI: https://doi.org/10.1515/jiip-2014-0003

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 H1-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 H1-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 L2-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).

Keywords: Inverse problems; Schrödinger equations; Carleman estimates

MSC: 35R30; 35Q40; 49K20

About the article

Received: 2014-01-20

Revised: 2014-05-12

Accepted: 2014-05-12

Published Online: 2015-07-07

Published in Print: 2015-12-01


Funding Source: National Science Foundation

Award identifier / Grant number: DMS-0104305

Funding Source: Air Force Office of Scientific Research

Award identifier / Grant number: FA9550-09-1-0459

Funding Source: National Science Foundation of China

Award identifier / Grant number: 11101166

Funding Source: National Science Foundation of China

Award identifier / Grant number: 61473126

Funding Source: Fundamental Research Funds for the Central Universities, HUST

Award identifier / Grant number: 2014TS067


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 23, Issue 6, Pages 587–609, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2014-0003.

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