Lipschitz stability for a hyperbolic inverse problem by finite local boundary data

Abstract

In this article we consider the inverse problem of determining the potential q in a wave equation in a bounded smooth domain Ω in from a finite number of data of the hyperbolic Dirichlet to Neumann map and we prove the Lipschitz stability in determining q. Our main result is stated as follows. Let T> diam Ω and Γ₀⊂∂Ω. For any k-dimensional space X in L ^∞(Ω), there exist 2k-functions f ₁, …, f _2k on (0,T)×∂Ω such that , provided that q₁, q₂∊X are uniformly bounded in a suitable Sobolev space. Here Λ_q is the Dirichlet to Neumann map for the coefficient q(x).

Keywords:

• Lipschitz stability
• Hyperbolic inverse problem
• Dirichlet to Neumann map
• Finite dimensionality

Keywords:

• 2000 Mathematics Subject Classifications:
• 35R30
• 35R25
• 65M32

Acknowledgements

The third named author was partly supported by Grant 15340027 from the Japan Society for the Promotion of Science and Grant 17654019 from the Ministry of Education, Cultures, Sports and Technology. The authors thank Professor V.G. Romanov for his precious comments and information of early related works, and they are very grateful to the anonymous referees for precise comments.