Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness

Pages: 1001 - 1014, Issue Special, September 2011

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Shitao Liu - Department of Mathematics, University of Virginia, Charlottesville, VA 22904, United States (email)
Roberto Triggiani - Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904, United States (email)

Abstract: We study a uniqueness inverse problem for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on $\Gamma_0 = \Gamma\\Gamma_1, T >0$, we establish uniqueness of both the damping and potential coecients for each equation. The proof uses critically the Carleman estimate in [11], together with a suggestion in [8, Thm 8.2.2, p.231]. A Riemannian version would also hold, this time by using the corresponding Carleman estimates in [19].

Keywords:  Global uniqueness; Carleman estimate; Coupled hyperbolic equations
Mathematics Subject Classification:  Primary: 35R30, 35L10; Secondary: 49K20

Received: July 2010;      Revised: April 2011;      Published: October 2011.