Inverse Problems for Single and Strongly Coupled PDEs via Boundary Measurements: a Carleman Estimates Approach

Liu, Shitao, Department of Mathematics, University of Virginia
Triggiani, Roberto, Department of Mathematics, University of Virginia

This dissertation addresses the inverse problems for single second-order hyperbolic equations and some strongly coupled systems by means of single boundary measurements. More precisely, we study the inverse problem of determining the interior damping and potential coefficients of the second-order hyperbolic equations, coupled hyperbolic equations and the structural acoustic interaction model, by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit sub-portion Γ 1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on the complementary part Γ 0 = Γ\Γ 1 , and T > 0, and under weak regularity requirements on the data. Two canonical results in inverse problems are established: global uniqueness and Lipschitz stability (at the L 2 -level). The proof relies on three main ingredients: (a) sharp Carleman estimates at the H 1 ×L 2 -level for second-order hyperbolic equations; (b) correspondingly implied continuous observability inequalities at the same energy level; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Neumann boundary data.

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PHD (Doctor of Philosophy)
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