Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

Fagueye Ndiaye, Idrissa Ly and Remi Léandre

International Journal of Mathematics and Mathematical Sciences, 2021, vol. 2021, 1-12

Abstract: In this paper, we are interested in the inverse problem of the determination of the unknown part ∂Ω,Γ0 of the boundary of a uniformly Lipschitzian domain Ω included in ℠N from the measurement of the normal derivative ∂nv on suitable part Γ0 of its boundary, where v is the solution of the wave equation ∂ttvx,t−Δvx,t+pxvx=0 in Ω×0,T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂Ω. From necessary conditions, we estimate a Lagrange multiplier kΩ which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:1716027

DOI: 10.1155/2021/1716027

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