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

Fagueye Ndiaye and Idrissa Ly

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

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 ℝN from the measurement of the normal derivative ∂nv on suitable part Γ0 of its boundary, where v is the solution of the wave equation ∂ttv(x, t) − Δv(x, t) + p(x)v(x) = 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 k(Ω) 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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https://doi.org/10.1155/2021/1716027

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jijmms:v:2021:y:2021:i:1:n:1716027

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