Inverse problem related to boundary shape identification for a hyperbolic differential equation
type: Item
instance of: scholarly article
title: Inverse problem related to boundary shape identification for a hyperbolic differential equation
zbMATH Open document ID: 1486.35468
DOI: 10.1155/2021/1716027
author: Fagueye Ndiaye
author: Idrissa Ly
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2022-01-19T00:00:00Z
review text: Summary: In this paper, we are interested in the inverse problem of the determination of the unknown part \(\partial\operatorname{\Omega},\operatorname{\Gamma}_0\) of the boundary of a uniformly Lipschitzian domain \(\operatorname{\Omega}\) included in \(\mathbb{R}^N\) from the measurement of the normal derivative \(\partial_nv\) on suitable part \(\operatorname{\Gamma}_0\) of its boundary, where \(v\) is the solution of the wave equation \(\partial_{tt}v \left(x, t\right)-\operatorname{\Delta}v \left(x, t\right)+p \left(x\right)v \left(x \right)=0\) in \(\operatorname{\Omega}\times \left(0, T\right)\) and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part \(\operatorname{\Gamma}\) of \(\partial\operatorname{\Omega}\). From necessary conditions, we estimate a Lagrange multiplier \(k\left(\operatorname{\Omega}\right)\) 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.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 7459468
zbMATH Keywords: Lagrange multiplier
zbMATH Keywords: inverse problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2021/1716027
OpenAlex ID: W3205642191
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Wikidata QID: Q115243733
description: scientific article; zbMATH DE number 7459468