Uniqueness and stability in multidimensional hyperbolic inverse problems
type: Item
instance of: scholarly article
title: Uniqueness and stability in multidimensional hyperbolic inverse problems
zbMATH Open document ID: 0923.35200
DOI: 10.1016/S0021-7824(99)80010-5
author: M. Yamamoto
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1999-10-14T00:00:00Z
review text: The author, first, considers the uniqueness and the stability for a multidimensional hyperbolic inverse problem, which is for determining \(y\) and \(f\) from the following system of equations \[ \partial^{2}_{t} y(x,t)=\Delta y -q(x)y+f(x)R(x,t), \quad x\in \Omega,\;0<t<T, \] \[ y(x,0)=\partial_{t}y(x,0)=0,\;x\in \Omega, \quad y|_{\partial \Omega}=0,\;t\in (0,T), \quad \frac{\partial y}{\partial \nu}\mid_{\partial \Omega}=z. \] Under the assumptions \(| R(x,0)| \geq r_0>0\) and \(R\in W^{3,\infty}(\Omega\times (0,T))\) the uniqueness is proved. Moreover, there exists a constant \(C>0\) such that \( C^{-1}\| f\| _{L^2(\Omega)}\leq \| \frac{\partial y(f)}{\partial\nu}\| _{H^1(0,T;L^2(\partial \Omega))}\leq C\| f\| _{L^2 (\Omega)}. \) The above results are used to deduce the uniqueness and the stability of determining the coefficient of the lowest-order derivative, as usual.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 1272013
zbMATH Keywords: uniqueness
zbMATH Keywords: stability
zbMATH Keywords: source reconstruction
reviewed by: Yu Wenhuan
MaRDI profile type: Publication
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full work available at URL: s0021-7824(99)80010-5
OpenAlex ID: W2071370548
description: scientific article; zbMATH DE number 1272013