Stability estimates for the inverse problem of finding the absorption constant
type: Item
instance of: scholarly article
title: Stability estimates for the inverse problem of finding the absorption constant
zbMATH Open document ID: 1028.35165
DOI: 10.1023/A:1012521812548
author: D. I. Glushkova
published in: Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: This paper considers the problem of reconstructing the spacewise dependent coefficient in an equation of hyperbolic type. Let \(u\) and \(\sigma\) satisfy \[ \begin{gathered} u_{tt}- u_{xx}- u_{yy}+ \sigma(x,y) u_t= \delta(y,t),\;u|_{t< 0}\equiv 0,\;(x,y,t)\in \mathbb{R}^3,\\ \text{supp}(\sigma)\subset\{(x, y)\in \mathbb{R}^2\mid y> 0\},\end{gathered}\tag{1} \] where \(\delta\) is the Dirac delta function. In a particular case of the paper, the inverse problem requires finding the absorption coefficient \(\sigma\) in a domain \(D\subset B(0, 2,1):= \{(x,y)\in \mathbb{R}^2\mid x^2+ (y- 2)^2< 1\}\) with smooth boundary, from the traces of the solution \(u\) of the problem (1) and its first-order partial derivatives with respect to \(x\), \(y\) and \(t\) given on \(S:= \{(x,y,t)\in \mathbb{R}^3\mid (x,y)\in\partial D, |y|< t< 5+|y|\}\), i.e. \[ D^k u|_S= f^k(x, y,t),\quad k= (k_1, k_2, k_3),\;|k|\leq 1.\tag{2} \] By defining the sets \[ P:= \{(x, y)\in \mathbb{R}^2\mid [x^2+ (y- 2)^2]^{1/2}+|y|< 9\}, \] \[ \Sigma(\sigma):= \{\sigma(x,y)\in H^8(P)\mid\|\partial^k\sigma/\partial^{k_1} x\partial^{k_2} y\|_{C(P)}\leq \sigma,\;k= (k_1,k_2), |k|\leq 6\} \] then it is shown that the following estimate holds: There exists \(0< \sigma\leq 1\) and \(C= C(\text{dist}(D,\partial B(0,2)))> 0\) such that \[ \int_D (\sigma_1- \sigma_2)^2 dD\leq C \sum_{|k|\leq 1} \int_S (f^k_1- f^k_2)^2 dS dt,\quad \forall\sigma_1,\sigma_2\in \Sigma(\sigma), \] where \(f^k_1\) and \(f^k_2\) are the data of the inverse problem (1) and (2) corresponding to \(\sigma_1\) and \(\sigma_2\), respectively.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35L10
zbMATH DE Number: 1915930
zbMATH Keywords: absorption coefficient
zbMATH Keywords: hyperbolic inverse problem
zbMATH Keywords: stability
reviewed by: Daniel Lesnic
MaRDI profile type: Publication
full work available at URL: a:1012521812548
OpenAlex ID: W57463580
description: scientific article; zbMATH DE number 1915930