An inverse problem for Helmholtz's equation. III
type: Item
instance of: scholarly article
title: An inverse problem for Helmholtz's equation. III
zbMATH Open document ID: 0713.35095
DOI: 10.1016/0893-9659(89)90128-6
author: A. G. Ramm
published in: Applied Mathematics Letters
publication date: 1989-01-01T00:00:00Z
review text: [For part II see Inverse Probl. 3, L 59-L 61 (1987; Zbl 0654.35088).] Consider the \(problem\) [\(\Delta\) \(+k^ 2n(z)]u=-\delta (x)h(k)\), \(x=(x_ 1,x_ 2,x_ 3)\), \(z=x_ 3\), \(h(k)=\int^{T}_{0}a(t)\exp (ikt)dt\), Im a\(=0\), \(a\in L^ 1[0,T]\), \(u'_+=\alpha u'_ -\), \(u_+=u_ -\) on \(P:=\{x|\) \(z=0\}\), \(u'_{\pm}:=(\partial u/\partial z)|_{z=\pm 0}\), \(\alpha =const>0.\) Given the surface data u(x,k) at \(z=0\) for all \(x\in P\) and all \(k>0\), one seeks to recover h(k), n(z) and \(\alpha\). The function n(z) is known for \(z>0\) and \(n(z)\in L^{\infty}(-\infty,0)\) is unknown. It is proved that the surface data determine uniquely and analytically \(\{\) n(z),h(k),\(\alpha\}\) if one of the following pieces of information is given: i) the total energy \(E=\int^{T}_{0}a^ 2dt\) of the wavelet, or ii) the value n(-0), or iii) the value n(-\(\infty).\) In formula (24) one term is missing so that the text below (24) should be modified.
reviewed by: A. G. Ramm
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 4173554
zbMATH Keywords: analytical dependence
zbMATH Keywords: energy
MaRDI profile type: Publication
full work available at URL: 0893-9659(89)90128-6
OpenAlex ID: W2087687322
cites work: Q3746172
cites work: Q3755908
cites work: Q3759327
cites work: Stability of an inverse hyperbolic problem
description: scientific article; zbMATH DE number 4173554