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Well-posedness of one-dimensional inverse acoustic problem in \(L_2\) for small depth or small data. (English) Zbl 1119.45303

Lavrent’ev, M. M. (ed.) et al., Ill-posed and non-classical problems of mathematical physics and analysis. Proceedings of the international conference, Samarkand State University, Samarkand, Uzbekistan, September 11–15, 2000. Utrecht: VSP (ISBN 90-6764-380-7/hbk). Inverse and Ill-Posed Problems Series, 57-69 (2003).
Summary: We investigate a system of nonlinear Volterra integral equations. In previous publications we showed that this system can be derived from an inverse acoustic problem and from an inverse electromagnetic problem respectively. Using the method of weighted estimates and the Banach fixed point theorem we prove that the inverse problem is well-posed if either the domain is sufficiently small or the data is sufficiently small. Moreover we find explicit constants which allow us to estimate the depth of well-posedness provided the data are arbitrary but fixed or to estimate the norm of the data which guarantees the existence while the depth is arbitrary but fixed.
For the entire collection see [Zbl 1099.34003].

MSC:

45Q05 Inverse problems for integral equations
45G15 Systems of nonlinear integral equations
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