Inverse logarithmic potential problem
type: Item
instance of: scholarly article
title: Inverse logarithmic potential problem
zbMATH Open document ID: 0876.31003
author: V. G. Cherednichenko
published in: Inverse and Ill-Posed Problems Series
publication date: 1996-07-16T00:00:00Z
review text: Most mathematical problems in science, technology and medicine are inverse problems, where inner parameters of a system inaccessible to measurement are to be determined. Such problems arise in geophysics, astrophysics, medicine, and so on. Studying such problems is the only complete way to analyze experimental results. Inverse problems may be considered among the pressing problems of current mathematical research. For more than one hundred years, many mathematicians and physicists have studied inverse problems, especially the inverse source problem relative to the Laplace equation. The Newton potential \((n\geq 3)\) or the logarithmic potential \((n=2)\) of a measure \(m\) is known on the boundary \(\partial\Omega\) of a bounded domain \(\Omega\subset \mathbb{R}^n\), where \(\text{supp }m\subset \Omega\cup\partial \Omega\). The measure \(m\) is to be determined by the potential on \(\partial\Omega\). In geophysics one can measure the gradient of the potential on \(\partial\Omega\) at finitely many points, from which \(m\) is to be determined. At least since 1911 (P. Pizzetti, G. Lauricella) it has been well known that this inverse source problem cannot be uniquely solved without additional restrictions on the measure to be determined. The measure \(m\) is only uniquely determined if the potential is known on the whole support of \(m\). Lack of information often results in ambiguity, so different inner parameters (measures) produce the same measured values, a very bad situation in applications. Only since 1960 do we know a sufficient number of results of basic mathematical theory to be able to treat certain inverse problems from a systematic point of view. We need in the future a large number of mathematicians who work on inverse problems and must, in this way, develop large new areas of mathematics. The bibliography of the book under review gives broad information about the papers of Russian mathematicians on inverse problems during the last 70 years. First importan…
reviewed by: Gottfried Anger
Mathematics Subject Classification ID: 31A25
Mathematics Subject Classification ID: 31-02
Mathematics Subject Classification ID: 35R30
zbMATH DE Number: 905017
zbMATH Keywords: inverse problems in the plane
zbMATH Keywords: Cauchy integral
zbMATH Keywords: nonlinear boundary value problem for analytic functions
zbMATH Keywords: conformal mapping
zbMATH Keywords: local existence and uniqueness
zbMATH Keywords: estimates of the gradient
zbMATH Keywords: univalent functions
zbMATH Keywords: gravity prospecting
zbMATH Keywords: magnetic prospecting
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 905017