Inverse logarithmic potential problem
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 important contributions on inverse source problems in \(\mathbb{R}^2\) were published by \textit{G. Herglotz} (Leipzig 1914), \textit{I. M. Rapoport} (1940-1950), \textit{P. S. Novikov} (1938), \textit{V. K. Ivanov} (1955-1962) using results of complex analysis. In the reviewer's book [Inverse problems in differential equations, Akademie Verlag/Plenum Press (1990; Zbl 0707.35157)], and in his latest papers one can find further references on inverse problems. Now we completely know the history of inverse problems during the last one hundred years. Let \(z=x+iy\in \mathbb{R}^2\), \(\mu_0(z,y)= \mu(x,\overline{z})\) be a function defined on a simply-connected domain \(D^+\) with Jordan boundary \(S\). The relation between exterior gravity potential and density \(\mu\) is given by the Cauchy integral \[ g(z,\overline{z})=- \frac{1}{\pi} \int_{D^+} \frac{\mu(\zeta,\overline{\zeta})}{\zeta-z} d\xi d\eta, \qquad \zeta=\xi+i\eta.\tag{1} \] The book under review contains detailed investigations on potentials of the form (1) and applications to inverse problems. There are different inverse problems related to (1). A first inverse problem consists in determining the domain \(D^+\) (constant density) from its exterior potential. In this case one has to solve a nonlinear boundary value problem for analytic functions using conformal mapping. Further, local existence and uniqueness theorems are established. The solution is looked for near a given one. One chapter deals with estimates of the gradient of the potential in \(\mathbb{R}^3\) and \(\mathbb{R}^2\). There are many relations between the inverse problem and univalent functions. In this way complex analysis can be applied to such problems. Another chapter deals with applications in gravity prospecting and in magnetic prospecting. Reviewer's remark: Books translated from Russian into English sometimes contain misprints in the author's name. The Russian language uses Cyrillic letters following the pronunciation of the words. Transcriptions of mathematicians of Western countries from Russian into English give, for instance, Cauchy \(\to\) Koshi, Brelot \(\to\) Brelo, Lichtenstein \(\to\) Likhtenshtein, and so on.
nonlinear boundary value problem for analytic functions
scientific article; zbMATH DE number 905017