Inverse Logarithmic Potential Problem (Inverse and Ill-posed Problems 5)
By: V. G. Cherednichenko (author)Hardback
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This monograph deals with the solvability of the inverse potential problem in the two-dimensional case. A new method based on the theory of boundary-value problems for analytic functions and univalent functions is constructed, and local existence theorems, a priori estimates, and a parameter continuation method are established. Furthermore, the smoothness of the inverse problem solution is investigated. The obtained results are applied to geologic interpretation of gravitational and magnetic fields. The following sections of mathematical analyses are used: harmonic functions; boundary-value problems for analytic functions; conformal mappings; non-linear analysis; problems with unknown boundaries; moments theory; variational inequality theory; ill-posed problems theory, in particular for differential equations; and numerical methods. This monograph should be of interest to mathematicians and geophysicists working with this theory.
Part 1 Formulation of inverse logarithmic potential problem - fundamental equation: formulation of inverse problem; nonlinear boundary value problem for mapping function; the analytic continuation of the potential across a boundary; the boundary analyticity of domain is a solution to an inverse problem; the structure of inverse problem solution, final solvability, examples. Part 2 Local solvability of an inverse problem: univalent function variation; local theorem of existence; linearization of the boundary value problem; the auxiliary problem; the Newton-Kantorovitch method; the explicit solution of the linear problem; the local uniqueness theorem; the density variation - the equivalent solution set of the inverse problems - remarks; the complex-valued density case; existence theorems for the inverse problems for small constant densities; proof of theorems. Part 3 The estimate of bounded univalent function co-efficients and univalent polynomials. Part 4 Mass potential estimates - necessary conditions of solvability - a priori estimates for inverse problem solution. Part 5 The continuation by the parameter of an inverse problem solution. Part 6 On the analyticity and smoothness of an inverse problem solution. Part 7 Inverse linear problem - determination of a density of the given domain by its exterior potential. Part 8 Conjugation of harmonic and analytic functions - direct and inverse problems. Part 9 Applications in gravity prospecting and in magnetic prospecting. (Part contents)
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- ID: 9789067642026
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