Multidimensional Inverse and Ill-Posed Problems for Differential Equations

Yurii E. Anikonov, Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia.

Part 1 Operator equations and inverse problems: definition of quasimonotonicity, the uniqueness theorem; inverse problems for hyperbolic equations; multidimensional inverse kinematic problems of seismics; on the uniqueness of the solution of the Fredholm and Volterra first-kind integral equations; on the uniqueness of a solution of integral equations of the first kind with entire kernel; existence and uniqueness of a solution to an inverse problem for a parabolic equation; formulas in multidimensional inverse problems for evolution equations. Part 2 Inverse problems for kinetic equations: kinetic equations; an example of an inverse problems for kinetic equation; one-dimensional inverse problems; multidimensional inverse problems; an uniqueness theorem for the solution of an inverse problem for a kinetic equation; the general uniqueness theorem; the effect of the "redundant" equation; problem of separation; differential and integro-differential identities; solution-existence problems; an inverse problem of mathematical biology. Part 3 Geometry of convex surfaces in the large and inverse problems of scattering theory: geometrical question of scattering theory; integral equation of the first kind; uniqueness; existence; stability. Part 4 Integral geometry: inversion formulas; the uniqueness and solvability; some applications; the structure of Riemann spaces and problems of the integral geometry; the solvability of a problem in integral geometry by integration along geodesics.