Multidimensional inverse and ill-posed problems for differential equations
The book is divided into four chapters dealing with inverse problems: 1) for second-order differential equations; 2) for kinetic equations; 3) related to scattering theory and the determination of unknown bodies; 4) related to integral geometry. 1) This chapter is mostly devoted to the question of uniqueness. The term to be recovered is usually a coefficient or the right-hand side of a linear differential equation. Hyperbolic problems are dealt with by the technique of the so-called quasimonotonic operators, while in the parabolic case the coefficients are assumed to be independent of one variable. When this is time, the inverse problem consists in recovering the coefficient of \(D_t u\), assuming that the equation is satisfied in the whole space-time cylinder \(D\times R\), \(D\) being a domain in \(\mathbb{R}^n\). Such a setting, which gives rise to a severely ill-posed direct problem, turns out to be appropriate in the identification case, when, in addition to the boundary data, we are given also an ``initial'' condition at some time \(t= t_0\). Using a Fourier \(t\)-transform and assuming that the boundary data admit a Fourier \(t\)-transform vanishing outside a given ball, the inverse problem can be reduced to a nonlinear integro-differential elliptic direct problem. The same can be done when the unknown coefficient is independent of one spatial variable, say \(z\), and the elliptic differential operator can be split as the sum of two differential operators acting on different variables. Another important technique developed by the author consists in deriving explicit representations for the unknowns in terms of the solution to some parabolic direct problem, when the data admit a factorization. In some specific cases it is possible to derive representations in terms of data, only. 2) After having introduced the general integrodifferential kinetic equation, the author begins to consider the case when such equation reduces to a differential one. For some simple one-dimensional equations it is again possible to express the solution to the inverse problem in a closed form, while the \(n\)-dimensional case is mainly concerned with uniqueness results. They can be proved by taking advantage of specific differential identities. Also existence results can be obtained, but under some, rather severe, restrictions on the data. 3) The first problem here consists in determining the scattering field \(u(x, p, \lambda)\) and the form of the scatterer \(K\), when the scattering amplitude \(A(\nu, p, \lambda)\) is known for particular directions \(\nu\) (\(\nu= p\) or \(\nu= -p\)). Such problems can be reduced to integral equations of the first kind of the form \[ f(n)= \int_\omega P(n, p) d\mu(p),\quad n\in \omega\tag{1} \] for the unknown measure \(\mu\). In (1) \(\omega\) denotes the unit sphere in \(\mathbb{R}^3\) and \(P(n, p)= a|\langle n, p\rangle|+ bY_+(|\langle n, p\rangle|)\), where \(a\), \(b\) and \(Y_+\) stand for two real constants and the Heaviside function, respectively. The basic results concern the uniqueness of the solution and an inversion formula. This latter can be derived when \(\mu\) is assumed to be absolutely continuous with respect to the Lebesgue measure with density \(z\) and \(f\) is smooth or analytic. In the latter case \(z\) itself is analytic. The previous results are used to prove the uniqueness (up to a translation) and the existence (under suitable assumptions) of a closed convex surface \(B\), when we are given some linear combination \(\phi(n)= aF(n)+ bS(n)\). The functionals \(F\) and \(S\) stand, respectively, for the area of the orthogonal projection of \(B\) on the plane with normal \(n\) and the area of the illuminated part of \(B\) in the direction \(n\). 4) The inverse problems for integral geometry are a natural generalization of the question of inverting the celebrated Radon transform. Basically, we are required to recover a function \(f\) from a family of integrals of it. Of course, the first question is the uniqueness of the solution. More can be derived in the two-dimensional case, when line integrals are involved. In that case, the author can obtain a representation of \(f\) in terms of the solution of an auxiliary integro-differential problem. It is worth noting that a basic role is played e.g. in seismics and geophysics by problems of integral geometry related to geodesy. Under the assumption that the corresponding Riemann metrics turns out to be analytic, uniqueness results and representation formulas can be proved in the framework of quotient spaces of analytic functions. The book is very rich in material and non-standard ideas: unusual formulations and interesting explicit representations are spread all over the text and are derived in the framework of the classical function theory. However, the text resembles more that of a survey than of an expository book. Consequently, it is more addressed to expert researchers than to younger ones. In the opinion of the reviewer some additional proofs would have allowed a deeper insight into the subject to a Western reader approaching such a topic, since many items in the references are in Russian or, if in English, are often without any proofs.
scientific article; zbMATH DE number 844030