The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

The main content of this book is composed from two doctoral theses: by V. V. Katrakhov (1989) and by S. M. Sitnik (2016). In our work, for the rst time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to dierential equations with singularities in coecients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Signicant part of these results is published with detailed proofs for the rst time. In the rst chapter, we give historical background, necessary notation, denitions, and auxiliary facts. In the second chapter, we give the detailed theory of Sonin and Poisson transmutations. In the third chapter, we describe an important special class of the Buschman-Erde´lyi transmutations and their applications. In the fourth chapter, we consider new weighted boundary-value problems with Sonin and Poisson transmutations. In the fth chapter, we consider applications of the Buschman-Erde´lyi transmutations of special form to new boundary-value problems for elliptic equations with signicant singularities of solutions. In the sixth chapter, we describe a universal compositional method for construction of transmutations and its applications. In the concluding seventh chapter, we consider applications of the theory of transmutations to dierential equations with variable coecients: namely, to the problem of construction of a new class of transmutations with sharp estimates of kernels for perturbed dierential equations with the Bessel operator, and to special cases of the well-known Landis problem on exponential estimates of the rate of growth for solutions of the stationary Schro¨dinger equation. The book is concluded with a brief biographic essay about Valeriy V. Katrakhov, as well as detailed bibliography containing 648 references.