Singular Integral Operators and Elliptic Boundary-Value Problems. Part I

Abstract

The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here.Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted Hölder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted Hölder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved.Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

Riemann problem for multiply connected domain in Besov spaces

“…Further, the problem of finding a sectionally analytic function for a given jump will be encountered several times, so it is convenient to formulate it in the form of the following: In the famous monographs of Muskhelishvili [9] and Gakhov [5], where the theories of singular integral equations (with the Cauchy kernel) and boundary value problems for analytic functions of a complex variable are summarized, and the main method of research is the apparatus of Cauchy-type integrals in the spaces C β , 0 < β < 1, and L p , p ≥ 1. The works of [1,2,[6][7][8]10] are devoted to the study of various cases of the specified range of problems in the same spaces (sometimes with weights). The results of these works can be generalized or refined by applying the above results to specified Besov spaces with relatively simplified data requirements.…”

On weak solutions of boundary value problems for some general differential equations

“…II, § § 2-3), and the later studies by Dezin [6] on smoothly generated general boundary value problems. At present, the study of general settings of boundary value problems takes place in the directions set by Petrovskii [7] in the mid-1960s, where the settings of boundary value problems were linked with a specific type of differential equation, and, among such settings, the relevant task is to single out the correct settings of boundary value problems (see, for example, the books by Bitsadze [8] and Soldatov [9]).…”

On a Topological Method for Calculating the Index of Quasi-Singular Integral Equation

“…Eigenvalue problems of the Cauchy-Riemann differential operator with regular boundary value conditions, in particular with nonlocal boundary value conditions [11], as well as with Dirichlet type homogeneous boundary value conditions [17,18], where the initial operator is proved to be Volterre, are reduced to the similar problem. The theory of singular integral equations is deeply studied in [19][20][21][22].…”