Open AccessSingular Integral Operators and Elliptic Boundary-Value Problems. I
Author(s) -
A. V. Soldatov,
Солдатов Александр Павлович
Publication year - 2017
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2017-63-1-1-189
Subject(s) - mathematics , singular integral , boundary value problem , elliptic operator , homogeneous , type (biology) , pure mathematics , order (exchange) , mathematical analysis , completeness (order theory) , cover (algebra) , cauchy distribution , integral equation , combinatorics , mechanical engineering , ecology , finance , engineering , economics , biology
The book consists of three Parts I-III and Part I is presented here. In this book, we develop a new approach mainly based on the author’s papers. Many results are published here for the rst time. Chapter 1 is introductory. The necessary background from functional analysis is given there for completeness. In this book, we mostly use weighted Ho¨lder spaces, and they are considered in Ch. 2. Chapter 3 plays the main role: in weighted Ho¨lder spaces we consider there estimates of integral operators with homogeneous dierence kernels, which cover potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Ch. 4, analogous estimates are established in weighted Lebesgue spaces. Integrals with homogeneous dierence 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 rst-order elliptic systems in terms of fundamental matrices or their parametrixes. Investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to rst-order elliptic systems.