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Second Kind Representations of Sobolev Space Solutions to a First Order General Elliptic Linear System in a Simply Connected Plane Domain

Authors
  • Klimentov, S. B.1, 2
  • 1 Southern Federal University, Rostov-on-Don, Russia , Rostov-on-Don (Russia)
  • 2 Southern Mathematical Institute, Vladikavkaz, Russia , Vladikavkaz (Russia)
Type
Published Article
Journal
Siberian Mathematical Journal
Publisher
Pleiades Publishing
Publication Date
May 27, 2021
Volume
62
Issue
3
Pages
434–448
Identifiers
DOI: 10.1134/S003744662103006X
Source
Springer Nature
Keywords
License
Yellow

Abstract

We consider a second kind representation for solutions to a first order general uniformly elliptic linear system in a simply connected plane domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ G $\end{document} with the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ W^{k-\frac{1}{p}}_{p} $\end{document}-boundary. We prove that the operator of the system is an isomorphism of Sobolev’s space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ W^{k}_{p}(\overline{G}) $\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ k\geq 1 $\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ p>2 $\end{document}, under appropriate assumptions about coefficients and the boundary. These results are new even for solutions to the canonical first order elliptic system (generalized analytic functions in the sense of Vekua).

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