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

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