Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya / Vestnik of Samara University. Natural Science SeriesVestnik Samarskogo universiteta. Estestvennonauchnaya seriya / Vestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University822210.18287/2541-7525-2020-26-1-7-13Research ArticleON THE NOETHER THEORY OF TWO-DIMENSIONAL SINGULAR OPERATORS AND APPLICATIONS TO BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF FOURTH-ORDER ELLIPTIC EQUATIONSDzhangibekovG.<p>Doctor of Physical and Mathematical Sciences, professor of Department of Functional Analysis and Differential Equations</p>gulkhoja@list.ruhttps://orcid.org/0000-0001-8804-4400ОdinabekovJ. M.<p>Candidate of Physical and Mathematical Sciences, associate professor, head of the Department of Fundamental and Natural Sciences</p>jasur_79@inbox.ruhttps://orcid.org/0000-0001-9851-9895Tajik National UniversityLomonosov Moscow State University in Dushanbe020320202617132512202025122020Copyright © 2020, Dzhangibekov G., Оdinabekov J.M.2020<p>It is known that the general theory of multidimensional singular integral operators over the entire space Em was constructed by S. G. Mikhlin. It is shown that in the two-dimensional case, if the operator symbol does not turn into zero, then the Fredholm theory holds. As for operators over a bounded domain, in this case the boundary of the domain significantly affects the solvability of such operator equations. In this paper we consider two-dimensional singular operators with continuous coefficients over a bounded domain. Such operators are widely used in many problems of the theory of partial differential equations. In this regard, it would be interesting to find criteria of Noetherity of such operators as explicit conditions for its coefficients. Depending on the 2m+ 1 connected components, necessary and sufficient conditions of Noetherity for such operators are obtained and a formula for the evaluation of the index is given. The results are applied to the Dirichlet problem for general fourth-order elliptic systems.</p>сингулярный интегральный оператор, индекс, символ, нетеровость оператора,
эллиптическая системаsingular integral operator, index, symbol, Noetherity of an operator, elliptic system.[Mikhlin S.G. Multidimensional singular integrals and integral equations. Moscow: Fizmatgiz, 1962, 254 р. Available at: https://booksee.org/book/578442. (In Russ.)][Stein E.M. Note on singular integrals. Proc. Amer. Math. Soc., 1957, vol. 8, pp. 250–254.][Vekua I.N. Generalized analytic functions. Moscow: Nauka, 1959, 652 p. Available at:][http://bookfi.net/book/442081. (In Russ.)][Boyarsky B.V. Studies on elliptic equations in the plane and boundary problems of function theory: Doctoral of Physical and Mathematical Sciences thesis. Moscow, 1960. (In Russ.)][Juraev A.D. Method of singular integral equations. Moscow: Nauka, 1987, 415 p. Available at: https://b-ok.africa/book/2929918/866958. (In Russ.)][Boymatov K.Kh., Dzhangibekov G. On a singular integral operator. Russian Mathematical Surveys, 1988, vol. 43, number 3, pp. 199–200. DOI: https://doi.org/10.1070/RM1988v043n03ABEH001746. (In Russ.)][Dzhangibekov G. On a class of two-dimensional singular integral operators and its applications to boundary value problems for elliptic systems of equations in the plane. Doklady Mathematics, 1993, vol. 47, №. 3, pp. 498–503. Available at: https://zbmath.org/?q=an:0881.47026. (In Russ.)][Jangibekov G., Khujanazarova G.Kh. On the Noether property and the index for some two-dimensional singular integral operators in a connected domain. Doklady Mathematics, 2004, vol. 396, №4, pp. 449–454. Available at:][https://www.elibrary.ru/item.asp?id=17352487. (In Russ.)][Dzhangibekov G., Odinabekov D.M., Khudzhanazarov G.Kh. The Noetherian Conditions and the Index of Some Class of Singular Integral Operators over a Bounded Simply Connected Domain. Moscow University Mathematics Bulletin, 2019, vol. 74, № 2, pp. 49–54. DOI: https://doi.org/10.3103/S0027132219020025. (In Russ.)][Duduchava R.V. On multidimensional singular integral operators II: The case of compact manifolds. Journal of Operator Theory, 1984, vol. 11, no. 1, pp. 41–76. Available at: https://www.researchgate.net/publication/266063688_On_multidimensional_singular_integral_operators_I_The_half-space_case.][Jangibekov G., Khujanazarova G.Kh. On the Dirichlet problem for elliptic systems of two equations of the fourth order on a plane. Doklady Mathematics, 2004, vol. 398, no. 2, pp. 151–155. Available at: https://www.elibrary.ru/item.asp?id=17352913. (In Russ.)][Dzhangibekov G., Odinabekov D.M. On the theory of two-dimensional singular integral operators with even characteristic over a bounded domains. In: Contemporary problems of mathematics and mechanics. Proceedings of the conference dedicated to the 80th anniversary of academician V.A. Sadovnichy. Moscow: МGU, 2019,][pp. 50–53. DOI: 10.29003/m978-5-317-06111-1. (In Russ.)]