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Differential Equations

, Volume 55, Issue 2, pp 231–242 | Cite as

On the Hölder Property of Solutions of a Generalized System of Beltrami Equations

  • M. M. SirazhudinovEmail author
  • S. V. Tikhomirova
Partial Differential Equations
  • 4 Downloads

Abstract

We define a generalized Beltrami system, which is a broad generalization of the scalar Beltrami equation to vector equations. The Riemann-Hilbert boundary value problem is considered for such a system under the assumption that it is elliptic (i.e., the roots of the characteristic equation belong to the interior of the unit disk centered at zero). A Cordes type condition on the location of the roots of the characteristic equation of the system is obtained; this is a sufficient condition for the solution of this problem to have the Hölder property. The proof is based on the properties of singular integral operators in a domain.

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References

  1. 1.
    Ladyzhenskaya, O.A. and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa (Linear and Quasilinear Equations of Elliptic Type), Moscow: Nauka, 1973.Google Scholar
  2. 2.
    Krylov, N.V., Nelineinye ellipticheskie i parabolicheskie uravneniya vtorogo poryadka (Nonlinear Second-Order Elliptic and Parabolic Equations), Moscow: Nauka, 1985.Google Scholar
  3. 3.
    Koshelev, A.N., Regulyarnost’ reshenii ellipticheskikh uravnenii i sistem (Regularity of Solutions of Elliptic Equations and Systems), Moscow: Nauka, 1986.Google Scholar
  4. 4.
    Boyarskii, B.V., Generalized solutions of a system of first-order differential equations of elliptic type with discontinuous coefficients, Mat. Sb., 1957, vol. 43 (85), no. 4, pp. 451–503.Google Scholar
  5. 5.
    Sirazhudinov, M.M., Aliev, Sh.G., and Sirazhudinova, S.P., On the H¨older property of solutions of a system of Beltrami equations, Vestn. Dagestan. Gos. Univ., 2016, no. 4, pp. 77–83.Google Scholar
  6. 6.
    Vinogradov, V.S., Boundary value problem for first-order elliptic system on the plane, Differ. Uravn., 1971, vol. 7, no. 8, pp. 1440–1448.zbMATHGoogle Scholar
  7. 7.
    Sirazhudinov, M.M., On the Riemann–Hilbert problem for first order elliptic systems in multiply connected domains, Russian Math. Surveys, 1993, vol. 80, no. 2, pp. 287–307.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Horn, R. and Jonson, Ch., Matrix Analysis, Cambridge: Cambridge Univ., 1985. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989.CrossRefGoogle Scholar
  9. 9.
    Vekua, I.N., Obobshchennye analiticheskie funktsii (Generalized Analytic Functions), Moscow: Nauka, 1988.Google Scholar
  10. 10.
    Sirazhudinov, M.M., A Riemann–Hilbert boundary-value problem (L2-theory), Differ. Equations, 1989, vol. 25, no. 8, pp. 999–1003.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Krein, S.G., Lineinye differentsial’nye uravneniya v banakhovom prostranstve (Linear Differential Equations in Banach Space), Moscow: Nauka, 1971.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Dagestan Scientific Center of the Russian Academy of SciencesMakhachkalaRussia
  2. 2.Dagestan State UniversityMakhachkalaRussia
  3. 3.Vladimir State UniversityVladimirRussia

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