Biharmonic Monogenic Functions and Biharmonic Boundary Value Problems

  • Serhii V. Gryshchuk
  • Sergiy A. Plaksa
Conference paper
Part of the Trends in Mathematics book series (TM)


We consider a commutative algebra B over the field of complex numbers with a basis {e1, e2} satisfying the conditions \((e_1^2+e_2^2)^2=0\), \(e_1^2+e_2^2\ne 0\). We consider a Schwarz-type boundary value problem for “analytic” B-valued functions in a simply connected domain. This problem is associated with BVPs for biharmonic functions. Using a hypercomplex analog of the Cauchy type integral, we reduce these BVPs to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property.


  1. 1.
    A. Douglis, A function-theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 6(2), 259–289 (1953)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S.V. Grishchuk, S.A. Plaksa, Monogenic functions in a biharmonic algebra (in Russian). Ukr. Mat. Zh. 61(12), 1587–1596 (2009). English transl. (Springer), in Ukr. Math. J. 61(12), 1865–1876 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S.V. Gryshchuk, One-dimensionality of the kernel of the system of Fredholm integral equations for a homogeneous biharmonic problem (Ukrainian. English summary). Zb. Pr. Inst. Mat. NAN Ukr. 14(1), 128–139 (2017)Google Scholar
  4. 4.
    S.V. Gryshchuk, S.A. Plaksa, Basic properties of monogenic functions in a biharmonic plane, in Complex Analysis and Dynamical Systems V, ed. by M.L. Agranovskii, M. Ben-Artzi, G. Galloway, L. Karp, V. Maz’ya. Contemporary Mathematics, vol. 591 (American Mathematical Society, Providence, 2013), pp. 127–134Google Scholar
  5. 5.
    S.V. Gryshchuk, S.A. Plaksa, Schwartz-type integrals in a biharmonic plane. Int. J. Pure Appl. Math. 83(1), 193–211 (2013)CrossRefGoogle Scholar
  6. 6.
    S.V. Gryshchuk, S.A. Plaksa, Monogenic functions in the biharmonic boundary value problem. Math. Methods Appl. Sci. 39(11), 2939–2952 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L.V. Kantorovich, V.I. Krylov, Approximate Methods of Higher Analysis, 3rd edn (Interscience publishers, Inc., New York, 1964). Transl. by Benser, C.D. from RussianGoogle Scholar
  8. 8.
    V.F. Kovalev, I.P. Mel’nichenko, Biharmonic functions on the biharmonic plane (in Russian). Rep. Acad. Sci. USSR A 8, 25–27 (1981)zbMATHGoogle Scholar
  9. 9.
    A.I. Lurie, Theory of Elasticity (Springer, Berlin, 2005). Engl. transl. by A. BelyaevCrossRefGoogle Scholar
  10. 10.
    I.P. Mel’nichenko, Biharmonic bases in algebras of the second rank (in Russian). Ukr. Mat. Zh. 38(2), 224–226 (1986). English transl. (Springer) in Ukr. Math. J. 38(2), 252–254 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S.G. Mikhlin, The plane problem of the theory of elasticity (Russian). Trans. Inst. of seismology, Acad. Sci. USSR. no. 65 (USSR Academy of Sciences Publishing House, Moscow, 1935)Google Scholar
  12. 12.
    S.G. Mikhlin, N.F. Morozov, M.V. Paukshto, The Integral Equations of the Theory of Elasticity. TEUBNER-TEXTE zur Mathematik Band, vol. 135. Transl. from Russ. by Rainer Radok, ed. by H. Gajewski (Springer, Stuttgart, 1995)Google Scholar
  13. 13.
    N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Fundamental Equations, Plane Theory of Elasticity, Torsion and Bending (Noordhoff International Publishing, Leiden, 1977). English transl. from the 4th Russian edition by R.M. RadokGoogle Scholar
  14. 14.
    L. Sobrero, Nuovo metodo per lo studio dei problemi di elasticità, con applicazione al problema della piastra forata. Ricerche di Ingegneria. 13(2), 255–264 (1934)MathSciNetzbMATHGoogle Scholar
  15. 15.
    V.I. Smirnov, A Course of Higher Mathematics, vol. 3, Part 2 (Pergamon Press, Oxford, 1964)Google Scholar

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Authors and Affiliations

  • Serhii V. Gryshchuk
    • 1
  • Sergiy A. Plaksa
    • 1
  1. 1.Institute of Mathematics, National Academy of Sciences of UkraineKievUkraine

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