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Biharmonic Monogenic Functions and Biharmonic Boundary Value Problems

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Book cover Analysis, Probability, Applications, and Computation

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

We consider a commutative algebra B over the field of complex numbers with a basis {e 1, e 2} 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.

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References

  1. A. Douglis, A function-theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 6(2), 259–289 (1953)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. 127–134

    Google Scholar 

  5. S.V. Gryshchuk, S.A. Plaksa, Schwartz-type integrals in a biharmonic plane. Int. J. Pure Appl. Math. 83(1), 193–211 (2013)

    Article  Google Scholar 

  6. S.V. Gryshchuk, S.A. Plaksa, Monogenic functions in the biharmonic boundary value problem. Math. Methods Appl. Sci. 39(11), 2939–2952 (2016)

    Article  MathSciNet  Google Scholar 

  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 Russian

    Google Scholar 

  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)

    MATH  Google Scholar 

  9. A.I. Lurie, Theory of Elasticity (Springer, Berlin, 2005). Engl. transl. by A. Belyaev

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. 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. Radok

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. V.I. Smirnov, A Course of Higher Mathematics, vol. 3, Part 2 (Pergamon Press, Oxford, 1964)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine

    Serhii V. Gryshchuk & Sergiy A. Plaksa

Authors
  1. Serhii V. Gryshchuk
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  2. Sergiy A. Plaksa
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Karl‐Olof Lindahl

  2. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Torsten Lindström

  3. Department of Mathematics, University of Torino, Torino, Italy

    Luigi G. Rodino

  4. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Joachim Toft

  5. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Patrik Wahlberg

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Gryshchuk, S.V., Plaksa, S.A. (2019). Biharmonic Monogenic Functions and Biharmonic Boundary Value Problems. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_14

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