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Boundary value problems for quaternionic monogenic functions on non-smooth sureaces

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Abstract

In this paper, analogous of the Compound Riemann-Hilbert boundary value problems are investigate for quaternionic monogenic functions. The solution (explicitly) of the problem is established over continuous surface, with little smoothness, which bounds a bounded domain of R3. In particular, smoothness property for high-dimensional Cauchy type integral are computed. We also use Zygmund type estimates to adapt existing one-variable complex results to ilustrate the Hölder-boundedness of the singular integral operator on 2-dimensional Ahlfors regular surfaces. At the end, uniqueness of solution for the Riemann boundary value problem have already built taking as a base the general Operator Theory.

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References

  1. R. Abreu and J. Bory (1998): Solvability of a Riemann linear conjugation problem on a fractal surface, Extracta Matematicae, Vol. 13, No. 1. (to appear)

  2. R. Abreu and J. Bory (1998): Removable singularirties forH-regular functions of Zygmund class, Proc. Royal. Irish. Acad. (submitted)

  3. F. Brackx, R. Delanghe and F. Sommen (1982): Clifford Analysis, Boston, Pitman.

    MATH  Google Scholar 

  4. A. A. Babaev and V. V. Salaev (1982): Boudary value problem and singular integral equation on a rectifiable contour, Mat. Zam, Vol. 31, No. 4, 571–580.

    MathSciNet  Google Scholar 

  5. N. K. Bari and S. B. Stieshkin (1956). Best approximations and differential properties of conjugate functions, Tr. Mosk. Obsh. Tom. 5, 483–522.

    MATH  Google Scholar 

  6. R.R. Coifman, A. McIntosh and Y. Meyer (1982): L’integrale de Cauchy définite un opérator bourné surL 2 pour les courves lipschitziennes. Anaals of Mathematic 116, 361–387.

    Article  MathSciNet  Google Scholar 

  7. G. David (1984): Opératours integraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup 17, 157–189.

    MATH  Google Scholar 

  8. — (1988). Morceaux de graphes Lipschitziens el intégrales singularies sur un surface, Rev. Mat. Iberoamericana 4, 73–114.

    MATH  MathSciNet  Google Scholar 

  9. E. P. Dolzhenko (1970): On the removal of singularities of analytic functions, Am. Math. Soc. Transl. 97 (2), 33–41.

    Google Scholar 

  10. K. J. Falconer (1985): The geometry of fractal sets, Cambriege. Univ. Press, Cambridge.

    MATH  Google Scholar 

  11. H. Federer (1969): Geometric Measure Theory, Springer-Verlag, Heidelberg/New York.

    MATH  Google Scholar 

  12. J. Feder (1988): Fractals, Plenum Press, New York.

    MATH  Google Scholar 

  13. F. D. Gajov (1977): Boundary value problems, 3rd ed; “Nauka”, Moskow; English transl. of 2 nd ed, Pergamon Press, Oxford, and Addison-Wesley, Reading, MA; 1966.

    Google Scholar 

  14. J. Gilbert and M. Murray (1991): Clifford algebras and Dirac operator in harmonic analysis, Cambridge University. Press, Cambridge.

    Google Scholar 

  15. K. Gürlebeck and W. Sprössig (1990): Quaternionic analysis and elliptic boundary value problems, Birkhausser, Boston.

    MATH  Google Scholar 

  16. K. Gürlebeck, U. Kähler, J. Ryan and W. Sprössig (1997): Clifford analysis over unbounded domains, Advances in Applied Mathematics 19, 216–239.

    Article  MATH  MathSciNet  Google Scholar 

  17. V. Iftimie (1965): Fonctions hypercomplexes. Bull. Math. Soc. Sci. Math. R. S. Rumanie 9, 279–332.

    MathSciNet  Google Scholar 

  18. B. A. Kats (1983): The Riemann problem on a closed Jordan curve, Sov. Math. (Iz VUZ), 27, 83–98.

    MATH  Google Scholar 

  19. — (1994): On the Riemann boundary value problem on a fractal curve, Russ. Acad. Sci. Dokl. Math. Vol. 48, No. 3, 559–561.

    MathSciNet  Google Scholar 

  20. — (1994): On a version of the Riemann boundary value problem on a fractal curve, Iz. Vyssh Ochebn Zaved. Mat, No. 4, (383), 10–20.

    MathSciNet  Google Scholar 

  21. A. N. Kolmogorov and V. M. Tikhomirov (1959): Uspekhi Mat. Nauk. 14, No. 2 (86), 3–86; English transl. in Amer. Math. Soc. Transl (2)17(1961).

    MATH  MathSciNet  Google Scholar 

  22. M. L. Lapidus (1991): Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjeture, Trans. Am. Math. Sci. 325, No. 2, 465–529.

    Article  MATH  MathSciNet  Google Scholar 

  23. D. L. Lord and A. G. O’Farrell (1991): Removable Singularities for Analytic Functions of Zygmund Class, Proc. Royal. Irish. Acad., Vol. 91A, No. 2, 195–204.

    MathSciNet  Google Scholar 

  24. J. K. Lu (1993): Boundary value problems for analytic functions. World Scientific Publish. Singapure. New Jersey. London. Hong Kong.

    MATH  Google Scholar 

  25. Alan McIntosh (1989): Clifford algebras and the higher dimensional Cauchy integral, Approximation Theory and Function Spaces, Banach Center. Publications 22, 253–267.

    MathSciNet  Google Scholar 

  26. B. B. Mandelbrot (1983): The fractal geometry of nature, rev. and enl. ed., W. H. Freeman, New York, N. Y.

    Google Scholar 

  27. Margaret Murray (1985): The Cauchy integral, Calderón commutators and conjugations of singular integrals inR m, Transactions of the Am. Math. Soc. 289, 497–518.

    Article  MATH  Google Scholar 

  28. N. I. Mushelisvili (1972): Singular integral equations, Nauka, Moskow, 1968, English transl. of 1st ed., Noodhoff, Groningen, 1953; reprint, 1972.

    Google Scholar 

  29. E. Oboloshvili (1988): Effective solution of some boundary value problems in two and three dimensional cases, Proceeding of Functional Analytic Methods in Complex Analysis and applications to Partial Diff Equations, World Scientific, 149–172

  30. J. Ryan (1993): Clifford algebras in analysis and related topics, CRC Press. Boca Raton, New York. London. Tokyo.

    Google Scholar 

  31. J. Ryan (1996): Dirac operator on spheres and hyperbolae, Bol. Soc.Mat. Mexicana (3) Vol. 3.

  32. V. V. Salaev (1976): Direct and inverse estimates for singular Cauchy integral over closed curves, Tom. 19, No. 3, 365–380.

    MATH  MathSciNet  Google Scholar 

  33. S. W. Semmes (1994): Finding structure in sets with little smoothness, Proceedings of the International Congress of Mathematics. Zurich, Switzerland, 875–885.

  34. — (1990): Analysis vs. Geometry on a class of rectifiable hypersurfaces inR n, Indiana Univ. Math. Journal, Vol. 39, No. 4, 1005–1035.

    Article  MATH  MathSciNet  Google Scholar 

  35. — (1990): Differentiable function theory on hypersurfaces inR n (Without bounds of their smoothness). Indiana Univ. Math. Journal, Vol. 39, No. 4, 1005–1035.

    Article  MATH  MathSciNet  Google Scholar 

  36. R. K. Seifulaev (1993): Boundary values of multidimensional integrals of potencial type, Russian Academy of Science. Dok. Math, Vol. 47, No. 1, 104–107.

    Google Scholar 

  37. E. M. Stein (1970): Singular integrals and differentiability property of functions, Princeton Univ, Press, Princeton, N J.

    Google Scholar 

  38. M. Shapiro and N. Vasilievsky (1995): Quaternionic ψ-hyperholomorphic functions, singular integral operator and boundary value problems I, ψ-hypercomplex function theory, Complex Variables 27, 17–46.

    MATH  Google Scholar 

  39. — (1995): Quaternionic ψ-hyperholomorphic functions, singular integral operator and boundary value problems II. Algebras of singular integral operators and Riemann type boundary value problems. Complex Variables 27, 67–96.

    MATH  Google Scholar 

  40. Z. Xu (1990): On linear and nonlinear Riemann-Hilbert problem for regular function with values in a Clifford algebras, Chin. Am. of Math. 11B:3, 349–358.

    Google Scholar 

  41. Z. Xu and C. Zhou (1993): On boundary value problems of Riemann-Hilbert type for monogenic functions in a half space ofR n, (n≥2), Complex Variables Theory Appl., 3–4.

  42. — (1996): On Riemann-Hilbert problems for nonhomogeneous Dirac equation in a half space ofR n, (n≥2), Clifford algebras in analysis and related topics. Edited by J. Ryan. CRC. Press. Boca Raton, New York. London. Tokyo, 285–295.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Oriente, 90500, Santiago of Cuba, Cuba

    Ricardo Abreu Blaya & Juan Bory Reyes

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  1. Ricardo Abreu Blaya
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  2. Juan Bory Reyes
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Blaya, R.A., Reyes, J.B. Boundary value problems for quaternionic monogenic functions on non-smooth sureaces. AACA 9, 1–22 (1999). https://doi.org/10.1007/BF03041934

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