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Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations

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Abstract

Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhotskii-Plemelj theorem for it as well as a necessary and sufficient condition for the possibility of extending a given Hölder function from such a surface up to a solution of the Moisil-Theodoresco system of partial differential equations in a domain. A formula for the square of a singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between the theory of the Moisil-Theodoresco system of partial differential equations and some versions of quaternionic analysis.

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References

  1. M. Zhdanov, Integral Transforms in Geophysics, Springer, Heidelberg (1998).

    Google Scholar 

  2. B. Schneider and M. Shapiro, “Some properties of the Cauchy-type integral for the Laplace vector fields theory,” Global Anal. Appl. Math., 274–280 (2004).

  3. B. Schneider and M. Shapiro, “Some properties of the Cauchy-type integral for the time-harmonic relativistic Dirac equation,” Math. Meth. Appl. Sci., 25, No. 16–18, 1441–1463 (2002).

    Article  MathSciNet  Google Scholar 

  4. B. Schneider and M. Shapiro, “Some properties of the quaternionic Cauchy-type integral for a piecewise Lyapunov surface of integration,” Contemp. Math., 364, 243–260 (2004).

    MathSciNet  Google Scholar 

  5. I. Mitelman and M. Shapiro, “Formulae of changing of integration order and of inversion for some multidimensional singular integrals and hypercomplex analysis,” J. Natural Geom., 5, 11–27 (1994).

    MathSciNet  Google Scholar 

  6. S. Mikhlin, Multidimensional Singular Integrals and Integral Equations [in Russian], Fizmatgiz, Moscow (1962).

    Google Scholar 

  7. K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley (1997).

  8. K. Gürlebeck and W. Sprössig, Quaternionic Analysis and Elliptic Boundary Value Problems, Akademie, Berlin (1989).

    Google Scholar 

  9. M. Shapiro and N. Vasilevski, “Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems. I. ψ-Hyperholomorphic function theory,” Complex Variables, Theory Appl., 27, 17–46 (1995).

    MathSciNet  Google Scholar 

  10. V. Kravchenko and M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Chapman & Hall (1996).

  11. M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Springer (1994).

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 105–112, January, 2006.

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Schneider, B. Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations. Ukr Math J 58, 118–126 (2006). https://doi.org/10.1007/s11253-006-0054-0

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  • DOI: https://doi.org/10.1007/s11253-006-0054-0

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