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Isomorphisms of Partial Differential Equations in Clifford Analysis

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Abstract

This paper studies some systems of second order partial differential equations associated to a Dirac type operator \({^\varphi \!\underline{\partial }}\) in \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\), with respect to an arbitrary structural set \(\varphi \). We develop a method by which we can transform these general systems into those connected to the standard Dirac operator \(\underline{\partial }\). One unexpected result is that there exists an isomorphism relating the above generalized equations to the original ones.

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References

  1. Abreu Blaya, R., Bory Reyes, J., Guzmán, A., Kähler, U.: Symmetries and Associated pairs in Quaternionic Analysis. Hypercomplex Analysis: New Perspectives and Applications. Trends in Mathematics, pp. 1–18 (2014)

  2. Abreu Blaya, R., Bory Reyes, J., Guzmán, A., Kähler, U.: On some structural sets and a quaternionic \((\varphi,\psi )-\)hyperholomorphic function theory. Math. Nachr. 288(13), 1451–1475 (2015)

    Article  MathSciNet  Google Scholar 

  3. Abreu Blaya, R., Bory Reyes, J., Guzmán, A., Kähler, U.: On the \({\Pi }\)-operator in Clifford analysis. J. Math. Anal. Appl. 434(2), 1138–1159 (2016)

    Article  MathSciNet  Google Scholar 

  4. Abreu Blaya, R., Bory Reyes, J., Guzmán, A., Kähler, U.: On the \(\phi \)-hyperderivative of the \(\psi \)-Cauchy-type integral in Clifford analysis. Comput. Meth. Funct. Theory 17, 101–119 (2017)

    Article  MathSciNet  Google Scholar 

  5. Bory Reyes, J., De Schepper, H., Adán Guzmán, A., Sommen, F.: Higher order Borel–Pompeiu representations in Clifford analysis. Math. Meth. Appl. Sci. 39(16), 4787–4796 (2016)

    Article  MathSciNet  Google Scholar 

  6. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)

    MATH  Google Scholar 

  7. Gürlebeck, K.: On some classes of \(\Pi \)-operators. In: Ryan, J., Struppa, D. (eds.) Dirac Operators in Analysis, Pitman Research Notes in Mathematics, vol. 394 (1998)

  8. Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser Verlag, Basel (2008)

    MATH  Google Scholar 

  9. Gürlebeck, K., Kähler, U., Shapiro, M.: On the \(\Pi \)-operator in hyperholomorphic function theory. Adv. Appl. Clifford Algebras 9(1), 23–40 (1999)

    Article  MathSciNet  Google Scholar 

  10. Gürlebeck, K., Nguyen, H.M.: \(\psi \)-Hyperholomorphic functions and an application to elasticity problems. AIP Conf. Proc. 1648(1), 440005 (2015)

    Article  Google Scholar 

  11. Gürlebeck, K., Nguyen, H.M.: On \(\psi \)-hyperholomorphic Functions and a Decomposition of Harmonics. Hyper Complex Analysis: New Perspectives and Applications. Trends in Mathematics, pp. 181–189 (2014)

  12. Malonek, H., Peña-Peña, D., Sommen, F.: Fischer decomposition by inframonogenic functions. CUBO Math. J. 12(02), 189–197 (2010)

    Article  MathSciNet  Google Scholar 

  13. Malonek, H., Peña-Peña, D., Sommen, F.: A Cauchy–Kowalevski theorem for inframonogenic functions. Math. J. Okayama Univ. 53, 167–172 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Moreno García, A., Moreno García, T., Abreu Blaya, R., Bory Reyes, J.: A Cauchy integral formula for inframonogenic functions in Clifford analysis. Adv. Appl. Clifford Algebras 27(2), 1147–1159 (2017)

    Article  MathSciNet  Google Scholar 

  15. Moreno García, A., Moreno García, T., Abreu Blaya, R., Bory Reyes, J.: Inframonogenic functions and their applications in three dimensional elasticity theory. Math. Meth. Appl. Sci. 41(10), 3622–3631 (2018)

    Article  Google Scholar 

  16. Moreno García, A., Moreno García, T., Abreu Blaya, R., Bory Reyes, J.: Decomposition of inframonogenic functions with applications in elasticity theory. Math. Meth. Appl. Sci. 43, 1915–1924 (2020)

    Article  MathSciNet  Google Scholar 

  17. Naser, M.: Hyperholomorphic functions. Siber. Math. J. 12, 959–968 (1971)

    Article  MathSciNet  Google Scholar 

  18. Nguyen, H.M.: \(\psi \)-Hyperholomorphic Function Theory in \({{\mathbb{R}}^3}\): Geometric Mapping Properties and Applications. Fakultat Bauingenieurwesen der Bauhaus-Universitat. Weimar. e-pub.uni-weimar.de (2015)(Habilitation Thesis)

  19. Nono, K.: Hyperholomorphic functions of a quaternion variable. Bull. Fukuoka Univ. Educ. III(32), 21–37 (1983)

    MATH  Google Scholar 

  20. Nono, K.: On the quaternion linearization of Laplacian \(\Delta \). Bull. Fukuoka Univ. Educ. 35, 5–10 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Shapiro, M.V., Vasilevsky, N.L.: On the Bergman kernel function in the Clifford analysis. In: Clifford Algebras and Their Applications in Mathematical Physics. Deinze, Fundam. Theor. Phys., vol. 55. Kluwer Acad. Publ., Dordrecht, pp. 183–192 (1993)

  22. Sommen, F., Van Acker, P.: \(SO(m)\)-Invariant Differential Operators on Clifford Algebra-Valued Functions. Found. Phys. 23(11) (1993)

  23. Vasilevski, N., Shapiro, M.: On an analog of the monogenity in the sense of Moisil–Theodoresko and some applications in the theory of the boundary value problems. Reports of the enlarged session of seminar of the Vekua Applied Mathematics Institute. Tibilis 1, 63–66 (1985)

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Acknowledgements

We thank the two anonymous reviewers whose comments, answers and suggestions helped improve and clarify this manuscript.

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Authors and Affiliations

  1. Universidad Autónoma de Guerrero, Lázaro Cárdenas 88, 39086, Chilpancingo de los Bravo, Mexico

    Daniel Alfonso Santiesteban & Ricardo Abreu Blaya

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  1. Daniel Alfonso Santiesteban
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  2. Ricardo Abreu Blaya
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Communicated by Hendrik De Bie.

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Alfonso Santiesteban, D., Abreu Blaya, R. Isomorphisms of Partial Differential Equations in Clifford Analysis. Adv. Appl. Clifford Algebras 32, 10 (2022). https://doi.org/10.1007/s00006-021-01191-y

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  • DOI: https://doi.org/10.1007/s00006-021-01191-y

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