Solutions of Inhomogeneous Generalized Moisil–Teodorescu Systems in Euclidean Space

  • Juan Bory-Reyes
  • Marco Antonio Pérez-de la RosaEmail author


Let \(\mathbb R_{0, m+1}^{(s)}\) be the space of s-vectors (\(0\le s\le m+1\)) in the Clifford algebra \(\mathbb R_{0, m+1}\) constructed over the quadratic vector space \(\mathbb R^{0, m+1}\), let \(r, p, q\in \mathbb N\) with \(0\le r\le m+1\), \(0\le p\le q\) and \(r+2q\le m+1\) and let \(\mathbb R_{0, m+1}^{(r,p,q)}=\sum _{j=p}^q\bigoplus \mathbb R_{0, m+1}^{(r+2j)}\). Then a \(\mathbb R_{0, m+1}^{(r,p,q)}\)-valued smooth function F defined in an open subset \(\Omega \subset \mathbb R^{m+1}\) is said to satisfy the generalized Moisil–Teodorescu system of type (rpq) if \(\partial _x F=0\) in \(\Omega \), where \(\partial _x\) is the Dirac operator in \(\mathbb R^{m+1}\). To deal with the inhomogeneous generalized Moisil–Teodorescu systems \(\partial _x F=G\), with a \(\sum _{j=p}^{q} \bigoplus {\mathbb {R}}^{(r+2j-1)}_{0,m+1}\)-valued continuous function G as a right-hand side, we embed the systems in an appropriate Clifford analysis setting. Necessary and sufficient conditions for the solvability of inhomogeneous systems are provided and its general solution described.


Clifford analysis Dirac operator Moisil–Teodorescu systems Conjugate harmonic pairs. 

Mathematics Subject Classification

Primary 30G359 Secondary 35F05 



  1. 1.
    Abreu Blaya, R., Bory Reyes, J., Delanghe, R., Sommen, F.: Generalized Moisil-Théodoresco systems and Cauchy integral decompositions. Int. J. Math. Math. Sci. 2008, 19 (2008). (Article ID746946 )CrossRefGoogle Scholar
  2. 2.
    Abreu Blaya, R., Bory Reyes, J., Luna-Elizarrarás, M.E., Shapiro, M.: \(\bar{\partial }\)-problem in domains of \(\mathbb{C}^2\) in terms of hyper-conjugate harmonic functions. Complex Var. Elliptic Equ. 57(7–8), 743–749 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abreu Blaya, R., Bory Reyes, J.: \({\overline{\partial }}\)-problem for an overdetermined system con two higher dimensional variables. Arch. Math. (Basel) 97(6), 579–586 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bory Reyes, J., Abreu Blaya, R., Pérez-de la Rosa, M.A., Schneider, B.: A quaternionic treatment of inhomogeneous Cauchy-Riemann type systems in some traditional theories. Compl. Anal. Oper. Theory. 11(5), 1017–1034 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bory Reyes, J., Delanghe, R.: On the structure of solutions of the Moisil-Théodoresco system in Euclidean space. Adv. Appl. Clifford Algebra 19(1), 15–28 (2009)CrossRefGoogle Scholar
  6. 6.
    Bory Reyes, J., Delanghe, R.: On the solutions of the Moisil Théodoresco system. Math. Methods Appl. Sci. 31(12), 1427–1439 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brackx, F., Delanghe, R., De Schepper, H.: Hardy spaces of solutions of generalized Riesz and Moisil-Teodorescu systems. Complex Var. Elliptic Equ. 57(7–8), 771–785 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman, Boston (1982)zbMATHGoogle Scholar
  9. 9.
    Brackx, F., Delanghe, R., Sommen, F.: On conjugate harmonic functions in Euclidean space. Math. Methods Appl. Sci. 25, 1553–1562 (2002)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Brackx, F., Delanghe, R., Sommen, F.: Differential forms and/or multi-vector functions. Cubo 7(2), 139–169 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Colombo, F., Luna-Elizarrarás, M.E., Sabadini, I., Shapiro, M., Struppa, D.C.: A quaternionic treatment of the inhomogeneous div-rot system. Mosc. Math. J. 12(1), 37–48 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cialdea, A.: On the theory of self-conjugate differential forms. Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996). Atti Sem. Mat. Fis. Univ. Modena 46(suppl.), 595–620 (1998)MathSciNetGoogle Scholar
  13. 13.
    Delanghe, R.: On homogeneous polynomial solutions of the Riesz system and their harmonic potentials. Complex Var. Elliptic Equ. 52(10–11), 1047–1061 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Delanghe, R.: On Moisil–Théodoresco systems in euclidean space. AIP Conf. Proc. 1048(1), 17–20 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    Delanghe, R.: On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space. Cubo 12(2), 145–167 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Delanghe, R., Lávička, R., Souček, V.: On polynomial solutions of generalized Moisil-Théodoresco systems and Hodge-de Rham systems. Adv. Appl. Clifford Algebr. 21(3), 521–530 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-valued Functions—A Function Theory for the Dirac Operator. Kluwer Academic, Dordrecht (1992)CrossRefGoogle Scholar
  18. 18.
    Delgado, B.B., Porter, M.R.: General solution of the inhomogeneous div-curl system and consequences. Adv. Appl. Clifford Algebra 27(4), 3015–3037 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Eelbode, D., Sommen, F.: Differential forms in Clifford analysis. Methods of complex and Clifford analysis, pp. 41–69. SAS, Delhi (2004)zbMATHGoogle Scholar
  20. 20.
    Fueter, R.: Die Funktionentheorie der Differentialgleichungen \(\Delta u=0\) und \(\Delta \Delta u=0\) mit vier reellen Variablen. (German) Comment. Math. Helv. 7(1), 307–330 (1934)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gilbert, J., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambrigde University Press, Cambridge (1991)CrossRefGoogle Scholar
  22. 22.
    Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Mathematical Methods in Practice. Wiley, Chichester (1997)zbMATHGoogle Scholar
  23. 23.
    Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser, Basel (2008)zbMATHGoogle Scholar
  24. 24.
    Lavicka, R.: Orthogonal Appell bases for Hodge-de Rham systems in Euclidean spaces. Adv. Appl. Clifford Algebr. 23(1), 113–124 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Malaspina, A.: The Rudin-Carleson theorem for non-homogeneous differential forms. Int. J. Pure Appl. Math. 1(2), 203–215 (2002)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Moisil, Gr, Théodoresco, N.: Fonctions holomorphes dans l’espace. Mathema-tica Cluj 5, 142–159 (1931)zbMATHGoogle Scholar
  27. 27.
    Nolder, C.A.: Conjugate harmonic functions and Clifford algebras. J. Math. Anal. Appl. 302(1), 137–142 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Porter, M.R., Shapiro, M., Vasilevski, N.L.: Quaternionic differential and integral operators and the \(\overline{\partial }\)-problem. J. Nat. Geom. 6(2), 101–124 (1994)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Porter, M.R., Shapiro, M., Vasilevski, N.L.: On the analogue of the \(\overline{\partial }\)-problem in quaternionic analysis. Clifford Algebras and Their Applications in Mathematical Physics (Deinze, 1993), Fundamental Theories of Physics, vol. 55, pp. 167–173. Kluwer Academic Publishers Group, Dordrecht (1993)CrossRefGoogle Scholar
  30. 30.
    Shapiro, M.: On the conjugate harmonic functions of M. Riesz–E. Stein–G. Weiss. Topics in complex analysis, differential geometry and mathematical physics (St. Konstantin, 1996), pp. 8–32. World Science, River Edge (1997)Google Scholar
  31. 31.
    Sirkka-Liisa, E., Heikki, O.: On Hodge-de Rham systems in hyperbolic Clifford analysis. AIP Conf. Proc. 1558, 492–495 (2013)ADSGoogle Scholar
  32. 32.
    Souchek, V.: On massless Field equation in higher dimensions. In: 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering. K. Gürlebeck and C. Könke (eds.) Weimar, Germany, 07–09 July (2009)Google Scholar
  33. 33.
    Souchek, V.: Representation theory in clifford analysis. In: Alpay, D. (ed.) Operator Theory, pp. 1509–1547. Springer, Basel (2015)CrossRefGoogle Scholar
  34. 34.
    Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85(2), 199–224 (1979)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Juan Bory-Reyes
    • 1
  • Marco Antonio Pérez-de la Rosa
    • 2
    Email author
  1. 1.ESIME-Zacatenco, Instituto Politécnico NacionalMexico CityMexico
  2. 2.Department of Actuarial Sciences, Physics and MathematicsUniversidad de las Américas PueblaSan Andrés CholulaMexico

Personalised recommendations