Abstract
The purpose of this paper is to solve a kind of Riemann–Hilbert problem for \(\Psi \)-hyperholomorphic functions, which are linked with the use of non-standard orthogonal basis of the Euclidean space \({\mathbb R}^m\). We approach this problem using the language of Clifford analysis for obtaining the explicit solution of the problem in a Jordan domain \(\Omega \subset {\mathbb R}^m\). Since our study is concerned with either a smooth or fractal boundary, the data of the problem involve Lipschitz class \({\text{ Lip }}(k-1+\alpha ,\Gamma )\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
Abreu, R., Ávila, R., Bory, J.: Boundary value problems with higher order Lipschitz boundary data for polymonogenic functions in fractal domains. Appl. Math. Comput. 269, 802–808 (2015)
Abreu, R., Bory, J., Guzmán, A., Kähler, U.: On the \(\varphi \)-hyperderivative of the \(\psi \)-Cauchy-type integral in Clifford analysis. Comput. Methods Funct. Theory 17(1), 101–119 (2017)
Abreu, R., Bory, J., Guzmán, A., Kähler, U.: On the \(\Pi \)-operator in Clifford analysis. J. Math. Anal. Appl. 434(2), 1138–1159 (2016)
Andersson, L.E., Elfving, T., Golub, G.H.: Solution of biharmonic equations with application to radar imaging. J. Comput. Appl. Math. 94, 153–180 (1998)
Balk, B.M.: On Polyanalytic Functions. Akademie Verlag, Berlin (1991)
Begehr, H., Chaudharyb, A., Kumarb, A.: Boundary value problems for bi-poly- analytic functions. Complex Variables Elliptic Equ. 55(1–3), 305–316 (2010)
Begehr, H., Hile, N.G.: A hierarchy of integral operators. Rocky Mt. J. Math. 27(3), 669–706 (1997)
Berdyshev, A.S., Cabada, A., Turmetov, BKh.: On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order, Acta Math. Sci. Ser. B Engl. Ed. 34, 1695–1706 (2014)
Bernstein, S.: On the index of Clifford algebra valued singular integral operators and the left linear Riemann problem. Complex Variables Theory Appl. 35(1), 33–64 (1998)
Bory Reyes, J., Abreu Blaya, R., Rodríguez Dagnino, R.M., Kats, B.A.: On Riemann boundary value problems for null solutions of the two dimensional Helmholtz equation. Anal. Math. Phys. 9(1), 483–496 (2019)
Bory, J., De Schepper, H., Guzmán, A., Sommen, F.: Higher order Borel-Pompeiu representations in Clifford analysis. Math. Methods Appl. Sci. 39(16), 4787–4796 (2016)
Brackx F., Delanghe, R., Sommen, F. Clifford analysis. In: Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, 1982
Cerejeiras, P., Kähler, U., Min, K.: On the Riemann boundary value problem for null solutions to iterated generalized Cauchy–Riemann operator in Clifford analysis. Results Math. 63(3–4), 1375–1394 (2013)
Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of Fermionic observables. Inventiones Mathematicae 189(3), 515–580 (2012)
Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992)
Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University Press, Cambridge (1991)
Gürlebeck, K., Habetha, K., Sprössig, W.: Application of Holomorphic Functions in Two and Higher Dimensions. Birkhäuser/Springer, Cham (2016)
Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic., functions in the plane and \(n\)-dimensional space. Translated from the, German Original, p. 2008. Birkhäuser Verlag, Basel (2006)
Harrison, J., Norton, A.: The Gauss-Green theorem for fractal boundaries. Duke Math. J. 67(3), 575–588 (1992)
Its, A.R.: The Riemann–Hilbert problem and integrable systems. Not. Am. Math. Soc. 50(11), 1389–1400 (2003)
Jinyuan, D., Yufeng, W.: On Riemann boundary value problems of polyanalytic functions and metaanalytic functions on the closed curves. Complex Var. 50(7–11), 521–533 (2005)
Lai, M.C., Liu, H.C.: Fast direct solver for the biharmonic equation on a disk and its application to incompressible flows. Appl. Math. Comput. 164, 679–695 (2005)
Le, J., Jinyuan, D.: Riemann boundary value problems for some k-regular functions in Clifford analysis. Acta Math. Sci. Ser. B Engl. Ed. 32(5), 2029–2049 (2012)
LeLongfei, G., Zunwei, F.: Riemann boundary value problem for H-2-monogenic function in Hermitian Clifford analysis. Bound. Value Probl. 2014, 81 (2014)
Li, D., Mao, J.-F.: A Koch-like sided fractal bow-tie dipole antenna. IEEE Trans. Antennas Propag. 60(5), 2242–2251 (2012)
McLaughlin, K., Miller, P.: The d-bar steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights. In: IMRP, p. 1–77 (2006)
Mushelisvili, N.I.: Singular Integral Equations. Nauka, M. (1968). (English transl. of 1st ed., Noodhoff, Groningen, (1953))
Muskhelishvili, N.I.: Singular integral equations. Boundary problems of function theory and their application to mathematical physics, Dover Publications Inc., New York, Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok, Corrected reprint of the 1953 English translation (1992)
Nono, K., Inenaga, Y.: On the Clifford linearization of Laplacian. J. Indian Inst. Sci. 67(5–6), 203–208 (1987)
Shapiro, M.V.: On the conjugate harmonic functions of M. Riesz-E. Stein-G. Weiss, In: Stancho D., et al. (Eds.), Topics in Complex Analysis, Differential Geometry and Mathematical Physics, Third International Workshop on Complex Structures and Vector Fields, St. Konstantin, Bulgaria, August 23–29, World Scientific, Singapore, 1997, pp. 8–32 (1996)
Stein, E.: Singular integrals and differentiability properties of functions. In: Princeton Mathematical Series, No. 30, Princeton University Press: Princeton (1970)
Vekua, I.N.: Generalized Analytic Functions. Nauka, Moscow (1959)
Vladimir, R.: On Hilbert and Riemann problems for generalized analytic functions and applications. Anal. Math. Phys. 11(1), 18 (2021)
Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63–89 (1934)
Zhang, Z., Gürlebeck, K.: Some Riemann boundary value problems in Clifford analysis (I). Complex Var. Elliptic Equ. 58(7), 991–1003 (2013)
Zhou-Zheng, K., Tie-Cheng, X., Wen-Xiu, M.: Riemann-Hilbert method for multi-soliton solutions of a fifth-order nonlinear Schrödinger equation. Anal. Math. Phys. 11(1), 13 (2021)
Funding
José Luis Serrano Ricardo gratefully acknowledges the financial support of the Postgraduate Study Fellowship of the Consejo Nacional de Ciencia y Tecnología (CONACYT) (grant number 1042069).
Author information
Authors and Affiliations
Contributions
All authors discussed the results and contributed equally to the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest regarding the publication of this paper.
Ethical approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ricardo, J.L.S., Blaya, R.A. & Ortiz, J.S. On a Riemann–Hilbert problem for \(\Psi \)-hyperholomorphic functions in \({\mathbb R}^m\). Anal.Math.Phys. 13, 84 (2023). https://doi.org/10.1007/s13324-023-00847-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-023-00847-1