Startseite On a Riemann--Hilbert boundary value problem for (ϕ,ψ)-harmonic functions in ℝ m
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On a Riemann--Hilbert boundary value problem for (ϕ,ψ)-harmonic functions in ℝ m

  • José Luis Serrano Ricardo ORCID logo , Ricardo Abreu Blaya ORCID logo , Juan Bory Reyes ORCID logo EMAIL logo und Jorge Sánchez Ortiz ORCID logo
Veröffentlicht/Copyright: 3. März 2022
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Abstract

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for ( φ , ψ ) -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space m . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω m with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ( 1 + α , Γ ) .

MSC 2010: 42C10; 30G35

Award Identifier / Grant number: SIP20211188

Award Identifier / Grant number: 1042069

Award Identifier / Grant number: 1011513

Award Identifier / Grant number: 100592

Funding statement: The work was supported by Secretaría de Investigación y Posgrado, Instituto Politécnico Nacional (grant number SIP20211188), and Consejo Nacional de Ciencia y Tecnología (grant numbers 1042069, 1011513, 100592).

References

[1] R. Abreu Blaya, J. Bory Reyes, A. Guzmán Adán and U. Kähler, On the Π-operator in Clifford analysis, J. Math. Anal. Appl. 434 (2016), no. 2, 1138–1159. 10.1016/j.jmaa.2015.09.038Suche in Google Scholar

[2] R. Abreu Blaya, J. Bory Reyes, A. Guzmán Adán and U. Kähler, On the φ-hyperderivative of the ψ-Cauchy-type integral in Clifford analysis, Comput. Methods Funct. Theory 17 (2017), no. 1, 101–119. 10.1007/s40315-016-0172-0Suche in Google Scholar

[3] E. P. Bellido, G. D. Bernasconi, D. Rossouw, J. Butet, O. J. F. Martin and G. A. Botton, Self-similarity of plasmon edge modes on Koch fractal antennas, ACS Nano 11 (2017), 11240–11249. 10.1021/acsnano.7b05554Suche in Google Scholar PubMed

[4] R. A. Blaya, R. A. Ávila and J. B. Reyes, Boundary value problems with higher order Lipschitz boundary data for polymonogenic functions in fractal domains, Appl. Math. Comput. 269 (2015), 802–808. 10.1016/j.amc.2015.08.012Suche in Google Scholar

[5] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Res. Notes Math. 76, Pitman, Boston, 1982. Suche in Google Scholar

[6] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990. 10.2307/2532125Suche in Google Scholar

[7] J. E. Gilbert and M. A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Stud. Adv. Math. 26, Cambridge University, Cambridge, 1991. 10.1017/CBO9780511611582Suche in Google Scholar

[8] K. Gürlebeck, K. Habetha and W. Spröß ig, Holomorphic Functions in the Plane and n-Dimensional Space, Birkhäuser, Basel, 2008. Suche in Google Scholar

[9] K. Gürlebeck, K. Habetha and W. Spröß ig, Application of Holomorphic Functions in Two and Higher Dimensions, Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-0348-0964-1Suche in Google Scholar

[10] J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Bull. Univ. Princeton 13 (1902), 49–52. Suche in Google Scholar

[11] J. Harrison and A. Norton, The Gauss-Green theorem for fractal boundaries, Duke Math. J. 67 (1992), no. 3, 575–588. 10.1215/S0012-7094-92-06724-XSuche in Google Scholar

[12] M. Karim, M. Rahim, H. Majid, O. Ayop, M. Abu and F. Zubir, Log periodic fractal Koch antenna for UHF band applications, Progr. Electromagn. Res. 100 (2010), 201–218. 10.2528/PIER09110512Suche in Google Scholar

[13] K. Kendig, Hassler Whitney: 1907–1989, Celebratio Math. (2013), http://celebratio.org/Whitney_H/article/245/. Suche in Google Scholar

[14] K. Nōno and Y. Inenaga, On the Clifford linearization of Laplacian, J. Indian Inst. Sci. 67 (1987), no. 5–6, 203–208. Suche in Google Scholar

[15] J. B. Reyes, H. De Schepper, A. G. Adán and F. Sommen, Higher order Borel–Pompeiu representations in Clifford analysis, Math. Methods Appl. Sci. 39 (2016), no. 16, 4787–4796. 10.1002/mma.3798Suche in Google Scholar

[16] J. L. S. Ricardo, J. Bory Reyes and R. Abreu Blaya, Singular integral operators and a ¯ -problem for ( φ , ψ ) -harmonic functions, Anal. Math. Phys. 11 (2021), no. 4, Paper No. 155. 10.1007/s13324-021-00590-5Suche in Google Scholar

[17] M. Shapiro, On the conjugate harmonic functions of M. Riesz–E. Stein–G. Weiss, Topics in Complex Analysis, Differential Geometry and Mathematical Physics (St. Konstantin 1996), World Scientific, River Edge (1997), 8–32. Suche in Google Scholar

[18] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Suche in Google Scholar

[19] D. Tumakov, D. Chikrin and P. Kokunin, Miniaturization of a Koch-type fractal antenna for Wi-Fi applications, Fractal Fract. 4 (2020), 10.3390/fractalfract4020025. 10.3390/fractalfract4020025Suche in Google Scholar

[20] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. 10.1007/978-1-4612-2972-8_14Suche in Google Scholar

[21] Z. Zhang and K. Gürlebeck, Some Riemann boundary value problems in Clifford analysis (I), Complex Var. Elliptic Equ. 58 (2013), no. 7, 991–1003. 10.1080/17476933.2011.613119Suche in Google Scholar

Received: 2021-09-26
Accepted: 2021-11-05
Published Online: 2022-03-03
Published in Print: 2022-06-01

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