Generalized integral representations for functions with values in C(V _3,3)

Abstract

By using the solution to the Helmholtz equation Δu − λu = 0 (λ ≥ 0), the explicit forms of the so-called kernel functions and the higher order kernel functions are given. Then by the generalized Stokes formula, the integral representation formulas related with the Helmholtz operator for functions with values in C(V _3,3) are obtained. As application of the integral representations, the maximum modulus theorem for function u which satisfies H u = 0 is given.

Keywords

Universal Clifford algebra / Helmholtz equation / Generalized Cauchy-Pompeiu formula

References

[1] de Almeida R., Kraußhar R. S.. On the asymptotic growth of entire monogenic functions. J. Anal. Appl., 2005, 24: 791-813

[2] Begehr H.. Iterations of Pompeiu operators. Mem. Diff. Eq. Math. Phys., 1997, 12: 3-21

[3] Begehr H.. Iterated integral operators in Clifford analysis. J. Anal. Appl., 1999, 18: 361-377

[4] Begehr H.. Representation formulas in Clifford analysis. Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, 2002, Singapore: World Scientific 8-13

[5] Begehr H., Dai D. Q., Li X.. Integral representation formulas in polydomains, Complex Variables. Theory Appl., 2002, 47: 463-484

[6] Begehr H., Zhang Z. X., Du J. Y.. On Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra. Acta Math. Sci., 2003, 23B(1): 95-103

[7] Begehr H., Du J. Y., Zhang Z. X.. On higher order Cauchy-Pompeiu formula in Clifford analysis and its applications. General Math., 2003, 11: 5-26

[8] Begehr H., Zhang Z. X., Vu T. N. H.. Generalized integral representations in Clifford analysis. Complex Var., Elliptic Eqns., 2006, 51: 745-762

[9] Brackx F., Delanghe R., Sommen F.. Clifford Analysis, Research Notes in Mathematics, 76, 1982, London: Pitman Books Ltd

[10] Constales D., de Almeida R., Kraußhar R. S.. On Cauchy estimates and growth orders of entire solutions of iterated Dirac and generalized Cauchy-Riemann equations. Math. Meth. Appl. Sci., 2006, 29: 1663-1686

[11] Delanghe R.. On regular analytic functions with values in a Clifford algebra. Math. Ann., 1970, 185: 91-111

[12] Delanghe R.. On the singularities of functions with values in a Clifford algebra. Math. Ann., 1972, 196: 293-319

[13] Delanghe R.. Morera’s theorem for functions with values in a Clifford algebra. Simon Stevin, 1969, 43: 129-140

[14] Delanghe R., Brackx F.. Hypercomplex function theory and Hilbert modules with reproducing kernel. Proc. London Math. Soc., 1978, 37: 545-576

[15] Du J. Y., Zhang Z. X.. A Cauchy’s integral formula for functions with values in a universal Clifford algebra and its applications. Complex Var. Theory Appl., 2002, 47: 915-928

[16] Franks E., Ryan J.. Bounded monogenic functions on unbounded domains. Contemp. Math., 1998, 212: 71-80

[17] Gürlebeck K.. On some operators in Clifford analysis. Contemp. Math., 1998, 212: 95-107

[18] Gürlebeck K., Kähler U., Ryan J. Clifford analysis over unbounded domains. Adv. Appl. Math., 1997, 2(19): 216-239

[19] Gürlebeck K., Sprössig W.. Quaternionic Analysis and Elliptic Boundary Value Problems, 1989, Berlin: Akademie-Verlag

[20] Gürlebeck K., Zhang Z. X.. Some Riemann boundary value problems in Clifford analysis. Math. Meth. Appl. Sci., 2010, 33: 287-302

[21] Iftimie V.. Functions hypercomplex. Bull. Math. Soc. Sci. Math. R. S. Romania., 1965, 9(57): 279-332

[22] Lu J. K.. Boundary value problems of analytic functions, 1993, Singapore: World Scientific

[23] Muskhelishvilli N. I.. Singular Integral Equations, 1968, Moscow: NauKa

[24] Obolashvili E.. Higher order partial differential equations in Clifford analysis, 2002, Boston, Basel, Berlin: Birkhauser

[25] Vu T. N. H.. Integral representations in quaternionic analysis related to Helmholtz operator. Comp. Var. Theory Appl., 2003, 12(48): 1005-1021

[26] Vu T. N. H.. Helmholtz Operator in Quaternionic Analysis, 2005, Berlin: Free University

[27] Vekua I. N.. Generalized Analytic Functions, 1962, Oxford: Pergamon Press

[28] Xu, Z. Y., Boundary Value Problems and Function Theory for Spin-invariant Differential Operators, Ph. D. Thesis, Ghent State University, 1989.

[29] Zhang Z. X.. Integral representations and its applications in Clifford analysis. Gen. Math. (in Romania), 2005, 13(3): 79-96

[30] Zhang Z. X.. On k-regular functions with values in a universal Clifford algebra. J. Math. Anal. Appl., 2006, 315(2): 491-505

[31] Zhang Z. X.. Some properties of operators in Clifford analysis. Complex Var., Elliptic Eqns., 2007, 6(52): 455-473

[32] Zhang Z. X.. On higher order Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra. Bull. Belg. Math. Soc., 2007, 14: 87-97