Riemann boundary value problem for H-2-monogenic function in Hermitian Clifford analysis
© Gu and Fu; licensee Springer. 2014
Received: 3 November 2013
Accepted: 28 March 2014
Published: 9 April 2014
Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions of two Hermitian Dirac operators. Using a circulant matrix approach, we will study the Riemann type problems in Hermitian Clifford analysis. We prove a mean value formula for the Hermitian monogenic function. We obtain a Liouville-type theorem and a maximum module for the function above. Applying the Plemelj formula, integral representation formulas, and a Liouville-type theorem, we prove that the Riemann type problems for Hermitian monogenic and Hermitian-2-monogenic functions are solvable. Explicit representation formulas of the solutions are also given.
The classical Riemann boundary value problem (BVP for short) theory in the complex plane has been systematically developed, see  and . It is natural to generalize the classical Riemann BVP theory to higher dimensions. Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis and a generation of complex in plane analysis. The theory is centered around the concept of monogenic functions, see [3–6], etc. Under the framework, in [7–12], many interesting results about BVP for monogenic functions in Clifford analysis were presented. In  and , Riemann BVP for harmonic functions (i.e., 2-monogenic functions) and biharmonic functions were studied, the solutions are given in an explicit way.
More recently, Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions of two Hermitian Dirac operators invariant under the action of the unitary group. This function theory can be found in  and , etc. In , based on the complex Clifford algebra , the Hermitian Cauchy integral formulas were constructed in the framework of circulant matrix functions, and the intimate relationship with holomorphic function theory of several complex variables was considered. For details, we refer to [17–20]. In  and , a matrix Hilbert transform in Hermitian Clifford analysis was studied, and analogs of characteristic properties of the matrix Hilbert transform in classical analysis and orthogonal Clifford analysis were given, for example by the usual Plemelj-Sokhotski formula. Under this setting it is natural to consider the Riemann BVP. In , the Riemann BVP for (left) Helmholtz H-monogenic functions (i.e., null solutions of perturbed Hermitian Dirac operators in the framework of Hermitian Clifford analysis). If the perturbed value vanishes, is , then the Riemann BVP for H-monogenic circulant matrix functions was solved. Also, we naturally consider Riemann BVP for H-monogenic circulant matrix functions (i.e., null solutions to ) and H-2-monogenic circulant matrix functions (i.e., null solutions to ). Roughly speaking Riemann BVP means that we prescribe that the solutions are bounded at infinity. Up to present, as far as we know, it is a new problem. In this paper, motivated by [8, 9, 13, 14, 17, 18], we will consider Riemann BVP for H-2-monogenic circulant matrix functions in Hermitian Clifford analysis. Applying the integral representation formulas of H-monogenic circulant matrix functions and H-2-monogenic circulant matrix functions, we get mean values formulas. Furthermore we prove a maximum modulus theorem and a Liouville theorem in Hermitian Clifford analysis. Finally we get explicit solutions for Riemann BVP for H-2-monogenic circulant matrix functions in Hermitian Clifford analysis. Some results of  and  are generalized in our paper.
called a Dirac operator. It is precisely this Dirac operator which underlies the notion of monogenicity of a function, a notion which is the higher dimensional counterpart of holomorphy in the complex plane. A function f defined and differentiable in an open region Ω of and taking values in is called (left) monogenic in Ω if . As the Dirac operator factorizes the Laplacian, , monogenicity can be regarded as a refinement of harmonicity. We refer to this setting as the orthogonal case, since the fundamental group leaving the Dirac operator invariant is the special orthogonal group , which is doubly covered by the group of the Clifford algebra . For this reason, the Dirac operator is also called rotation invariant. When allowing for complex constants, the set of generators produces the complex Clifford algebra , being the complexification of the real Clifford algebra , i.e. . Any complex Clifford number may be written as , , an observation leading to the definition of the Hermitian conjugation , where the bar notation stands for the usual Clifford conjugation in , i.e. the main anti-involution for which , . This Hermitian conjugation also leads to a Hermitian inner product and its associated norm on is given by and .
involving the classical Cauchy-Riemann operators and their complex conjugates in the complex -planes, .
which will play the role of the differential form.
The ring of such matrix functions over is denoted by . In what follows, O will be denoting the matrix in with zero entries.
Definition 2.2 The matrix function is called (left) H-monogenic in Ω if and only if it satisfies in Ω the system .
where denotes the Clifford norm.
Definition 2.3 The matrix function () is called (left) H-2-monogenic in Ω if and only if it satisfies in Ω the system .
3 Some properties for H-monogenic circulant matrix functions
for each such that .
The result follows. □
The notions of continuity, differentiability, and integrability of have the usual component-wise meaning.
Theorem 3.2 (Liouville theorem)
If the matrix function is H-monogenic in and satisfies for all then must be a constant circulant matrix in .
where denotes the symmetric difference of and , is Lebesgue volume measure on , so that . The last expression above tends to 0 as . Thus and so is a constant circulant matrix. □
Theorem 3.3 (Maximum modulus theorem)
for all , then must be constant circulant matrix in Ω.
Since , then . So let ; this implies that . As is continuous in Ω, there exists an such that . This means that is relatively closed in Ω.
which yields for all , this means that and hence that is relatively open in Ω. As Ω is supposed to be connected it follows that .
we have () in Ω for all all . Thus , are constants in Ω. The result follows. □
4 Higher order Hermitian Borel-Pompeiu formula in Hermitian Clifford analysis
Integral representation formulas in Clifford analysis have been well developed in [3, 23–25], etc. These integral representation formulas are powerful tools. In this section, we get the explicit expression of the kernel function for and then get the explicit integral representation formulas for functions in Hermitian Clifford analysis. These explicit integral representation formulas play an important role in studying the further properties of the functions in Hermitian Clifford analysis.
where denotes the area of the unit sphere in .
Lemma 4.1 Let be as in (4.5). Then .
Proof The identity is obtained by straightforward calculation. □
Proof In view of Lemma 4.2, the identity is obtained by straightforward calculation. □
Theorem 4.4 (Higher order Hermitian Borel-Pompeiu formula)
then the left-hand side of the stated formula apparently equals zero.
where is defined as in (4.2).
Then the result follows. □
Proof Since is H-2-monogenic in Ω, in view of Theorem 4.4, the result follows. □
Theorem 4.7 (Mean value theorem for H-2-monogenic matrix function)
for each such that .
The proof is done. □
Corollary 4.8 If the matrix function is H-2-monogenic in and satisfies for all , then must be a constant circulant matrix in .
Proof The proof is similar to the method in Theorem 3.2. □
Suppose Ω is an open bounded non-empty subset of with a Liapunov boundary ∂ Ω, we usually write and . The notations and will be reserved for Clifford vectors associated to points , while their Hermitian counterparts are denoted and . By means of the matrix approach sketched above, the following Hermitian Plemelj-Sokhotski formula.
Theorem 4.10 Let be an open ball centered at , with radius R in , , in , , . Then in .
Therefore , and the result follows. □
where , then in .
Proof In view of the weak singularity of , combining Theorem 4.6 with Lemma 4.9, the theorem can be similarly proved similarly to Theorem 4.10. □
where be a constant circulant matrix.
5 Riemann boundary value problem for H-monogenic functions
where is any invertible constant circulant matrix, we denote by an invertible element for . Here is a given circulant matrix function in , .
Combining Theorem 3.2 with Theorem 4.10, there exists a constant circulant matrix such that .
On the other hand, it can be directly proved that (5.2) is the solution of (5.1), and the proof is done. □
Remark 5.2 If (5.1) is solved in , i.e. is required, then the problem has the unique solution (5.2) (taking ).
6 Riemann boundary value problem for H-2-monogenic function in Hermitian Clifford analysis
where , are invertible constant circulant matrices and , are given circulant matrix functions in , . We shall give the explicit expression of solutions for (6.1).
where is denoted as in (6.4).
Combining (6.7) with (6.10) we arrive at the proposed result.
On the other hand, it can be directly proved that (6.2) are the solution of (6.1) and the proof is done. □
Remark 6.2 If (6.1) is solved in , i.e. is required, then the problem has the unique solution (6.2) (taking ).
This paper is supported by National Natural Science Foundation of China (11271175), the AMEP, and DYSP of Linyi University. The authors would like to thank the referees for their valuable suggestion and comments.
- Muskhelishvilli NI: Singular Integral Equations. Nauka, Moscow; 1968.Google Scholar
- Lu J: Boundary Value Problems of Analytic Functions. World Scientific, Singapore; 1993.Google Scholar
- Brackx F, Delanghe R, Sommen F Research Notes in Mathematics 76. In Clifford Analysis. Pitman, London; 1982.Google Scholar
- Delanghe R, Sommen F, Souček V: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic, Dordrecht; 1992.MATHView ArticleGoogle Scholar
- Delanghe R: On the regular analytic functions with values in a Clifford algebra. Math. Ann. 1970, 185: 91-111. 10.1007/BF01359699MathSciNetView ArticleGoogle Scholar
- Delanghe R: On the singularities of functions with values in a Clifford algebra. Math. Ann. 1972, 196: 293-319. 10.1007/BF01428219MATHMathSciNetView ArticleGoogle Scholar
- Bernstein S: On the left linear Riemann problem in Clifford analysis. Bull. Belg. Math. Soc. Simon Stevin 1996, 3: 557-576.MATHMathSciNetGoogle Scholar
- Zhang Z, Du J: On certain Riemann boundary value problems and singular integral equations in Clifford analysis. Chin. Ann. Math., Ser. A 2000, 22: 421-426.MathSciNetGoogle Scholar
- Xu Z: On linear and nonlinear Riemann-Hilbert problems for regular functions with values in a Clifford algebra. Chin. Ann. Math., Ser. B 1990, 11(3):349-358.MATHGoogle Scholar
- Abreu Blaya R, Bory Reyes J: On the Riemann Hilbert type problems in Clifford analysis. Adv. Appl. Clifford Algebras 2001, 11(1):15-26. 10.1007/BF03042036MATHMathSciNetView ArticleGoogle Scholar
- Bory Reyes J, Abreu Blaya R: The quaternionic Riemann problem with natural geometric condition on the boundary. Complex Var. Theory Appl. 2000, 42: 135-149. 10.1080/17476930008815278MATHMathSciNetView ArticleGoogle Scholar
- Abreu Blaya R, Bory Reyes J: Boundary value problems for quaternionic monogenic functions on non-smooth surfaces. Adv. Appl. Clifford Algebras 1999, 9(1):1-22. 10.1007/BF03041934MATHMathSciNetView ArticleGoogle Scholar
- Gürlebeck K, Zhang Z: Some Riemann boundary value problems in Clifford analysis. Math. Methods Appl. Sci. 2010, 33: 287-302.MATHMathSciNetGoogle Scholar
- Zhang Z, Gürlebeck K: Some Riemann boundary value problems in Clifford analysis (I). Complex Var. Elliptic Equ. 2013, 58: 991-1003. 10.1080/17476933.2011.613119MATHMathSciNetView ArticleGoogle Scholar
- Brackx F, Bureš J, De Schepper H, Eelbode D, Sommen F, Soucěk V: Fundaments of Hermitean Clifford analysis part I: complex structure. Complex Anal. Oper. Theory 2007, 1(3):341-365. 10.1007/s11785-007-0010-5MATHMathSciNetView ArticleGoogle Scholar
- Brackx F, Bureš J, De Schepper H, Eelbode D, Sommen F, Soucěk V: Fundamentals of Hermitean Clifford analysis part II: splitting of h -monogenic equations. Complex Var. Elliptic Equ. 2007, 52: 1068-1078.View ArticleGoogle Scholar
- Brackx F, De Knock B, De Schepper H, Sommen F: On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis. Bull. Braz. Math. Soc. 2009, 40(3):395-461. 10.1007/s00574-009-0018-8MATHMathSciNetView ArticleGoogle Scholar
- Brackx F, De Knock B, De Schepper H: A matrix Hilbert transform in Hermitean Clifford analysis. J. Math. Anal. Appl. 2008, 344: 1068-1078. 10.1016/j.jmaa.2008.03.043MATHMathSciNetView ArticleGoogle Scholar
- Kytmanov AM: The Bochner-Martinelli Integral and Its Applications. Brikhäuser, Basel; 1995.MATHView ArticleGoogle Scholar
- Rocha-Chavez R, Shapiro M, Sommen F Res. Notes Math. 428. In Integral Theorem for Functions and Differential Forms in . Chapman & Hall/CRC, New York; 2002.Google Scholar
- Abreu Blaya R, Bory Reyes J, Brackx F, De Knock B, De Schepper H, Peña Peña D, Sommen F: Hermitean Cauchy integral decomposition of continuous functions on hypersurfaces. Bound. Value Probl. 2008., 2008: Article ID 425256Google Scholar
- Abreu Blaya R, Bory Reyes J, Brackx F, De Schepper H, Sommen F: Boundary value problems associated to a Hermitian Helmholtz equation. J. Math. Anal. Appl. 2012, 389: 1268-1279. 10.1016/j.jmaa.2012.01.006MATHMathSciNetView ArticleGoogle Scholar
- Gürlebeck K, Sprössig W: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester; 1997.MATHGoogle Scholar
- Gürlebeck K, Habetha K, Sprössig W: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser, Basel; 2008.MATHGoogle Scholar
- Kravchenko VV, Shapiro MV Pitman Research in Mathematics Series 351. In Integral Representations for Spatial Models of Mathematical Physics. Longman, Harlow; 1996.Google Scholar
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