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Riemann boundary value problem for H-2-monogenic function in Hermitian Clifford analysis

Longfei Gu and Zunwei Fu*

Author Affiliations

Department of Mathematics, Linyi University, Linyi, Shandong, 276005, P.R. China

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Boundary Value Problems 2014, 2014:81  doi:10.1186/1687-2770-2014-81


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/81


Received:3 November 2013
Accepted:28 March 2014
Published:9 April 2014

© 2014 Gu and Fu; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> Riemann type problems for Hermitian monogenic and Hermitian-2-monogenic functions are solvable. Explicit representation formulas of the solutions are also given.

Keywords:
Hermitian Clifford analysis; Riemann type problems; Hermitian monogenic function

1 Introduction

The classical Riemann boundary value problem (BVP for short) theory in the complex plane has been systematically developed, see [1] and [2]. 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 [13] and [14], 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 [15] and [16], etc. In [17], based on the complex Clifford algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3">View MathML</a>, the Hermitian Cauchy integral formulas were constructed in the framework of circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions, and the intimate relationship with holomorphic function theory of several complex variables was considered. For details, we refer to [17-20]. In [18] and [21], 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 [22], 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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M5">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M6">View MathML</a>, then the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M7">View MathML</a> Riemann BVP for H-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions was solved. Also, we naturally consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> Riemann BVP for H-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions (i.e., null solutions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M11">View MathML</a>) and H-2-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions (i.e., null solutions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M13">View MathML</a>). Roughly speaking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> Riemann BVP for H-2-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions in Hermitian Clifford analysis. Applying the integral representation formulas of H-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions and H-2-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> Riemann BVP for H-2-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions in Hermitian Clifford analysis. Some results of [14] and [22] are generalized in our paper.

2 Preliminaries

In this section we recall some basic facts about Clifford algebras and Hermitian Clifford analysis which will be needed in the sequel. More details can also be found in [4] and [5].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M21">View MathML</a> be an 2n-dimensional (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M22">View MathML</a>) real linear space with basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24">View MathML</a> be the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M25">View MathML</a>-dimensional real linear space with basis

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M26">View MathML</a>

where N stands for the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M27">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M28">View MathML</a> denote the family of all order-preserving subsets of N in the above way. Now denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M29">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M31">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M32">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M33">View MathML</a>. The product on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M35">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M36">View MathML</a> is the cardinal number of the set A, the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M38">View MathML</a>, the symmetric difference set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M39">View MathML</a> is also order-preserving in the above way, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M40">View MathML</a> is the coefficient of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M32">View MathML</a>-component of the Clifford number λ. Also, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M42">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M43">View MathML</a>. It follows at once from the multiplication rule (2.1) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M30">View MathML</a> is the identity element written now as 1 and, in particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M45">View MathML</a>

(2.2)

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24">View MathML</a> is a real linear, associative, but non-commutative algebra and it is called the Clifford algebra over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M21">View MathML</a>. An involution is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M48">View MathML</a>

(2.3)

From (2.1) and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M49">View MathML</a>

(2.4)

The Euclidean space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a> is embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M51">View MathML</a> by identifying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M52">View MathML</a> with the Clifford vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M54">View MathML</a>

Note that the square of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53">View MathML</a> is scalar valued and equals the norm squared up to a minus sign: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M56">View MathML</a>. The dual of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53">View MathML</a> is the vector-valued first order differential operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M58">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a> and taking values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24">View MathML</a> is called (left) monogenic in Ω if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M61">View MathML</a>. As the Dirac operator factorizes the Laplacian, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M62">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M63">View MathML</a> invariant is the special orthogonal group <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M64">View MathML</a>, which is doubly covered by the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M65">View MathML</a> group of the Clifford algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24">View MathML</a>. For this reason, the Dirac operator is also called rotation invariant. When allowing for complex constants, the set of generators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M67">View MathML</a> produces the complex Clifford algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M68">View MathML</a>, being the complexification of the real Clifford algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M70">View MathML</a>. Any complex Clifford number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M71">View MathML</a> may be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M73">View MathML</a>, an observation leading to the definition of the Hermitian conjugation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M74">View MathML</a>, where the bar notation stands for the usual Clifford conjugation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M24">View MathML</a>, i.e. the main anti-involution for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M77">View MathML</a>. This Hermitian conjugation also leads to a Hermitian inner product and its associated norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M80">View MathML</a>.

The above will be the framework for the so-called Hermitian Clifford analysis, yet a refinement of orthogonal Clifford analysis. An elegant way for introducing this setting consists in considering a so-called complex structure, i.e. a specific <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M81">View MathML</a>-element J for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M82">View MathML</a> (see [15-17]). Here, J is chosen to act upon the generators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M83">View MathML</a> of the Clifford algebra as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M84">View MathML</a>

With J one may associate two projection operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M85">View MathML</a> which will produce the main protagonists of the Hermitian setting by acting upon the corresponding objects in the orthogonal framework. First of all, the so-called Witt basis elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M86">View MathML</a> for the complex Clifford algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3">View MathML</a> are obtained through the action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M88">View MathML</a> on the orthogonal basis elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M89">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M90">View MathML</a>

These Witt basis elements satisfy the Grassmann identities,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M91">View MathML</a>

and the duality identities,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M92">View MathML</a>

Next we identify a vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M93">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a> with the Clifford vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M95">View MathML</a> and we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M96">View MathML</a> the action of the complex structure J on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53">View MathML</a>, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M98">View MathML</a>

Note that the vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M96">View MathML</a> are orthogonal, the Clifford vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M96">View MathML</a> anti-commute. The actions of the projection operators on the Clifford vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M53">View MathML</a> then produce the Hermitian Clifford variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M104">View MathML</a> and its Hermitian conjugate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M105">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M106">View MathML</a>

which can be rewritten in terms of the Witt basis elements as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M107">View MathML</a>

where n complex variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M108">View MathML</a> have been introduced, with complex conjugates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M110">View MathML</a>. Finally, the Hermitian Dirac operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M112">View MathML</a> are derived from the orthogonal Dirac operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M63">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M114">View MathML</a>

where we have introduced

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M115">View MathML</a>

In view of the Witt basis, the Hermitian Dirac operators are expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M116">View MathML</a>

involving the classical Cauchy-Riemann operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M117">View MathML</a> and their complex conjugates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M118">View MathML</a> in the complex <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M119">View MathML</a>-planes, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M110">View MathML</a>.

Finally observe that the Hermitian vector variables and Dirac operators are isotropic, since the Witt basis elements are, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M121">View MathML</a>

whence the Laplacian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M122">View MathML</a> allows for the decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M123">View MathML</a>

while also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M124">View MathML</a>

For further use, we introduce the Hermitian oriented surface elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M126">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M127">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M128">View MathML</a> denotes the vector-valued oriented surface element and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M129">View MathML</a>. They are explicitly given by means of the following differential forms of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M130">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M131">View MathML</a>

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M132">View MathML</a>

and the corresponding oriented volume elements then read

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M133">View MathML</a>

We also consider the associated volume element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M134">View MathML</a>, defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M135">View MathML</a>

reflecting integration over the respective complex <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M119">View MathML</a>-planes, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M110">View MathML</a>. One has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M138">View MathML</a>

We still introduce the matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M139">View MathML</a>

which will play the role of the differential form.

Definition 2.1 A continuously differentiable function f on an open region Ω of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a> with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3">View MathML</a> is called a (left) h-monogenic function in Ω, iff it satisfies in Ω the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M142">View MathML</a>

or, equivalently, the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M143">View MathML</a>

The respective fundamental solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M145">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M146">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M147">View MathML</a> denotes the area of the unit sphere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M148">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>. The transition from Hermitian Clifford analysis to a circulant matrix approach is essentially based on the following observation. Introducing the particular circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrices

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M151">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M153">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M154">View MathML</a>, where δ is the diagonal matrix with the Dirac delta distribution δ on the diagonal, may be considered as a fundamental solution of the matrix operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M6">View MathML</a>. This has also led to a theory of H-monogenic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> circulant matrix functions, the framework for this theory being as follows. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M158">View MathML</a> be continuously differentiable functions defined in Ω and taking values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3">View MathML</a>, and consider the corresponding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M160">View MathML</a> circulant matrix function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M161">View MathML</a>

The ring of such matrix functions over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M3">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M163">View MathML</a>. In what follows, O will be denoting the matrix in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M164">View MathML</a> with zero entries.

Definition 2.2 The matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M165">View MathML</a> is called (left) H-monogenic in Ω if and only if it satisfies in Ω the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M166">View MathML</a>.

The notions of continuity, differentiability, and integrability of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M167">View MathML</a> have the usual component-wise meaning. In particular, we will need to defined in this way the classes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M169">View MathML</a>, of r times continuously differentiable functions over some suitable subset Ω of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M171">View MathML</a> stands for Hölder continuous circulant matrix functions over Ω. We introduce the non-negative function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M172">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M173">View MathML</a> denotes the Clifford norm.

Definition 2.3 The matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M174">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M175">View MathML</a>) is called (left) H-2-monogenic in Ω if and only if it satisfies in Ω the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M176">View MathML</a>.

In what follows we suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M177">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178">View MathML</a>.

3 Some properties for H-monogenic circulant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> matrix functions

Theorem 3.1If the matrix functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M180">View MathML</a>isH-monogenic in Ω then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M181">View MathML</a>

(3.1)

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183">View MathML</a>.

Proof Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183">View MathML</a>. Apply Hermitian Cauchy’s integral formula I (in [17]). On the ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M186">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M187">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M188">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M189">View MathML</a>, we apply the Hermitian Clifford-Stokes theorem (in [17]),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M190">View MathML</a>

The result follows. □

The notions of continuity, differentiability, and integrability of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M191">View MathML</a> have the usual component-wise meaning.

Theorem 3.2 (Liouville theorem)

If the matrix function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M165">View MathML</a>isH-monogenic in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>and satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M194">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M195">View MathML</a>then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M196">View MathML</a>must be a constant circulant matrix in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>.

Proof By Theorem 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M198">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M199">View MathML</a> denotes the symmetric difference of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M202">View MathML</a> is Lebesgue volume measure on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M204">View MathML</a>. The last expression above tends to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M205">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M206">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M207">View MathML</a> is a constant circulant matrix. □

Theorem 3.3 (Maximum modulus theorem)

Let the matrix functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M191">View MathML</a>be aH-monogenic in the open and connected set Ω. If there exists a point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M209">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M210">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212">View MathML</a>must be constant circulant matrix in Ω.

Proof Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M213">View MathML</a> and consider the subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M214">View MathML</a> of Ω given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M215">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M216">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M217">View MathML</a>. So let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M218">View MathML</a>; this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M219">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M220">View MathML</a> is continuous in Ω, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M221">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M222">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M214">View MathML</a> is relatively closed in Ω.

Now take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M226">View MathML</a>. By Theorem 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M227">View MathML</a>

(3.3)

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M228">View MathML</a>

we then have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M229">View MathML</a>

(3.4)

Applying Hölder’s inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M230">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M231">View MathML</a>

which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M232">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M233">View MathML</a>, this means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M234">View MathML</a> and hence that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M214">View MathML</a> is relatively open in Ω. As Ω is supposed to be connected it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M236">View MathML</a>.

Now if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M237">View MathML</a> then clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M238">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M240">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M241">View MathML</a> is H-monogenic in Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M242">View MathML</a>

(3.5)

then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M245">View MathML</a>. Hence we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M246">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M247">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211">View MathML</a>. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M211">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M250">View MathML</a>

(3.6)

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M251">View MathML</a>

(3.7)

and by (3.7), differentiating twice, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M252">View MathML</a>

(3.8)

Summing up over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M253">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M254">View MathML</a>

(3.9)

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M255">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M256">View MathML</a>) in Ω for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M253">View MathML</a> all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M258">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M260">View MathML</a> are constants in Ω. The result follows. □

Corollary 3.4Let Ω be a bounded open set in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>and suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M263">View MathML</a>are functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M264">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M265">View MathML</a>isH-monogenic in Ω. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M266">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M267">View MathML</a> 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.

In what follows, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M268">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M269">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M270">View MathML</a>

(4.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M271">View MathML</a>

(4.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M272">View MathML</a>

(4.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M147">View MathML</a> denotes the area of the unit sphere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>.

Lemma 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M275">View MathML</a>be as in (4.5). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M276">View MathML</a>.

Proof The identity is obtained by straightforward calculation. □

Lemma 4.2Denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M278">View MathML</a>, then

1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M279">View MathML</a>

(4.6)

2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M280">View MathML</a>

(4.7)

Lemma 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M281">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M282">View MathML</a>be as in (4.4) and (4.3). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M283">View MathML</a>

(4.8)

Proof In view of Lemma 4.2, the identity is obtained by straightforward calculation. □

Theorem 4.4 (Higher order Hermitian Borel-Pompeiu formula)

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M284">View MathML</a>is a 2n-dimensional compact differentiable and oriented manifold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M285">View MathML</a>smooth boundaryΓ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M157">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M158">View MathML</a>are functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M288">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M289">View MathML</a>is the matrix function. It then follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M290">View MathML</a>

(4.9)

Proof First let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M291">View MathML</a>. It then follows from the Stokes formula, which can be found in [17], that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M292">View MathML</a>

(4.10)

then the left-hand side of the stated formula apparently equals zero.

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M293">View MathML</a> and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M295">View MathML</a>. Invoking the previous case, we may then write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M296">View MathML</a>

(4.11)

Here we take the limits for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M297">View MathML</a>. In view of the weak singularity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M298">View MathML</a> the third term of (4.11) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M299">View MathML</a>

(4.12)

since the integrand only contains functions which are integrable on Γ. Furthermore we may write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M300">View MathML</a>

(4.13)

we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M301">View MathML</a>

(4.14)

Combining the Stokes formula in Hermitian Clifford analysis with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M302">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M303">View MathML</a>

(4.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M304">View MathML</a> is defined as in (4.2).

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M305">View MathML</a>

(4.16)

Then the result follows. □

Theorem 4.5If the matrix function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212">View MathML</a>isH-2-monogenic in Ω then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M307">View MathML</a>

(4.17)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212">View MathML</a> is H-2-monogenic in Ω, in view of Theorem 4.4, the result follows. □

Theorem 4.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309">View MathML</a>be an open ball centered at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M310">View MathML</a>with radiusRin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M312">View MathML</a>and the matrix function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212">View MathML</a>isH-2-monogenic in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309">View MathML</a>, then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M315">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M316">View MathML</a>

(4.18)

Theorem 4.7 (Mean value theorem for H-2-monogenic matrix function)

If the matrix function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M317">View MathML</a>isH-2-monogenic in Ω then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M318">View MathML</a>

(4.19)

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183">View MathML</a>.

Proof Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M183">View MathML</a>, by Theorem 4.5 we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M323">View MathML</a>

(4.20)

Combining with the Stokes formula in Hermitian Clifford analysis, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212">View MathML</a> is H-2-monogenic in Ω, Lemma 4.3 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M325">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M326">View MathML</a>

(4.21)

The proof is done. □

Corollary 4.8If the matrix function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M327">View MathML</a>isH-2-monogenic in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>and satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M194">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M195">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M196">View MathML</a>must be a constant circulant matrix in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>.

Proof The proof is similar to the method in Theorem 3.2. □

Suppose Ω is an open bounded non-empty subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a> with a Liapunov boundary Ω, we usually write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M334">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M335">View MathML</a>. The notations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M337">View MathML</a> will be reserved for Clifford vectors associated to points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M338">View MathML</a>, while their Hermitian counterparts are denoted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M339">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M340">View MathML</a>. By means of the matrix approach sketched above, the following Hermitian Plemelj-Sokhotski formula.

We shall introduce the following matrix operators:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M341">View MathML</a>

(4.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M342">View MathML</a>

(4.23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M343">View MathML</a>.

Lemma 4.9[18,21]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M344">View MathML</a>. Then the boundary values of the Hermitian Cauchy integral<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M345">View MathML</a>are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M346">View MathML</a>

Theorem 4.10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309">View MathML</a>be an open ball centered at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M310">View MathML</a>, with radiusRin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M351">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M352">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M353">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M355">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>.

Proof We only need to prove that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M357">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M358">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M359">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M360">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M357">View MathML</a>, taking constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M363">View MathML</a> is a ball with the center at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M364">View MathML</a> and radius δ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M365">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M366">View MathML</a> is a Liapunov boundary. Using the Hermitian Borel-Pompeiu formula, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M367">View MathML</a>

(4.24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M368">View MathML</a>

(4.25)

Using Lemma 4.9, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M369">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M370">View MathML</a>

(4.26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M371">View MathML</a>

(4.27)

Combining (4.26) with (4.27), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M372">View MathML</a>

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M358">View MathML</a>, and the result follows. □

Theorem 4.11Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M309">View MathML</a>be an open ball centered at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M310">View MathML</a>, with radiusRin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M377">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M379">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M380">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M212">View MathML</a>satisfies the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M382">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M383">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M384">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M385">View MathML</a>.

Proof In view of the weak singularity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M386">View MathML</a>, combining Theorem 4.6 with Lemma 4.9, the theorem can be similarly proved similarly to Theorem 4.10. □

Theorem 4.12Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M387">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M388">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M389">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M390">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M383">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M392">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M393">View MathML</a>), then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M394">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M395">View MathML</a>

(4.28)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M396">View MathML</a>be a constant circulant matrix.

5 Riemann boundary value problem for H-monogenic functions

An <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> Riemann boundary value problem for H-monogenic functions is denoted as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M398">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M399">View MathML</a> is any invertible constant circulant matrix, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M400">View MathML</a> an invertible element for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M399">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M402">View MathML</a> is a given circulant matrix function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354">View MathML</a>.

Theorem 5.1The Riemann boundary value problem (5.1) is solvable and the solution can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M405">View MathML</a>

(5.2)

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M406">View MathML</a>

(5.3)

Furthermore, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M407">View MathML</a>

(5.4)

and we then have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409">View MathML</a>. The transmission condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M410">View MathML</a>

can be changed into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M411">View MathML</a>

(5.5)

and if we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M412">View MathML</a>

(5.6)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M413">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M415">View MathML</a>. Using Lemma 4.9, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M416">View MathML</a>

(5.7)

From (5.5) and (5.7) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M417">View MathML</a>

(5.8)

Combining Theorem 3.2 with Theorem 4.10, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M4">View MathML</a> circulant matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M396">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M420">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M7">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M422">View MathML</a> is required, then the problem has the unique solution (5.2) (taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M423">View MathML</a>).

6 Riemann boundary value problem for H-2-monogenic function in Hermitian Clifford analysis

In this section, we shall consider the following <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M1">View MathML</a> Riemann boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M425">View MathML</a>

(6.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M427">View MathML</a> are invertible constant circulant matrices and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M428">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M429">View MathML</a> are given circulant matrix functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354">View MathML</a>. We shall give the explicit expression of solutions for (6.1).

Theorem 6.1The Riemann boundary value problem (6.1) is solvable and the solution is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M432">View MathML</a>

(6.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M433">View MathML</a>

(6.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M434">View MathML</a>

(6.4)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M207">View MathML</a> be the solution of (6.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409">View MathML</a>. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M437">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M438">View MathML</a>

(6.5)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M439">View MathML</a> and Theorem 5.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M440">View MathML</a>

(6.6)

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M441">View MathML</a>

(6.7)

Combining (6.6) with (6.7) we then get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M442">View MathML</a>

(6.8)

If we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M443">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M409">View MathML</a> and use

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M445">View MathML</a>

then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M446">View MathML</a>

(6.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M447">View MathML</a> is denoted as in (6.4).

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M448">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M354">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M450">View MathML</a>, using Theorem 5.1 we get the following representation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M451">View MathML</a>

(6.10)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M7">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M422">View MathML</a> is required, then the problem has the unique solution (6.2) (taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/81/mathml/M423">View MathML</a>).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Acknowledgements

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.

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