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H λ -regular vector functions and their boundary value problems

Piwen Yang* and Dan Li

Author Affiliations

Department of Mathematics, Sichuan Normal University, Chengdu, 610066, P.R. China

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Boundary Value Problems 2012, 2012:75  doi:10.1186/1687-2770-2012-75

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/75


Received:22 February 2012
Accepted:2 July 2012
Published:19 July 2012

© 2012 Yang and Li; 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 cited.

Abstract

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M2">View MathML</a>, where λ is a positive real constant. In this paper, by using the methods from quaternion calculus, we investigate the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions, that is, the complex vector solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M4">View MathML</a> of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M5">View MathML</a>, and work out a systematic theory analogous to quaternionic regular functions. Differing from that, the component functions of quaternionic regular functions are harmonic, the component functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular functions satisfy the modified Helmholtz equation, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M8">View MathML</a>. We give out a distribution solution of the inhomogeneous equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a> and study some properties of the solution. Moreover, we discuss some boundary value problems for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular functions and solutions of equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a>.

MSC: 30G35, 35J05.

Keywords:
quaternion calculus; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function; modified Helmholtz equation; Riemann-Hilbert type boundary value problem

Research

It is well known that the theories of holomorphic functions of one complex variable and regular functions of quaternion as well as Clifford calculus are closely connected with the theory of harmonic functions, i.e., their component functions are all harmonic. But side by side with the Laplace operator is the Helmholtz operator and modified Helmholtz operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M13">View MathML</a>

which play an important role and are often met in application. In recent years, it has been considered that by replacing the harmonic function with the solutions of Helmholtz equation and modified Helmholtz equation, the theory of regular functions is naturally generalized in quaternion calculus and Clifford calculus. The theory has been well developed and has been applied to the research of some partial differential equations such as Helmholtz equation, Klein-Cordon equation, and Schroding equation. The corresponding results can be found in [1-3,5-11,13-15].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15">View MathML</a> denote the real and complex quaternion space respectively. Their basis elements 1, i, j, k satisfy the following relations: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M18">View MathML</a>.

In [2,3], the authors introduced a differential operator of first order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M19">View MathML</a>, where λ is a positive real constant. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M21">View MathML</a> is namely the 3-dimensional Helmholtz operator. A quaternion function theory associated with the operator was established which involved the Pompeiu formula corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M22">View MathML</a>, the Cauchy integral formula for solutions of equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M23">View MathML</a>, the Plemelj formula of Cauchy type integral and the theory of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M24">View MathML</a>. By using these results, the authors investigated the Dirichlet boundary problems for Helmholtz equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M25">View MathML</a>

Since the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M26">View MathML</a> can not be factorized into the product of two differential operators of first order in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M27">View MathML</a>, the quaternion function theory about modified Helmholtz equation was developed in complex quaternion space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15">View MathML</a>, namely the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M30">View MathML</a> and some related equations were directly investigated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15">View MathML</a>. However, different from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M15">View MathML</a> is a Euclidean 8-space; and since there exists a set of zero divisors in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M34">View MathML</a>, a non-zero complex quaternion is not necessarily invertible. There exist many differences between the two theories.

In this article, we shall use the quasi-quaternion space introduced in [18,19] and transform the modified Helmholtz operator into matric form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M35">View MathML</a>. By using the quaternion technique, we obtain a systematic theory about the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M36">View MathML</a>-regular vector functions, that is, the complex vector solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M37">View MathML</a> of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M5">View MathML</a>, analogous to the quaternion regular function. Because the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions are two-dimensional complex vector functions, this is more similar to the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M14">View MathML</a>.

For applications of partial differential equations, the research of boundary value problems is very important. How should appropriate boundary data be chosen for the Helmholtz equation or modified Helmholtz equation of first order? So far, there have been very few research works on the aspect. In this article, we introduce and investigate some Riemann-Hilbert type boundary value problems for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions and solutions of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a>, obtain general solutions and solvable conditions respectively in different cases.

1 Some notations and definitions

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M43">View MathML</a>

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M44">View MathML</a>

Henceforth we shall abbreviate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M45">View MathML</a> to 1.

Introduce the three-dimensional modified Helmholtz operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M46">View MathML</a> of first order, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M47">View MathML</a>, λ is a positive real constant. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M48">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M49">View MathML</a>, where △ is the three-dimensional Laplace operator. The matrix forms of D, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M50">View MathML</a> are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M51">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M52">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M53">View MathML</a>

Let Ω be a region in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a> which identifies with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M55">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M56">View MathML</a> is a complex vector function defined in Ω. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M57">View MathML</a> and satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M58">View MathML</a>

(1)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a> will be called <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function in Ω.

2 Pompeiu formula and Cauchy integral formula of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M36">View MathML</a>-regular vector function

Let Ω be a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M62">View MathML</a> with piecewise smooth boundary S. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M64">View MathML</a> are two-dimensional complex vector functions defined in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M65">View MathML</a>. By the divergence theorem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M66">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M68">View MathML</a> denotes the unit outward normal to the surface S. From the equality (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M69">View MathML</a>

(3)

It is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M71">View MathML</a>, is a fundamental solution of the modified Helmholtz operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M26">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M74">View MathML</a>. We write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M75">View MathML</a>

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M76">View MathML</a> is a complex vector function defined in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M77">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M78">View MathML</a> be a fixed point in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M79">View MathML</a> be an open ball whose center is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M80">View MathML</a>, and the radius ε is so small that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M81">View MathML</a>. Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M82">View MathML</a>. Using the formula (3) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M83">View MathML</a> and replacing U, V by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M85">View MathML</a> respectively, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M86">View MathML</a>

(4)

Where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M87">View MathML</a>

It is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M88">View MathML</a>

Then letting ε tend to zero in (4), we obtain the following Pompeiu formula corresponding to the operator D.

Theorem 1Let Ω be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>with piecewise smooth boundaryS. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M90">View MathML</a>is a complex vector function defined in Ω and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M77">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M92">View MathML</a>

(5)

By applying Theorem 1, we can deduce the following Cauchy integral formula of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function.

Theorem 2If a complex vector function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M94">View MathML</a>and satisfies the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M95">View MathML</a>in Ω, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M96">View MathML</a>

(6)

and if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M97">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M98">View MathML</a>

(7)

Proof The formula (6) follows directly from the Pompeiu formula (5) and the equality (7) can easily be derived from (3). □

3 Cauchy type integral and Plemelj formula

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a> be a complex vector function defined on a closed smooth surface S in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M103">View MathML</a>

(8)

and call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M104">View MathML</a> the Cauchy type integral with respect to the operator D. In the following, we shall simply call it the Cauchy type integral. In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a> is called the density function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a>.

For arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M107">View MathML</a>, there exists a neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M108">View MathML</a> of p which does not intersect with S. In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M108">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M110">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular in the exterior of S. In addition, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a> converges to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M114">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M115">View MathML</a>, we provide that the integral on the right-hand side of (8) represents Cauchy’s principal value.

Lemma 1Let Ω be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>with smooth boundaryS. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M117">View MathML</a>, in the sense of Cauchy’s principal value, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M118">View MathML</a>

(9)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M119">View MathML</a> be an open ball with the radius ε and the center p, write the component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M120">View MathML</a> lying in the exterior of Ω as Γ. Then x is an interior point of the region inclosed by the closed surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M121">View MathML</a>. By the Pompeiu formula (5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M122">View MathML</a>

(10)

Similarly to the proof of Theorem 1, we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M123">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M124">View MathML</a> in (10), it follows that (9) holds. □

By using Lemma 1, we can obtain the following Plemelj formula of the Cauchy type integral (8).

Theorem 3Write the domain Ω as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M125">View MathML</a>and the complementary domain of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M126">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M127">View MathML</a>. Whenptends to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M128">View MathML</a>from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M125">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M127">View MathML</a>respectively, the limits of the Cauchy type integral (8) exist, which will be written as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M131">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M132">View MathML</a>respectively, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M133">View MathML</a>

(11)

The above formula can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M134">View MathML</a>

(12)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102">View MathML</a>, therefore the improper integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M137">View MathML</a> is convergent. By Lemma 1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M138">View MathML</a>

The Cauchy type integral (8) can be written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M139">View MathML</a>

(13)

By the Pompeiu formula, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M140">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M141">View MathML</a>, using the method similar to one complex variable [12,13], we can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M142">View MathML</a>

Moreover, by using the Hölder inequality, it is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M143">View MathML</a>

Thus letting p tend to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M128">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M127">View MathML</a> respectively in (13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M147">View MathML</a>

This is (11), and (12) is easily deduced from (11). □

The following result follows directly from Theorem 3.

Corollary 1Let Ω be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>whose boundary is a closed smooth surfaceS. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a>is a complex vector function defined on the surfaceS, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102">View MathML</a>. Then the Cauchy type integral (8) whose density function is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a>is a Cauchy integral if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M154">View MathML</a>

4 Operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M156">View MathML</a> be a complex vector function defined in a bounded domain Ω of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M158">View MathML</a>

(14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M159">View MathML</a>

In this section, we shall get that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a> is a distribution solution of the inhomogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M162">View MathML</a>

(15)

and shall discuss some properties of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a>.

Similarly to the quaternion calculus [3,17], we can obtain the following results.

Theorem 4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a>exists for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M166">View MathML</a>in the exterior of Ω. Beside<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular in the exterior of Ω and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M169">View MathML</a>

Theorem 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a>exists almost everywhere on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M62">View MathML</a>and belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M174">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M175">View MathML</a>denotes any bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>.

For complex vector functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M178">View MathML</a> given on Ω, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M179">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M181">View MathML</a>, it is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M182">View MathML</a> is a distribution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M183">View MathML</a>.

Theorem 6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M160">View MathML</a>. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M185">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M186">View MathML</a>

holds.

Proof From the equality (2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M187">View MathML</a>

In the above equality replacing U, V by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M85">View MathML</a> respectively, by using the method analogous to the proof of Pompeiu formula (5), we can derive the Pompeiu formula corresponding to the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M50">View MathML</a>, i.e., if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M77">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M192">View MathML</a>

(16)

Thus for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M193">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M194">View MathML</a>

holds.

In accordance with Theorem 5, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M195">View MathML</a>. Thereby by the Fubini theorem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M196">View MathML</a>

the desired result follows. □

Let complex vector functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M197">View MathML</a>. If for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M193">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M199">View MathML</a>

then f is called a generalized derivative corresponding to the operator D of g. The derivative is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M200">View MathML</a>. From Theorem 6 and the definition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M201">View MathML</a>.

Theorem 7If a complex vector function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M202">View MathML</a>and satisfies the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M203">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M204">View MathML</a>

This shows that if the complex vector functiongis a classical solution of the equation (15), then it is also a distributional solution of the equation.

Proof It follows by the definition and the divergence theorem. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M205">View MathML</a> be a complex vector. The model of a is defined

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M206">View MathML</a>

It is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M207">View MathML</a>

By using similar methods to those used when proving the Hölder continuous of the operator T in quaternion calculus [16,17], we can prove the following theorem.

Theorem 8Let Ω be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>, the complex vector function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210">View MathML</a>.

(a) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M211">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M212">View MathML</a>

(17)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M213">View MathML</a>is a positive real constant depending only onp, Ω.

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M214">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M215">View MathML</a>

(18)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M216">View MathML</a>is a positive real constant depending only onp.

The inequalities (17) and (18) imply that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a>is a compact mapping from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210">View MathML</a>into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M221">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M222">View MathML</a>

(19)

Proof (a) From the definition (14) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M155">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M224">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M225">View MathML</a>

we have by Hölder’s inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M226">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M227">View MathML</a>. By hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M229">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M231">View MathML</a>, namely d, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M232">View MathML</a> denote the diameter of a bounded domain Ω and the distance between ζ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M126">View MathML</a> respectively.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M234">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M235">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M236">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M237">View MathML</a>

The last inequality is immediate from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M238">View MathML</a>

In fact, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M239">View MathML</a>, it is easy to see that the real function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M240">View MathML</a> is a monotone decreasing function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M242">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M243">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M244">View MathML</a>

Hence we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M245">View MathML</a>

(20)

Noting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M246">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M247">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M248">View MathML</a>

(21)

The inequality (17) follows immediately from (20) and (21).

(b) Without loss of generality, we may take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M249">View MathML</a>. We write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M250">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M251">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M252">View MathML</a>.

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M253">View MathML</a>

Here we use the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M254">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M255">View MathML</a>

We get by Hölder’s inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M256">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M257">View MathML</a>

Using the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M258">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M260">View MathML</a> are positive real constants, and noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M229">View MathML</a>, we then obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M262">View MathML</a>

(22)

By simple computation we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M263">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M264">View MathML</a>

By using a similar method, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M265">View MathML</a>

(23)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M266">View MathML</a>

(24)

The required estimate then follows by combining the resulting inequalities. □

5 Some boundary value problems for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular functions

It is well known that the Dirichlet problem for analytic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M268">View MathML</a> in a bounded domain of the complex plane, boundary value of which is a given complex value function, is overdetermined, thereby being unsolvable in general. In the theory of boundary value problems for analytic functions, the boundary condition is replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M269">View MathML</a>, and a more general problem is the so-called Riemann-Hilbert problem with boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M270">View MathML</a>. Analogously to this, the Dirichlet problem for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular functions, boundary value of which is a given complex value vector function, is also overdetermined, and we have therefore to consider new boundary conditions. In this section, we introduce and discuss some Riemann-Hilbert type boundary value problems for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions.

Let Ω be a bounded domain with smooth boundary S in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M274">View MathML</a>. S satisfies the exterior sphere condition, that is, for every point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M275">View MathML</a>, there exists a ball B satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M276">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277">View MathML</a> denotes the transversal domain of Ω on the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278">View MathML</a>, its boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M279">View MathML</a> is a closed smooth curve and the projection of every point of Ω on the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277">View MathML</a>. We consider the following boundary value problems:

Find a continuous solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M282">View MathML</a> of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M283">View MathML</a>

(25)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M126">View MathML</a>, satisfying the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M285">View MathML</a>

(26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M286">View MathML</a>

(27)

where φ is a given complex value function on S, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M288">View MathML</a> is a given complex value function on L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M290">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291">View MathML</a> is a given real value function on L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M102">View MathML</a>. This problem is called problem H of the equation (25), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M295">View MathML</a> is called index of the problem H.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M296">View MathML</a>, if u satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M297">View MathML</a>

(28)

besides the above boundary conditions, where a is a real constant, then the problem is called problem D.

In particular, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M298">View MathML</a> in the equation (25), the above problems are namely the problem H and problem D for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions.

Lemma 2Suppose complex value functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M300">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M301">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M300">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M303">View MathML</a>satisfy compatible condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M304">View MathML</a>

(29)

then the following overdetermined system with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M76">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M306">View MathML</a>

(30)

has the general solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M307">View MathML</a>

(31)

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M308">View MathML</a>is any analytic function in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M310">View MathML</a>

Proof Noting the compatible condition and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M311">View MathML</a> is an analytic function with respect to z, using the Pompeiu formula [12], it is not difficult to verify by direct calculation that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M76">View MathML</a> expressed by (31) is the general solution of the system (30). □

As a special case of Theorem 6.13 in [4], we can derive the following result.

Lemma 3Ifφis continuous onS, then the Dirichlet problem with the boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M313">View MathML</a>

for the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314">View MathML</a>in Ω has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M315">View MathML</a>.

Similarly to harmonic function, we have the following result.

Lemma 4For the Dirichlet problem of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314">View MathML</a>in Ω, Green functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M317">View MathML</a>exist such that the solutions of the problem can be represented by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M317">View MathML</a>, namely we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M319">View MathML</a>

(32)

These Green functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M317">View MathML</a>are unique.

Proof Suppose functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M321">View MathML</a>. By Green’s second identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M322">View MathML</a>

where ν denotes the unit outward normal to the surface S, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M323">View MathML</a>

(33)

Let p be a fixed point in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M324">View MathML</a> be an open ball whose radius ε is so small that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M325">View MathML</a>. Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M326">View MathML</a>. Replacing v by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M327">View MathML</a>, using the formula (33) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M83">View MathML</a> and letting ε tend to zero, similarly to the proof of Theorem 1, we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M329">View MathML</a>

Thus when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M330">View MathML</a> and satisfies the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M332">View MathML</a>

(34)

For a given p in Ω, find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M333">View MathML</a> which satisfies the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M334">View MathML</a> in Ω and the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M335">View MathML</a> on S. By virtue of Lemma 3, this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M333">View MathML</a> is existential and unique. Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M337">View MathML</a>. When w satisfies the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M314">View MathML</a> in Ω, from (33) we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M339">View MathML</a>

Subtracting this from (34), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M340">View MathML</a>

A simple approximation argument shows that this formula continues to hold for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M341">View MathML</a>. □

With the aid of the methods of conformal mapping and standardizing boundary condition from complex analysis (see [12,13]), we can map conformally <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277">View MathML</a> into the unit disk on the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278">View MathML</a>, and transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M288">View MathML</a> in the condition (33) into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M345">View MathML</a>. Hence without loss of generality, we shall directly suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M277">View MathML</a> is the unit disk <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M347">View MathML</a> on the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M278">View MathML</a> and replace (27) by the following condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M349">View MathML</a>

(35)

Using these results, we can discuss the solvability of the problem H and the problem D for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions and the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M351">View MathML</a>.

Theorem 9 (1) If the index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M352">View MathML</a>, the problem H for the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions in Ω is solvable. The problem has the general solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M56">View MathML</a>, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M355">View MathML</a>

(36)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M356">View MathML</a>

(37)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M357">View MathML</a>

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M359">View MathML</a>are arbitrary complex constants, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M360">View MathML</a>

(2) If the index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M361">View MathML</a>, the problem H for the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions in Ω is solvable if and only if the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291">View MathML</a>in the boundary conditions (27) satisfies the following conditions

(38)

When the conditions (38) hold, the solution then has the same expression as (1), except that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M365">View MathML</a>

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M4">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a> satisfies the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M95">View MathML</a> which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M370">View MathML</a>

(39)

From Lemma 4, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M371">View MathML</a> expressed in (36) is the unique solution of the Dirichlet problem with the boundary condition (26) for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M372">View MathML</a> in Ω, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M300">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M303">View MathML</a> satisfy the compatible condition of Lemma 2

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M375">View MathML</a>

(40)

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M376">View MathML</a> can be given by the formula (37). Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M376">View MathML</a> expressed in (37) satisfies the boundary condition (35) if and only if the analytic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M308">View MathML</a> satisfies the following boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M379">View MathML</a>

(41)

By means of the results about the Riemann-Hilbert boundary value problem for analytic function in the unit disk [13], we can derive the solvable conditions and the expression of solutions. □

Corollary 2The problem D for the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions in Ω has a unique solution, and the solution is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M381">View MathML</a>which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M371">View MathML</a>is given by (36) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M376">View MathML</a>expressed as (37) where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M384">View MathML</a>

Proof The result follows immediately from Theorem 9 and the results of the Dirichlet boundary value problem for analytic function in the unit disk. □

Since the solution u of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a> can be expressed as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M386">View MathML</a>, where Ψ is any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector functions in Ω, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M388">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M390">View MathML</a>, therefore the problem H of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a> in Ω can be transformed into the problem H of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M56">View MathML</a> in Ω with the following boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M394">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M395">View MathML</a>

namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M396">View MathML</a>. Using Theorem 10, we obtain the following result about the problem H for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a> in Ω.

Theorem 10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M388">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M210">View MathML</a>.

(a) If the index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M352">View MathML</a>, the problem H for the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a>in Ω has the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M402">View MathML</a>, where the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a>is expressed as (a) of Theorem 9 with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406">View MathML</a>replacing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291">View MathML</a>respectively.

(b) If the index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M361">View MathML</a>, replacing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a>by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405">View MathML</a>, the problem H for the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a>in Ω is solvable if and only if the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406">View MathML</a>satisfies the conditions (38). When the conditions (38) hold, the problem then has the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M414">View MathML</a>, where the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a>is expressed as (b) of Theorem 9 with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406">View MathML</a>replacing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291">View MathML</a>respectively.

In the same way, we can obtain the result about the problem D for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a> in Ω.

Corollary 3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M422">View MathML</a>. The problem D for the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M9">View MathML</a>in Ω has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M414">View MathML</a>, where the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M1">View MathML</a>-regular vector function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M59">View MathML</a>is expressed as Corollary 2 with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M406">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M429">View MathML</a>replacing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/75/mathml/M291">View MathML</a>andarespectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

PWY has presented the main purpose of the article. Both authors read and approved the final version of the manuscript.

Acknowledgement

This work is supported by National Natural Science Foundation of China (61173121), the Foundation of Doctor Education of China (20095134110001), and the Key Project Foundation of the Education Department of Sichuan Province of China (12ZA136). The authors would like to thank the referee for helpful comments and suggestions.

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