-regular vector functions and their boundary value problems
© Yang and Li; licensee Springer 2012
Received: 22 February 2012
Accepted: 2 July 2012
Published: 19 July 2012
Let , where λ is a positive real constant. In this paper, by using the methods from quaternion calculus, we investigate the -regular vector functions, that is, the complex vector solutions of the equation , 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 -regular functions satisfy the modified Helmholtz equation, that is , . We give out a distribution solution of the inhomogeneous equation and study some properties of the solution. Moreover, we discuss some boundary value problems for -regular functions and solutions of equation .
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 and denote the real and complex quaternion space respectively. Their basis elements 1, i, j, k satisfy the following relations: , , .
Since the operator can not be factorized into the product of two differential operators of first order in , the quaternion function theory about modified Helmholtz equation was developed in complex quaternion space , namely the operator , and some related equations were directly investigated by . However, different from , is a Euclidean 8-space; and since there exists a set of zero divisors in , 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 . By using the quaternion technique, we obtain a systematic theory about the -regular vector functions, that is, the complex vector solutions of the equation , analogous to the quaternion regular function. Because the -regular vector functions are two-dimensional complex vector functions, this is more similar to the case of .
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 -regular vector functions and solutions of the equation , obtain general solutions and solvable conditions respectively in different cases.
1 Some notations and definitions
Henceforth we shall abbreviate to 1.
then will be called -regular vector function in Ω.
2 Pompeiu formula and Cauchy integral formula of -regular vector function
Then letting ε tend to zero in (4), we obtain the following Pompeiu formula corresponding to the operator D.
By applying Theorem 1, we can deduce the following Cauchy integral formula of the -regular vector function.
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
and call the Cauchy type integral with respect to the operator D. In the following, we shall simply call it the Cauchy type integral. In addition, is called the density function of .
Consequently, is -regular in the exterior of S. In addition, it is easy to see that converges to 0 as .
When , we provide that the integral on the right-hand side of (8) represents Cauchy’s principal value.
Letting in (10), it follows that (9) holds. □
By using Lemma 1, we can obtain the following Plemelj formula of the Cauchy type integral (8).
This is (11), and (12) is easily deduced from (11). □
The following result follows directly from Theorem 3.
and shall discuss some properties of the operator .
Theorem 5 Let , then exists almost everywhere on and belongs to , , where denotes any bounded domain in .
When , , it is easy to show that is a distribution on .
the desired result follows. □
then f is called a generalized derivative corresponding to the operator D of g. The derivative is denoted by . From Theorem 6 and the definition, .
This shows that if the complex vector function g is 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. □
- (a)For any ,(17)
where is a positive real constant depending only on p.
where . By hypothesis , we have . Let , , namely d, denote the diameter of a bounded domain Ω and the distance between ζ and respectively.
In fact, from , it is easy to see that the real function is a monotone decreasing function in and , so that .
- (b)Without loss of generality, we may take . We write
It is easy to see that .
The required estimate then follows by combining the resulting inequalities. □
5 Some boundary value problems for -regular functions
It is well known that the Dirichlet problem for analytic functions 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 , and a more general problem is the so-called Riemann-Hilbert problem with boundary condition . Analogously to this, the Dirichlet problem for -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 -regular vector functions.
Let Ω be a bounded domain with smooth boundary S in , . S satisfies the exterior sphere condition, that is, for every point , there exists a ball B satisfying . denotes the transversal domain of Ω on the plane , its boundary is a closed smooth curve and the projection of every point of Ω on the plane is in . We consider the following boundary value problems:
where φ is a given complex value function on S, , is a given complex value function on L, , . is a given real value function on L, , , . This problem is called problem H of the equation (25), and is called index of the problem H.
besides the above boundary conditions, where a is a real constant, then the problem is called problem D.
In particular, when in the equation (25), the above problems are namely the problem H and problem D for the -regular vector functions.
Proof Noting the compatible condition and that is an analytic function with respect to z, using the Pompeiu formula , it is not difficult to verify by direct calculation that expressed by (31) is the general solution of the system (30). □
As a special case of Theorem 6.13 in , we can derive the following result.
for the equation in Ω has a unique solution .
Similarly to harmonic function, we have the following result.
These Green functions are unique.
A simple approximation argument shows that this formula continues to hold for . □
Using these results, we can discuss the solvability of the problem H and the problem D for the -regular vector functions and the equation .
By means of the results about the Riemann-Hilbert boundary value problem for analytic function in the unit disk , we can derive the solvable conditions and the expression of solutions. □
Proof The result follows immediately from Theorem 9 and the results of the Dirichlet boundary value problem for analytic function in the unit disk. □
namely . Using Theorem 10, we obtain the following result about the problem H for the equation in Ω.
Theorem 10 Let , .
(a) If the index , the problem H for the equation in Ω has the solution , where the -regular vector function is expressed as (a) of Theorem 9 with , replacing , respectively.
(b) If the index , replacing by , the problem H for the equation in Ω is solvable if and only if the function satisfies the conditions (38). When the conditions (38) hold, the problem then has the solution , where the -regular vector function is expressed as (b) of Theorem 9 with , replacing , respectively.
In the same way, we can obtain the result about the problem D for the equation in Ω.
Corollary 3 Suppose that . The problem D for the equation in Ω has a unique solution , where the -regular vector function is expressed as Corollary 2 with , and replacing , and a respectively.
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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