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 .
MSC: 30G35, 35J05.
Keywords:quaternion calculus; -regular vector function; modified Helmholtz equation; Riemann-Hilbert type boundary value problem
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
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].
where is namely the 3-dimensional Helmholtz operator. A quaternion function theory associated with the operator was established which involved the Pompeiu formula corresponding to , the Cauchy integral formula for solutions of equation , the Plemelj formula of Cauchy type integral and the theory of operator . By using these results, the authors investigated the Dirichlet boundary problems for Helmholtz equation
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
It is easy to see that
Introduce the three-dimensional modified Helmholtz operator of first order, where , λ is a positive real constant. Define , then , where △ is the three-dimensional Laplace operator. The matrix forms of D, are
Suppose is a complex vector function defined in Ω and . Let be a fixed point in Ω and be an open ball whose center is , and the radius ε is so small that . Write . Using the formula (3) in and replacing U, V by , respectively, we have
It is easy to show that
Then letting ε tend to zero in (4), we obtain the following Pompeiu formula corresponding to the operator D.
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
Proof Let be an open ball with the radius ε and the center p, write the component of lying in the exterior of Ω as Γ. Then x is an interior point of the region inclosed by the closed surface . By the Pompeiu formula (5), we have
Similarly to the proof of Theorem 1, we can derive
By using Lemma 1, we can obtain the following Plemelj formula of the Cauchy type integral (8).
The above formula can be rewritten as
The Cauchy type integral (8) can be written in the following form:
By the Pompeiu formula, we obtain
Moreover, by using the Hölder inequality, it is easy to show that
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 inwhose boundary is a closed smooth surfaceS. is a complex vector function defined on the surfaceS, and, . Then the Cauchy type integral (8) whose density function isis a Cauchy integral if and only if,
Proof From the equality (2), we get
In the above equality replacing U, V by , respectively, by using the method analogous to the proof of Pompeiu formula (5), we can derive the Pompeiu formula corresponding to the operator , i.e., if , then
the desired result follows. □
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. □
It is easy to show that
we have by Hölder’s inequality
The last inequality is immediate from
Hence we obtain
The inequality (17) follows immediately from (20) and (21).
Here we use the estimates
We get by Hölder’s inequality
Using the inequality
By simple computation we have
By using a similar method, we can obtain
The required estimate then follows by combining the resulting inequalities. □
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.
has the general solution
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.
Lemma 3Ifφis continuous onS, then the Dirichlet problem with the boundary condition
Similarly to harmonic function, we have the following result.
where ν denotes the unit outward normal to the surface S, we obtain
Let p be a fixed point in Ω and be an open ball whose radius ε is so small that . Write . Replacing v by , using the formula (33) in and letting ε tend to zero, similarly to the proof of Theorem 1, we can derive
For a given p in Ω, find which satisfies the equation in Ω and the boundary condition on S. By virtue of Lemma 3, this is existential and unique. Write . When w satisfies the equation in Ω, from (33) we derive
Subtracting this from (34), we get
With the aid of the methods of conformal mapping and standardizing boundary condition from complex analysis (see [12,13]), we can map conformally into the unit disk on the plane , and transform in the condition (33) into . Hence without loss of generality, we shall directly suppose that is the unit disk on the plane and replace (27) by the following condition
When the conditions (38) hold, the solution then has the same expression as (1), except that
From Lemma 4, the function expressed in (36) is the unique solution of the Dirichlet problem with the boundary condition (26) for the equation in Ω, so that , satisfy the compatible condition of Lemma 2
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. □
Since the solution u of the equation can be expressed as , where Ψ is any -regular vector functions in Ω, if , , then , therefore the problem H of the equation in Ω can be transformed into the problem H of the -regular vector function in Ω with the following boundary conditions
(b) If the index, replacingby, the problem H for the equationin Ω is solvable if and only if the functionsatisfies the conditions (38). When the conditions (38) hold, the problem then has the solution, where the-regular vector functionis expressed as (b) of Theorem 9 with, replacing, respectively.
The authors declare that they have no competing interests.
PWY has presented the main purpose of the article. Both authors read and approved the final version of the manuscript.
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.
Caçåo, I, Constales, D, Kraußhar, RS: A unified approach for the treatment of some higher dimensional Dirac type equations on spheres. Proceedings of the 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, 07-09 July (2009)
Ha, TVN: Integral representation in quaternionic analysis related to the Helmholtz operator. Complex Var. Theory Appl.. 48, 1005–1021 (2003). Publisher Full Text
Huang, LD: The existence and uniqueness theorems of the linear and nonlinear Riemann-Hilbert problems for the generalized holomorphic vectors of the second kind. Acta Math. Sci.. 10(2), 185–199 (1990)
Kravchenko, VV, Sharpiro, MV: Helmholtz operator with a quaternionic wave number and associated function theory, II. Integral representations. Acta Appl. Math.. 32(3), 243–265 (1993). Publisher Full Text
Sharpiro, MV: On the properties of a class of singular integral equations connected with the three-dimensional Helmholtz equation. In: Abstracts of lectures at the 14 school on Operator Theory in Functional Spaces, Novgorod, U.S.S.R. (1989)
Xu, ZY: A function theory for the operator . Complex Var. Theory Appl.. 16(1), 27–42 (1991). Publisher Full Text
Yang, PW, Yang, S, Li, ML: An initial-boundary value problem for the Maxwell equations. J. Differ. Equ.. 249, 3003–3023 (2010). Publisher Full Text