Abstract
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; RiemannHilbert type boundary value problemResearch
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, KleinCordon equation, and Schroding equation. The corresponding results can be found in [13,511,1315].
In [2,3], the authors introduced a differential operator of first order , where λ is a positive real constant. It is easy to see that
where is namely the 3dimensional 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 8space; and since there exists a set of zero divisors in , a nonzero complex quaternion is not necessarily invertible. There exist many differences between the two theories.
In this article, we shall use the quasiquaternion 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 twodimensional 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 RiemannHilbert 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
Denote
It is easy to see that
Henceforth we shall abbreviate to 1.
Introduce the threedimensional modified Helmholtz operator of first order, where , λ is a positive real constant. Define , then , where △ is the threedimensional Laplace operator. The matrix forms of D, are
where
and then
Let Ω be a region in which identifies with . is a complex vector function defined in Ω. If and satisfies the equation
2 Pompeiu formula and Cauchy integral formula of regular vector function
Let Ω be a bounded domain in with piecewise smooth boundary S. , are twodimensional complex vector functions defined in Ω and . By the divergence theorem
where , denotes the unit outward normal to the surface S. From the equality (2), we have
It is easy to show that , , is a fundamental solution of the modified Helmholtz operator . When , . We write
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
Where
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.
Theorem 1Let Ω be a bounded domain inwith piecewise smooth boundaryS. Ifis a complex vector function defined in Ω and, then
By applying Theorem 1, we can deduce the following Cauchy integral formula of the regular vector function.
Theorem 2If a complex vector functionand satisfies the equationin Ω, then
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 be a complex vector function defined on a closed smooth surface S in , , . Denote
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 .
For arbitrary , there exists a neighborhood of p which does not intersect with S. In ,
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 righthand side of (8) represents Cauchy’s principal value.
Lemma 1Let Ω be a bounded domain inwith smooth boundaryS. If, in the sense of Cauchy’s principal value, then
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
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).
Theorem 3Write the domain Ω asand the complementary domain ofas. Whenptends tofromandrespectively, the limits of the Cauchy type integral (8) exist, which will be written asandrespectively, and
The above formula can be rewritten as
Proof Since , , therefore the improper integral is convergent. By Lemma 1, we have
The Cauchy type integral (8) can be written in the following form:
By the Pompeiu formula, we obtain
When , using the method similar to one complex variable [12,13], we can show that
Moreover, by using the Hölder inequality, it is easy to show that
Thus letting p tend to from and respectively in (13), we get
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,
4 Operator
Let be a complex vector function defined in a bounded domain Ω of . Denote
where
In this section, we shall get that if , then is a distribution solution of the inhomogeneous equation
and shall discuss some properties of the operator .
Similarly to the quaternion calculus [3,17], we can obtain the following results.
Theorem 4If, thenexists for allin the exterior of Ω. Besideisregular in the exterior of Ω and
Theorem 5Let, thenexists almost everywhere onand belongs to, , wheredenotes any bounded domain in.
For complex vector functions , given on Ω, define
When , , it is easy to show that is a distribution on .
holds.
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
holds.
In accordance with Theorem 5, . Thereby by the Fubini theorem,
the desired result follows. □
Let complex vector functions . If for any ,
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, .
Theorem 7If a complex vector functionand satisfies the equation, then
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 be a complex vector. The model of a is defined
It is easy to show that
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, the complex vector function, .
whereis a positive real constant depending only onp, Ω.
whereis a positive real constant depending only onp.
The inequalities (17) and (18) imply thatis a compact mapping from, into, , and
Proof (a) From the definition (14) of ,
Since
we have by Hölder’s inequality
where . By hypothesis , we have . Let , , namely d, denote the diameter of a bounded domain Ω and the distance between ζ and respectively.
The last inequality is immediate from
In fact, from , it is easy to see that the real function is a monotone decreasing function in and , so that .
Let
Hence we obtain
Noting
thus
i.e.,
The inequality (17) follows immediately from (20) and (21).
(b) Without loss of generality, we may take . We write
where
We have
Here we use the estimates
and
We get by Hölder’s inequality
and
Using the inequality
where , are positive real constants, and noting , we then obtain
By simple computation we have
and
By using a similar method, we can obtain
and
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 socalled RiemannHilbert 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 RiemannHilbert 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:
Find a continuous solution of the equation
in , satisfying the boundary conditions
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.
When , if u satisfies the condition
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.
Lemma 2Suppose complex value functions, . If, satisfy compatible condition
then the following overdetermined system with respect to
has the general solution
hereis any analytic function in,
Proof Noting the compatible condition and that is an analytic function with respect to z, using the Pompeiu formula [12], 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 [4], we can derive the following result.
Lemma 3Ifφis continuous onS, then the Dirichlet problem with the boundary condition
for the equationin Ω has a unique solution.
Similarly to harmonic function, we have the following result.
Lemma 4For the Dirichlet problem of the equationin Ω, Green functionsexist such that the solutions of the problem can be represented by, namely we have
These Green functionsare unique.
Proof Suppose functions . By Green’s second identity
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
Thus when and satisfies the equation ,
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
A simple approximation argument shows that this formula continues to hold for . □
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
Using these results, we can discuss the solvability of the problem H and the problem D for the regular vector functions and the equation .
Theorem 9 (1) If the index, the problem H for theregular vector functions in Ω is solvable. The problem has the general solutions, with
where
here, are arbitrary complex constants, satisfying
(2) If the index, the problem H for theregular vector functions in Ω is solvable if and only if the functionin the boundary conditions (27) satisfies the following conditions
When the conditions (38) hold, the solution then has the same expression as (1), except that
Proof If is a regular vector function, then satisfies the equation which is equivalent to
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
Consequently, can be given by the formula (37). Furthermore, expressed in (37) satisfies the boundary condition (35) if and only if the analytic function satisfies the following boundary condition
By means of the results about the RiemannHilbert 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 theregular vector functions in Ω has a unique solution, and the solution iswhichis given by (36) andexpressed as (37) where
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
where
namely . Using Theorem 10, we obtain the following result about the problem H for the equation in Ω.
(a) If the index, the problem H for the equationin Ω has the solution, where theregular vector functionis expressed as (a) of Theorem 9 with, replacing, respectively.
(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 theregular vector functionis 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 3Suppose that. The problem D for the equationin Ω has a unique solution, where theregular vector functionis expressed as Corollary 2 with, andreplacing, 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.
References

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, 0709 July (2009)

Cürlebeck, K: Hypercomplex factorization of the Helmholtz equation. Z. Anal. Anwend.. 5(2), 125–131 (1986)

Cürlebeck, K, Sprößig, W: Quaternionic Analysis and Elliptic Boundary Value Problem, Birkhäuser, Boston (1990)

Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (2001)

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 modified Helmholtz equations and their boundary value problems. Acta Math. Sci.. 8(2), 165–179 (1988)

Huang, LD: The existence and uniqueness theorems of the linear and nonlinear RiemannHilbert problems for the generalized holomorphic vectors of the second kind. Acta Math. Sci.. 10(2), 185–199 (1990)

Kravchenko, VV: Applied Quaternionic Analysis, Heldermann, Berlin (2003)

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

Kravchenko, VV, Sharpiro, MV: Hypercomplex factorization of the multidimensional Helmholtz operator and some it’s applications. Dokl. Sem. Inst. I. N. Vekua, Tbilisi. 5(1), 106–109 (1990)

Sharpiro, MV: On the properties of a class of singular integral equations connected with the threedimensional Helmholtz equation. In: Abstracts of lectures at the 14 school on Operator Theory in Functional Spaces, Novgorod, U.S.S.R. (1989)

Vekya, IN: Generalized Analytic Functions, Pergamon, Oxford (1962)

Wen, GC: Conformal Mappings and Boundary Value Problems, Amer. Math. Soc., Providence (1992) Translated from the Chinese by Kuniko Weltin

Xu, ZY: A function theory for the operator . Complex Var. Theory Appl.. 16(1), 27–42 (1991). Publisher Full Text

Xu, ZY: Helmholtz equations and boundary value problems. Partial Differential Equations with Complex Analysis, pp. 204–214. Longman Sci. Tech., Harlow (1992)

Yang, PW: Hölder continuity of and RiemannHilbert boundary value problem in quaternion calculus. Acta Math. Sin. Chinese Ser.. 46(5), 993–998 (2003)

Yang, PW: Quaternion Calculus and Partial Differential Equations, Science Press, Beijing (2009)

Yang, PW, Li, ML, Chen, Y: Initialboundary value problems of some hyperbolic systems of first order equations. Acta Math. Appl. Sin.. 31(1), 61–70 (2008)

Yang, PW, Yang, S, Li, ML: An initialboundary value problem for the Maxwell equations. J. Differ. Equ.. 249, 3003–3023 (2010). Publisher Full Text