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-regular vector functions and their boundary value problems
Boundary Value Problems volume 2012, Article number: 75 (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 .
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].
Let and denote the real and complex quaternion space respectively. Their basis elements 1, i, j, k satisfy the following relations: , , .
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
Henceforth we shall abbreviate to 1.
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
Let Ω be a region in which identifies with . is a complex vector function defined in Ω. If and satisfies the equation
then will be called -regular vector function in Ω.
2 Pompeiu formula and Cauchy integral formula of -regular vector function
Let Ω be a bounded domain in with piecewise smooth boundary S. , are two-dimensional 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
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 1 Let Ω be a bounded domain in with piecewise smooth boundary S. If is 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 2 If a complex vector function and satisfies the equation in Ω, then
and if , 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 right-hand side of (8) represents Cauchy’s principal value.
Lemma 1 Let Ω be a bounded domain in with smooth boundary S. 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 3 Write the domain Ω as and the complementary domain of as . When p tends to from and respectively, the limits of the Cauchy type integral (8) exist, which will be written as and respectively, 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
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 1 Let Ω be a bounded domain in whose boundary is a closed smooth surface S. is a complex vector function defined on the surface S, and , . Then the Cauchy type integral (8) whose density function is is a Cauchy integral if and only if ,
Let be a complex vector function defined in a bounded domain Ω of . Denote
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 .
Theorem 4 If , then exists for all in the exterior of Ω. Beside is -regular in the exterior of Ω and
Theorem 5 Let , then exists almost everywhere on and belongs to , , where denotes any bounded domain in .
For complex vector functions , given on Ω, define
When , , it is easy to show that is a distribution on .
Theorem 6 Let . Then for any ,
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
Thus for any ,
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 7 If a complex vector function and satisfies the equation , then
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. □
Let be a complex vector. The model of a is defined
It is easy to show that
Theorem 8 Let Ω be a bounded domain in , the complex vector function , .
For any ,(17)
where is a positive real constant depending only on p, Ω.
where is a positive real constant depending only on p.
The inequalities (17) and (18) imply that is a compact mapping from , into , , and
Proof (a) From the definition (14) of ,
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 .
Hence we obtain
The inequality (17) follows immediately from (20) and (21).
Without loss of generality, we may take . We write
It is easy to see that .
Here we use the estimates
We get by Hölder’s inequality
Using the inequality
where , are positive real constants, and noting , we then obtain
By simple computation we have
By using a similar method, we can obtain
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:
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 2 Suppose complex value functions , . If , satisfy compatible condition
then the following overdetermined system with respect to
has the general solution
here is any analytic function in ,
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 3 If φ is continuous on S, then the Dirichlet problem with the boundary condition
for the equation in Ω has a unique solution .
Similarly to harmonic function, we have the following result.
Lemma 4 For the Dirichlet problem of the equation in Ω, Green functions exist such that the solutions of the problem can be represented by , namely we have
These Green functions are 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 the -regular vector functions in Ω is solvable. The problem has the general solutions , with
here , are arbitrary complex constants, satisfying
If the index , the problem H for the -regular vector functions in Ω is solvable if and only if the function 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
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 Riemann-Hilbert boundary value problem for analytic function in the unit disk , we can derive the solvable conditions and the expression of solutions. □
Corollary 2 The problem D for the -regular vector functions in Ω has a unique solution, and the solution is which is given by (36) and expressed 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
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.
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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.
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.