One class of generalized boundary value problem for analytic functions
- Pingrun Li^{1}Email author
https://doi.org/10.1186/s13661-015-0301-0
© Li; licensee Springer. 2015
Received: 5 August 2014
Accepted: 4 February 2015
Published: 24 February 2015
Abstract
In this paper, a boundary value problem for analytic functions with two unknown functions on two parallel straight lines is studied, the general solutions in the different domains as well as the conditions of solvability are obtained in class \(\{1\}\), and the behaviors of solutions are discussed at \(z=\infty\) and in the different domains, respectively. Therefore, the classic Riemann boundary value problem is extended further.
Keywords
1 Introduction and preliminaries
Many mathematicians have studied the boundary value problems of analytic functions and formed a perfect theoretical system; see [1–7]. The boundary value problem of analytic functions on an infinite straight line has been studied in the literature, and there has been a brief description of boundary value problems of analytic function with an unknown function on several parallel lines. In this paper, we will put forward the boundary value problems of analytic functions with two unknown functions on two parallel lines and a general method different from the one in classical boundary value theory. Moreover, we will give and discuss the general solution and solvability conditions, which will generalize the classical theory of boundary value problems of analytic functions.
Let us describe the definitions of Plemelj formula and function class \(\{1\}\) on an infinite straight line.
Definition 1.1
- (1)
For any sufficiently large positive number M, \(\omega(x)\) satisfies \(\omega(x)\in H\) on \([-M,M]\) (see [8] for the definition of H).
- (2)
\(|\omega(x_{1})-\omega(x_{2})|\leq A|\frac{1}{x_{1}}-\frac{1}{x_{2}}|\), for any \(|x_{j}|>M\) (\(j=1,2\)) and some positive real number A.
Under condition (2), we say that \(\omega(x)\) satisfies the Hölder condition on \(N_{\infty}\) and denote it \(\omega(x)\in H(N_{\infty})\), where \(N_{\infty}=\{x: |x|>M\}\) is a neighborhood of ∞.
Definition 1.2
Assume that \(\omega(x)\) is continuous on \((-\infty, \infty)\) and \(\int_{-\infty}^{\infty}|\omega(x)|\, dx<+\infty\), then we say that \(\omega(x)\in L_{1}(-\infty,\infty)\).
Definition 1.3
If \(\omega(x)\) satisfies: (1) \(\omega(x)\in\hat {H}\), (2) \(\omega(x)\in L_{1}(-\infty,\infty)\), then we say that \(\omega (x)\) belongs to the function class \(\{1\}\).
Definition 1.4
Assume that \(\omega(x)\in\{1\}\), then the integrals \(\Omega^{+}(z)=\frac{1}{\sqrt{2\pi}}\int_{0}^{+\infty}\omega (t)e^{itz}\, dt\) and \(\Omega^{-}(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{0}\omega(t)e^{itz}\, dt\) are called the left and right one-sided Fourier integral, respectively.
Lemma 1.1
(see [1])
If \(\omega(z)\in H\) with respect to any finite part of some infinite domain D, and \(\omega(z)\) is analytic in any neighborhood of infinity, then \(\omega(z)\in\hat{H}\).
Lemma 1.2
(see [2])
If \(\omega(t)\) belongs to the class \(\{1\} \), then the left and right one-sided Fourier integrals defined in Definition 1.4 are analytic when \(\operatorname{Im} z>0\) and \(\operatorname{Im} z<0\), respectively.
Lemma 1.3
(see [8])
2 Problem presentation
Now, we put forward the boundary value problem of analytic functions on two parallel lines.
Without loss of generality, we assume that the two lines are parallel to the X-axis (otherwise, we can translate them into this case by a linear transformation), and denote them by \(L_{1}\), \(L_{2}\), where \(L_{j}\) can be expressed by \(\zeta=x+il_{j}\) (\(x\in(-\infty,\infty)\), \(l_{2}< l_{1}\) are real numbers) and take the direction from left to right as the positive direction. Let \(L=L_{1}+L_{2}\).
Actually, (2.1) is a boundary value problem on two parallel straight lines \(\operatorname{Im} z=l_{1}\), \(\operatorname{Im} z=l_{2}\) with ∞ as a pole. Here \(\Phi^{+}(\zeta )\) is the boundary value of analytic function \(\Phi^{+}(z)\) which is analytic in \(\{z: \operatorname{Im} z> l_{1}\}\) and belongs to the class \(\{1\}\) on \(L_{1}\), \(\Phi^{-}(\zeta)\) is the boundary value of analytic function \(\Phi ^{-}(z)\) which is analytic in \(\{z: \operatorname{Im} z< l_{2}\}\) and belongs to the class \(\{1\}\) on \(L_{2}\), and \(\Psi^{\pm}(\zeta)\) is the boundary value of analytic function \(\Psi(z)\) which is analytic in \(\{z: l_{2} <\operatorname{Im} z < l_{1}\} \) and belongs to the class \(\{1\}\) on \(L_{1}\), \(L_{2}\), respectively. The functions \(D_{1}(\zeta)\) and \(D_{2}(\zeta)\) belong to \(\hat{H}\) on \(L_{1}\), \(L_{2}\), respectively. The functions \(G_{1}(\zeta)\) and \(G_{2}(\zeta)\) belong to the class \(\{1\}\) on \(L_{1}\), \(L_{2}\), respectively. Hence, for the functions appearing in (2.1) the one-sided limits exist when \(x\rightarrow\infty\) on \(L_{1}\), \(L_{2}\).
It can be seen from (2.1) that the order of \(\Phi(z)\) is equal to that of \(\Psi(z)\) at infinity. Therefore, if the orders of \(\Phi(z)\) and \(\Psi(z)\) are m at infinity, then such a problem can be denoted as \(R_{m}\). Actually, problem \(R_{0}\) and problem \(R_{-1}\) are often discussed. On the problem \(R_{0}\), both \(\Phi(\infty)\) and \(\Psi(\infty)\) are supposed to be finite and nonzero. On \(R_{-1}\), both \(\Phi(\infty)\) and \(\Psi(\infty)\) are assumed to be zero. Such a problem R is called regular if \(D_{j}(\zeta)\) is not zero on L; otherwise, it is called irregular or of exception type.
Remark 2.1
Since the positive direction of \(L_{j}\) is the direction from left to right, when the observer moves from left to right on \(L_{j}\), the boundary values of left region of \(L_{j}\) is positive boundary value, i.e., the positive boundary value of \(\Phi(z)\) is the boundary value above \(L_{1}\), and the negative boundary value of \(\Phi(z)\) is ones below \(L_{2}\). The positive or negative boundary values of \(\Psi(z)\) can be defined in a similar way.
3 Resolution
(1) We firstly consider the solutions of \(M_{1}^{+}(z)\) and \(M_{1}^{-}(z)\), respectively in \(\operatorname{Im} z>l_{1}\) and \(\operatorname{Im} z< l_{1}\).
Case: \(k\geq0\).
Case: \(k<0\).
It follows from similar arguments as above that \(M_{1}^{+}(z)\) is analytic in \(\{z: \operatorname{Im} z>l_{1}\}\).
(2) We secondly consider the solutions of \(M_{2}^{+}(z)\) and \(M_{2}^{-}(z)\), respectively, in \(\operatorname{Im} z>l_{2}\) and \(\operatorname{Im} z< l_{2}\).
Case: \(k \geq0\).
Case: \(k<0\).
Hence we get the solution of the boundary value problem (2.1).
Theorem 3.1
The boundary value problem (3.1) with two unknown functions \(\Psi(z)\) and \(\Phi(z)\) on two parallel lines has a solution in \(\{z: l_{2}< \operatorname{Im} z<l_{1}\}\) and \(\{\operatorname{Im} z>l_{1}\}\cup\{\operatorname{Im} z< l_{2}\}\), respectively. Moreover, the general solution can be expressed by (3.33)-(3.36), where \(Y_{j}^{\pm }(z)\) (\(j=1,2\)) is defined by (3.3) and \(F_{j}(z)\) (\(j=1,2\)) are defined by (3.8) and (3.9). When \(\kappa>-1\), \(p_{k}(z)\) is a polynomial with κ order, and when \(\kappa\leq-1\), the necessary conditions for solvability still are (3.18). In all, the degree of freedom of the solution is \(\kappa+1\).
4 Further discussion on solution and solvability conditions
In this section, we say more about the solution (3.33)-(3.36) of (2.1) and the solvability conditions.
(1) The case that the solution lies in \(\operatorname{Im} z>l_{1}\) and \(\operatorname{Im} z< l_{2}\).
As in (3.27) and (3.28), \(\Phi^{+}(z)\) is analytic in \(\{z: \operatorname{Im} z> l_{1}\}\) and \(\Phi^{-}(z)\) is analytic in \(\{z: \operatorname{Im} z< l_{2}\}\). No matter how we choose κ, the boundary value problem (2.1) is solvable and its solution can be expressed by (3.35)-(3.36).
(2) The case that the solution lies in \(l_{2}<\operatorname{Im} z<l_{1}\).
Then \(\Psi(z)\) is analytic in \(\{z: l_{2}<\operatorname{Im} z<l_{1}\}\) and has a bounded solution. When \(\kappa>0\), \(z_{0}\) is a κ-order pole of \(Y_{1}^{-}(z)Y_{2}^{+}(z)\), and therefore \(Y_{1}^{-}(z)Y_{2}^{+}(z)\frac{p_{\kappa }(z)}{(z-z_{0})^{\kappa}}\) is analytic in \(\{z: l_{2}<\operatorname{Im} z< l_{1}\}\). Hence, \(\Psi(z)\) is analytic in \(\{z: l_{2}<\operatorname{Im} z<l_{1}\}\) and \(\Psi(z)\) is a constant while \(z=\infty\).
For \(z\in\{z: l_{2}< \operatorname{Im} z< l_{1}\}\), \(\Psi(z)\) can be defined by (3.33), (3.34) if \(D_{1}(z)\) and \(D_{2}^{-1}(z)\) are not zero. Otherwise, if \(z_{1}^{*},z_{2}^{*},\ldots,z_{n}^{*}\) are common zero-points of \(D_{1}(z)\) and \(D_{2}^{-1}(z)\) with the orders \(s_{1},s_{2},\ldots,s_{n}\), respectively, then \(\Psi^{(j)}(z_{q}^{*})=0\) (\(1\leq q\leq n\), \(1\leq j\leq s_{q}\)). Let \(s=\sum_{q=1}^{n} s_{q}\). Then the following solvability conditions must be augmented.
(3) The case of solutions at \(z=\infty\).
For \(\kappa<0\), the condition (4.2) should hold and \(j=1,2,\ldots ,|\kappa|\).
If \(z=\infty\) is a special node, i.e., \(\mu_{\infty}=0\), one can translate it into the case that \(\mu_{\infty}\leq\frac{1}{2}\) as a common node. For the rest, similar arguments can be used [9].
As for the boundary value problem with n unknown functions on n (\(n>2\)) parallel lines, there is no essential difference for the solving method with the case \(n=2\). We will not elaborate.
5 Example
Declarations
Acknowledgements
The author expresses sincere thanks to the referee(s) for the careful and details reading of the manuscript and very helpful suggestions, which improved the manuscript substantially.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Lu, JK: Boundary Value Problems for Analytic Functions. World Scientific, Singapore (2004) Google Scholar
- Muskhelishvilli, NI: Singular Integral Equations. Nauka, Moscow (2002) Google Scholar
- Lu, JK: On methods of solution for some kinds of singular integral equations with convolution. Chin. Ann. Math., Ser. B 8(1), 97-108 (1987) MATHMathSciNetGoogle Scholar
- Du, JY: On quadrature formulae for singular integrals of arbitrary order. Acta Math. Sci., Ser. B 24(1), 9-27 (2004) MATHMathSciNetGoogle Scholar
- Abreu-Blaya, R, Bory-Reyes, J, Brackx, F, De Schepper, H, Sommen, F: Boundary value problems for the quaternionic Hermitian in \(\mathbb{R}^{4n}\) analysis. Bound. Value Probl. (2012). doi:10.1186/1687-2770-2012-74 MathSciNetMATHGoogle Scholar
- Lin, CC, Lin, YC: Boundary values of harmonic functions in spaces of Triebel-Lizorkin type. Integral Equ. Oper. Theory 79, 23-48 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Lu, JK: Some classes boundary value problems and singular integral equations with a transformation. Adv. Math. 23(5), 424-431 (1994) MathSciNetGoogle Scholar
- Li, PR: On the method of solving two kinds of convolution singular integral equations with reflection. Ann. Differ. Equ. 29(2), 159-166 (2013) MATHMathSciNetGoogle Scholar
- Li, PR: The integral equations containing both cosecant and convolution kernel with periodicity. J. Syst. Sci. Math. Sci. 30(8), 1148-1155 (2010) MATHGoogle Scholar