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Riemann boundaryvalue problem for doublyperiodic bianalytic functions
 Huili Han^{1}Email authorView ORCID ID profile,
 Hua Liu^{2} and
 Yufeng Wang^{3}
 Received: 26 November 2017
 Accepted: 23 May 2018
 Published: 31 May 2018
Abstract
In this paper, we study Riemann boundaryvalue problem for doublyperiodic bianalytic functions. By the decomposition of doublyperiodic polyanalytic functions, the problem is transformed into two equivalent and independent Riemann boundaryvalue problems of doublyperiodic analytic functions, which has been discussed according to growth order of functions at the origin by Jianke Lu. Finally, we obtain the explicit expression of solutions and the conditions of solvability for the doublyperiodic bianalytic functions.
Keywords
 Doublyperiodic bianalytic function
 Riemann boundaryvalue problem
 Growth order
 Canonical function
 Freedom
MSC
 30G30
 45E05
1 Introduction
An extension of analytic function leads to polyanalytic function, which is usually defined as solutions of simple complex partial differential equation \(\partial_{\bar{z}}^{n}f=0\), where \(\partial_{\bar{z}}\) is the classical Cauchy–Riemann operator \(\partial_{\bar{z}}=1/2[\partial/\partial x+i(\partial/\partial y)]\). Polyanalytic function stemmed from planar elasticity problems and was first investigated by Kolossov in 1908. A good overview of polyanalytic function is included in Balk’s excellent monograph [1] or the literature [2]. Recently, various boundaryvalue problems (BVP) of polyanalytic functions and other functions determined by the general partial differential equations have been widely investigated by Begehr, Schmersau, Hile, Vanegas, Kumar, Jinyuan Du, Yufeng Wang, Ying Wang, Zhihua Du and others (see, for example, [1–21]). The general partial differential equations include the inhomogeneous polyanalytic equation [5], the higher order Poisson equation [6], and polyharmonic equations [7, 8].
For the compact Riemann surfaces of finite genus, the classical BVP of analytic functions was discussed in [22, 23]. However, for the very important doublyperiodic problem, it is essential to have an effective method of solution which has been systematically investigated by Jianke Lu [24]. Later, BVP of automorphic analytic functions has been first discussed by Gakhov and Chibrikova [25, 26].
Up to now, Riemann BVP for doublyperiodic polyanalytic function has not been wellposed and systematically investigated. In this article, our main objective is to set up the theory of doublyperiodic bianalytic function. The way to solve this problem is the conversion method used in [16]. Riemann BVP for doublyperiodic polyanalytic functions will be presented in the forthcoming paper.
This article is organized as follows. In Sect. 2, we give a decomposition of doublyperiodic polyanalytic functions, which will be used to solve BVPs of doublyperiodic bianalytic functions. It is worth mentioning that the decomposition obtained here is distinct from the classical decomposition described in [2]. In Sect. 3, the growth order of doublyperiodic polyanalytic functions at the origin is defined. To pose the reasonable BVPs of doublyperiodic bianalytic functions, the definition of growth order at the origin is needed. In the classical monographs [24, 25], the growth order of doublyperiodic functions is not explicitly defined. In Sect. 4, Riemann BVP of doublyperiodic bianalytic functions is presented. The solutions and conditions of solvability of this kind of problem are obtained by Jianke Lu [24]. By the decomposition of doublyperiodic bianalytic functions, the problem is transformed into two independent Riemanntype BVPs of doublyperiodic analytic functions. Finally, the solution is explicitly expressed as an integral representation.
2 Doublyperiodic polyanalytic functions
Without loss of generality, we always assume that \(\operatorname{Im}(\omega _{2}/\omega_{1})>0\) in the following. The parallelogram with vertices \(\omega_{1}+\omega_{2}\), \(\omega_{1}+\omega_{2}\), \(\omega_{1}\omega_{2}\), and \(\omega _{1}\omega_{2}\) is denoted by \(S_{0}\), which is usually called the fundamental cell. Obviously, the origin is the center of the fundamental cell \(S_{0}\).
If the open set Ω on the complex plane \(\mathbb{C}\) satisfies the condition \(z+2k\omega_{1}+2\ell \omega_{2} \in \Omega\) for \(\forall z\in\Omega\), \(\forall k,\ell\in\mathbb{Z}\), then Ω is called a doublyperiodic open set with periods \(2\omega_{1}\), \(2\omega_{2}\). Similar to the definition of singleperiodic polyanalytic function in [10], we give the following definition.
Definition 2.1
Finally, one arrives at the decomposition of doublyperiodic polyanalytic functions, used to solve Riemann BVP of doublyperiodic bianalytic functions in the sequel.
Theorem 2.1
Proof
3 Growth order of doublyperiodic bianalytic functions
First, the following definition is analogous to that in [10]. This is just the definition of growth order for the general polyanalytic function at the origin [15].
Definition 3.1
Now, one has the following result needed in the sequel.
Lemma 3.1
Proof
Corollary 3.1
Proof
Lemma 3.2
Suppose Ω to be a doublyperiodic open set with periods \(2\omega_{1}\), \(2\omega_{2}\), \(0\in\Omega\) and \(f\in\mathit {DPH}_{2}(\Omega;2\omega_{1},2\omega_{2})\). If \(\operatorname{Ord}(f,0)\leq m\), then \(\operatorname{Ord}(f_{1},0)\leq m+2\) and \(\operatorname{Ord}(f_{2},0)\leq m+1\), where \(f_{j}\) is jcomponent of f.
Proof
Finally, if \(m\leq1\), by Corollary 3.1 in [15], similar to the discussion above, it is not difficult to know that the conclusion remains true. □
Theorem 3.1
Proof
The sufficiency is obvious. Finally, if \(m\leq3\), analogously to the discussion above, the conclusion is also true. This completes the proof of this theorem. □
4 Riemann BVP for doublyperiodic analytic functions
In this section, we will give the solutions and conditions of solvability for Riemann BVP for doublyperiodic analytic functions, which was investigated in detail by Jianke Lu [24].
Let \(L_{0}\) be a closed smooth Jordan curve, oriented counterclockwise. The fundamental cell \(S_{0}\) is divided into two domains denoted by \(S_{0}^{+}\) and \(S_{0}^{}\), respectively. Without loss of generality, we always assume \(0\in S_{0}^{+}\). Let \(L_{k,\ell}=2k\omega_{1}+2\ell\omega_{2}+L_{0}\) for \(k,\ell\in \mathbb{Z}\), \(S^{+}=\bigcup_{k,\ell\in\mathbb{Z}} (2k\omega_{1}+2\ell\omega _{2}+S_{0}^{+} )\), \(S^{}=\mathbb{C}\setminus\overline{S^{+}} \), and we assume that \(L_{k,\ell}\) has the same orientation as \(L_{0}\) for every \(k,\ell\in\mathbb{Z}\). For the convenience, we set \(L=\bigcup_{k,\ell\in\mathbb{Z}} L_{k,\ell}\).
4.1 The case \(G_{*}=2k\omega_{1}+2\ell\omega_{2}\) for some \(k,\ell\in \mathbb{Z}\)
 (1)
\(X\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\);
 (2)
\(X^{+}(t)=G(t)X^{}(t)\), \(t\in L\);
 (3)
\(X^{\pm}(t)\in H(L)\);
 (4)
\(X(z)\neq0\), \(z\neq2(p\omega_{1}+q\omega_{2})\) with \(p,q\in \mathbb{Z}\) and \(X^{\pm}(t)\neq0\) for \(t\in L\);
 (5)
\(X(z)\) has a pole of order −κ at the origin, or say \(\operatorname{Ord}(X,0)=\kappa\).
If the function Y also satisfies five properties from (1) to (5) above, then there exists \(C\in\mathbb{C}\) such that \(Y(z)=CX(z)\), where X is given by (4.6).
Theorem 4.1
 (1)When \(\kappa+m>0\), \(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be expressed aswhere \(\Pi_{\kappa+m1}(\zeta)\) is defined by (4.9) and X is given by (4.6).$$\begin{aligned} \Phi(z) =& \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta (tz)+\zeta(z)\bigr]\, \mathrm{d}t \\ &{}+X(z)p_{\kappa+m1}(z),\quad p_{\kappa+m1}\in\Pi _{\kappa+m1}( \zeta), \end{aligned}$$(4.10)
 (2)When \(\kappa+m\leq0\), if and only if\(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be written as$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0,\quad k=1,0,1,2,\ldots,\kappam1, $$(4.11)We assume \(\zeta^{(1)}(t)=\zeta^{(0)}(t)=1\) in (4.11).$$ \Phi(z) = \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta (tz)\zeta(t)\bigr]\, \mathrm{d}t+CX(z),\quad C\in\Pi_{\kappa+m}(\zeta). $$(4.12)
In general, the freedom of solutions is \(\kappa+m\).
4.2 The case \(G_{*}\neq2k\omega_{1}+2\ell\omega_{2}\) for any \(k,\ell \in\mathbb{Z}\)
 (5′):

\(X(z)\) has a pole of order \(\kappa1\) at the origin, precisely \(\operatorname{Ord}(X,0)=\kappa1\).
The following result is also obtained by Jianke Lu. X used in the following theorem is defined by (4.6) with (4.13).
Theorem 4.2
 (1)When \(\kappa+m+1\geq0\), if and only if the condition of solvabilityis fulfilled, \(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be expressed as$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0, \quad k=0,1,2,\ldots,\kappam1, $$(4.14)with \(p_{\kappa+m}\in\Pi_{\kappa+m}(\zeta)\). We assume \(\zeta ^{(0)}(t)=1\) and \(\zeta^{(j)}(t)=0\) for \(j<0\) in (4.14).$$\begin{aligned} \Phi(z) =& \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)} \bigl[\zeta (tz)+\zeta(z) \zeta(tG_{0})\zeta(G_{0}) \bigr]\,\mathrm{d}t \\ &{}+X(z) \bigl[p_{\kappa +m}(z)p_{\kappa+m}(G_{0})\bigr], \end{aligned}$$(4.15)
 (2)When \(\kappa+m+1<0\), if and only ifand$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta(tG_{0}) \zeta (t)\bigr]\,\mathrm{d}t=0 $$(4.16)are satisfied, \(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be written as (4.15). We assume \(\zeta^{(0)}(t)=1\) in (4.17).$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0,\quad k=0,1,2,\ldots,\kappam2, $$(4.17)
In general, the freedom of solutions is \(\kappa+m\).
5 Riemann BVP for doublyperiodic bianalytic functions
Lemma 5.1
Proof
Conversely, if V is the solution of \(\mathit{DBR}_{m}\) problem (5.1), obviously boundary conditions in (5.3) and (5.4) are valid. By Theorem 3.1, \(\operatorname{Ord}(V,0) \leq m\) implies \(\operatorname {Ord}(V_{1},0) \leq m+2\) and \(\operatorname{Ord}(V_{2},0) \leq m+1\), and the validity of relation (5.6). This completes the proof. □
Analogously to the preceding section, we will discuss \(\mathit{DBR}_{m}\) problem (5.1) in two cases according to \(G_{*}=0\) (\(\operatorname{mod} 2\omega_{1},2\omega_{2}\)) or \(G_{*}\neq0\) (\(\operatorname{mod} 2\omega_{1},2\omega_{2}\)).
5.1 The case \(G_{*}=2k\omega_{1}+2\ell\omega_{2}\) for some \(k,\ell\in \mathbb{Z}\)
In this case, \(G_{*}=0\) (\(\operatorname{mod} 2\omega_{1},2\omega_{2}\)). And we will discuss \(\mathit {DBR}_{m}\) problem (5.1) in three subcases.
Theorem 5.1
Proof
Remark 5.1
Theorem 5.2
Proof
Theorem 5.3
Proof
5.2 The case \(G_{*}\neq2k\omega_{1}+2\ell\omega_{2}\) for any \(k,\ell \in\mathbb{Z}\)
In this case, there exists \(G_{0}\in S_{0}\) such that \(G_{*}=G_{0}\) (\(\operatorname{mod} 2\omega_{1},2\omega_{2}\)), and \(G_{0}\neq0\). We will investigate the problem in four subcases.
Theorem 5.4
Proof
Theorem 5.5
Proof
Theorem 5.6
Proof
Remark 5.2
To sum up the discussion above, the freedom of solutions of \(\mathit{DBR}_{m}\) problem (5.1) is \(2(\kappa+m)+1\).
6 Conclusion
In this article, we define doublyperiodic polyanalytic functions and growth order of doublyperiodic polyanalytic functions at the origin. Riemann BVP of doublyperiodic bianalytic functions is presented. The problem is transformed into two independent Riemanntype BVPs of doublyperiodic analytic functions. Finally, the solution is explicitly expressed as the integral representation.
Boundary value problems are always related with the theory of elasticity (see, for example, [24, 27, 28]). If the stresses and the elastic region are doubly periodic, BVPs of doublyperiodic functions can be applied to the theory of planar elasticity. Furthermore, the number and the shape of cracks in the socalled fundamental periodic parallelogram described in Sect. 2 could be arbitrary. In some sense, the results obtained here could contribute to the investigation of planar elasticity of doublyperiodic functions.
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable suggestions which helped to improve this work.
Availability of data and materials
Not applicable.
Funding
This research is supported by Major Innovation Projects for Building Firstclass Universities in China’s Western Region (ZKZD2017009).
Authors’ contributions
HH carried out theoretical calculation, participated in the design of the study, and drafted the manuscript. HL conceived of the study and participated in the design of the study. YW participated in its design and helped to draft the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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