 Research
 Open access
 Published:
Riemann boundaryvalue problem for doublyperiodic bianalytic functions
Boundary Value Problems volumeÂ 2018, ArticleÂ number:Â 88 (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.
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 Cauchyâ€“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 Balkâ€™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, [1â€“21]). The general partial differential equations include the inhomogeneous polyanalytic equation [5], the higher order Poisson equation [6], and polyharmonic equations [7, 8].
Riemann BVP of singleperiodic polyanalytic functions has been investigated [10, 11]. Analogously, a Riemann BVP of rotationinvariant polyanalytic functions has been discussed in [20]. Actually, a singleperiodic polyanalytic function is defined by a translationinvariant group
which is generated by two elements \(\{\tau_{1}, \tau_{1}\}\) with \(\tau_{\pm1}(z)=z\pm\omega\). Generally speaking, the singleperiodic polyanalytic function is translationinvariant under the group \(\mathcal{T}\), and the rotationinvariant polyanalytic function is invariant under a rotation group
In general, singleperiodic polyanalytic function and rotationinvariant polyanalytic function are automorphic.
In 1935, Natanzon first made use of doublyperiodic bianalytic function to deal with a problem on stresses deriving from a stretched plate. In 1957, Erwe also studied other classes of doublyperiodic polyanalytic functions. In 1982, Pokazeev further considered a general form of doublyperiodic polyanalytic functions. A concise history of investigation of doublyperiodic polyanalytic function has been introduced in the literature [2]. Doublyperiodic polyanalytic function is also automorphic and is determined by another translationinvariant group
Investigation of Riemann BVP for this kind of functions is a spontaneous thing.
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 \(S_{0}\).
The classical Weierstrassâ€™s Î¶function is defined by
with \(\Omega_{k\ell}=2k\omega_{1}+2\ell\omega_{2}\), and k, l are integers. Clearly,
where \(\eta_{j}=\zeta(\omega_{j})\) satisfies the relation
Let
with
By simple computation, one has
This implies that Ï• is a doublyperiodic bianalytic function.
If the open set Î© 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 Î© 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
Suppose f to be a polyanalytic function [1] of order n on Î©, where Î© is a doublyperiodic open set with periods \(2\omega_{1}\), \(2\omega _{2}\). If
then we say that f is a doublyperiodic polyanalytic function of order n with periods \(2\omega_{1}\), \(2\omega_{2}\) on Î©, or simply doublyperiodic polyanalytic function. The collection of all the doublyperiodic polyanalytic functions on Î© is denoted by \(\mathit{DPH}_{n}(\Omega)\).
By Definition 2.1, \(\mathit{DPH}_{1}(\Omega)\) is just a set of all the doublyperiodic analytic functions on the doublyperiodic open set Î©. The function \(f\in\mathit{DPH}_{1}(\Omega)\) is called doublyperiodic bianalytic function. \(\mathit{DPH}_{n}(\Omega)\) is a subset of the collection of polyanalytic functions on Î© denoted by \(H_{n}(\Omega)=\{f: \partial_{\bar{z}}^{n}f(z)=0, z\in\Omega\}\). Equation (2.5) is equivalent to
Now we introduce the subset of \(\mathit{DPH}_{n}(\Omega)\) as follows:
which is an object of investigation in the following.
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
Let Î© be a doublyperiodic open set with periods \(2\omega_{1}\), \(2\omega_{2}\) and \(0\notin\Omega\). Then
with \([\overline{z}\lambda z\delta\zeta(z)]^{j}\mathit{DPH}_{1}(\Omega ;2\omega_{1},2\omega_{2})=\{[\overline{z}\lambda z\delta\zeta (z)]^{j}f(z):f\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\}\) for \(j=0,1,\ldots,n1\), where Î¶ is defined by (2.1), and Î», Î´ are given by (2.4).
Proof
We only need to verify the relation âŠ† by induction. Obviously, if \(n=1\), the theorem is straightforward. Suppose that the relation
is valid. Next one has to verify
Let \(f\in\mathit{DPH}_{n}(\Omega;2\omega_{1},2\omega_{2})\). Then \(\partial_{\bar{z}}f\in\mathit{DPH}_{n1}(\Omega;2\omega_{1},2\omega _{2})\). And hence, by the inductive hypothesis, there exist \(g_{j}(z)\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega _{2})\), \(j=0,1,\ldots,n2\), such that
Setting
one has, by (2.9),
Therefore, \(h\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\). And (2.10) is rewritten as
which implies that (2.8) remains true.â€ƒâ–¡
Theorem 2.1 indicates that \(f\in\mathit{DPH}_{n}(\Omega;2\omega_{1},2\omega _{2})\) admits a unique decomposition
where \(f_{j}\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\) is called jcomponent of f with respect to the base \(\{[\overline{z}\lambda z\delta\zeta(z)]^{j}: j=0,1,2,\ldots ,n1\}\). Specially, \(g\in\mathit{DPH}_{2}(\Omega;2\omega_{1},2\omega_{2})\) has the unique expansion
with \(g_{j}\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\), \(j=1,2\).
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
Suppose Î© to be a doublyperiodic open set with periods \(2\omega_{1}\), \(2\omega_{2}\), \(0\in\Omega\) and \(f\in \mathit{DPH}_{n}(\Omega;2\omega_{1},2\omega_{2})\). If there exists an integer m such that
then we say that f possesses order m at the origin, denoted by \(\operatorname{Ord}(f,0)=m\). If
then we say that f has order +âˆž at the origin, denoted by \(\operatorname{Ord}(f,0)=+\infty\). We assume \(\operatorname{Ord}(f,0)=\infty\) if and only if \(f=0\).
Now, one has the following result needed in the sequel.
Lemma 3.1
Suppose Î© to be a doublyperiodic open set with periods \(2\omega_{1}\), \(2\omega_{2}\), \(0\in\Omega\) and \(f\in\mathit {DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\). At the deleted neighbor of the origin, one has
where Î¶ is defined by (2.1), the order of \(c_{0}(z)\in\mathit {DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\) is not more than 1 and \(c_{j}\in \mathbb{C}\), \(j=1,2,\ldots \)â€‰.
Proof
Since \(0\in\Omega\) and \(f\in\mathit{DPH}_{1}(\Omega ;2\omega_{1},2\omega_{2})\), at the deleted neighbor of the origin, one has Laurentâ€™s expansion
for sufficiently small \(r>0\). Let
where \(\zeta_{0}(0)=0\) and \(\zeta_{0}(z)\) is analytic at the origin. Thus one gets
which is equivalent to
Inserting (3.4) into (3.2), we get
with
The order of \(d(z)\) is obviously not more than 1. This completes the proof.â€ƒâ–¡
Corollary 3.1
Suppose that \(f\in\mathit{DPH}_{1}(\mathbb {C};2\omega_{1},2\omega_{2})\) possesses a uniquely possible singular point \(z=0\) in the fundamental cell \(S_{0}\). Then one has
where Î¶ is defined by (2.1), and \(c_{j}\in\mathbb{C}\), \(j=0,1,2,\ldots \)â€‰.
Proof
By Lemma 3.1, at the deleted neighbor of the origin, one has
where Î¶ is defined by (2.1), the order of \(c_{0}(z)\in\mathit {DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\) is not more than 1 and \(c_{j}\in \mathbb{C}\), \(j=1,2,\ldots \)â€‰. And \(c_{0}(z)\in\mathit{DPH}_{1}(\mathbb {C};2\omega_{1},2\omega_{2})\) implies that \(c_{0}(z)\) is an elliptic function. By Liouvilleâ€™s theorem, \(c_{0}(z)\) is a constant function.â€ƒâ–¡
Lemma 3.2
Suppose Î© 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
First, this theorem is verified under \(m>1\). By Theorem 2.1, there exist \(f_{1},f_{2}\in\mathit{DPH}_{1}(\Omega;2\omega _{1},2\omega_{2})\) such that \(f(z)=f_{1}(z)+\phi(z)f_{2}(z)\). This leads to
near the origin, where \(a_{j}\), \(b_{k}\) are constants.
Let \(\ell\in\mathbb{Z}^{+}\) and \(\ell\geq m\). We choose sufficiently small \(r>0\) such that \(D_{r}=\{z: z< r\}\subseteq\Omega\). By (3.6), one has
Now \(\operatorname{Ord}(f,0)\leq m\) implies that there exist \(M>0\), \(r_{1}>0\) such that
We assume \(0< r< r_{1}\), and one has the estimation
Combining (3.7) with (3.8), we get
for sufficiently small \(r>0\). Let \(r\rightarrow0^{+}\), and one has
This leads to
Therefore,
and
where Ï• is defined by (2.3).
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.â€ƒâ–¡
In what follows, we need the operators
where \(d_{j}\) is the jcoefficient of Laurentâ€™s expansion of the function f defined in (3.2).
Theorem 3.1
Suppose Î© to be a doublyperiodic open set with periods \(2\omega_{1}\), \(2\omega_{2}\), \(0\in\Omega\) and \(f\in\mathit {DPH}_{2}(\Omega;2\omega_{1},2\omega_{2})\). Then \(\operatorname{Ord}(f,0)\leq m\) if and only if
and
where \(\mathcal{L}_{j}\) is the operator defined by (3.9), and Î´ is given in (2.4).
Proof
First, we assume \(m>3\). And we prove the necessity. By Theorem 2.1, there exist \(f_{1},f_{2}\in\mathit{DPH}_{1}(\Omega;2\omega _{1},2\omega_{2})\) such that \(f(z)=f_{1}(z)+\phi(z)f_{2}(z)\). By Lemma 3.2, \(\operatorname{Ord}(f,0)\leq m\) implies \(\operatorname {Ord}(f_{1},0)\leq m+2\) and \(\operatorname{Ord}(f_{2},0)\leq m+1\). Also by Lemma 3.1,
Inserting (3.12) and (3.13) into the expression \(f(z)=f_{1}(z)+\phi (z)f_{2}(z)\), one easily gets
Thus, \(\operatorname{Ord}(f,0)\leq m\), where f given in (3.14), if and only if
where \(\mathit{P.P} (h,0)\) denotes the principal part of h. Equation (3.15) is equivalent to
Inserting this expression into (3.14), one has
where \(c_{0}(z),d_{0}(z)\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\), \(\operatorname{Ord}(c_{0},0)\leq1\), \(\operatorname{Ord}(d_{0},0)\leq1\), and \(d_{j}, c_{k}\in \mathbb{C}\). Therefore, (3.11) is true.
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.â€ƒâ–¡
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 \(L=\bigcup_{k,\ell\in\mathbb{Z}} L_{k,\ell}\).
Now, our problem is to find a sectionally doublyperiodic analytic function \(\Phi(z)\in\mathit {DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\) satisfying a boundary condition and a growth condition
where the given HÃ¶ldercontinuous functions G, g satisfy \(G(t+2\omega_{j})=G(t)\), \(g(t+2\omega_{j})=g(t)\), \(j=1,2\) and \(G(t)\neq0\), \(t\in L\). This problem is simply called \(\mathit{DR}_{m}\) problem.
Introduce the function
with
Then Î¼ is an elliptic function with periods \(2\omega_{1}\), \(2\omega_{2}\) and \(\operatorname{Ord}(\mu,0)=1\). Let
which is called the index, and
Without loss of generality, we assume \(G_{*}\notin L\) in the following. The solutions and conditions of solvability of \(\mathit{DR}_{m}\) problem (4.1) are presented in two cases.
4.1 The case \(G_{*}=2k\omega_{1}+2\ell\omega_{2}\) for some \(k,\ell\in \mathbb{Z}\)
In this case, \(G_{*}=0\) (\(\operatorname{mod} 2\omega_{1},2\omega_{2}\)). Let
where
and
In (4.8), \(\eta_{j}=\zeta(\omega_{j})\), \(j=1,2\). X defined by (4.6) is called the canonical function which possesses the following five properties:

(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 âˆ’Îº 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).
For convenience, we introduce the set of elliptic functions of order k with an exclusive singular point \(z=0\) as follows:
Now we state the results obtained by Jianke Lu.
Theorem 4.1
Under this case, two subcases arise:

(1)
When \(\kappa+m>0\), \(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be expressed as
$$\begin{aligned} \Phi(z) =& \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta (tz)+\zeta(z)\bigr]\, \mathrm{d}t \\ &{}+X(z)p_{\kappa+m1}(z),\quad p_{\kappa+m1}\in\Pi _{\kappa+m1}( \zeta), \end{aligned}$$(4.10)where \(\Pi_{\kappa+m1}(\zeta)\) is defined by (4.9) and X is given by (4.6).

(2)
When \(\kappa+m\leq0\), if and only if
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0,\quad k=1,0,1,2,\ldots,\kappam1, $$(4.11)\(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be written as
$$ \Phi(z) = \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta (tz)\zeta(t)\bigr]\, \mathrm{d}t+CX(z),\quad C\in\Pi_{\kappa+m}(\zeta). $$(4.12)We assume \(\zeta^{(1)}(t)=\zeta^{(0)}(t)=1\) in (4.11).
In general, the freedom of solutions is \(\kappa+m\).
4.2 The case \(G_{*}\neq2k\omega_{1}+2\ell\omega_{2}\) for any \(k,\ell \in\mathbb{Z}\)
In this case, there exists \(G_{0}\in S_{0}\) such that \(G_{*}=G_{0}\) (\(\operatorname{mod} 2\omega_{1},2\omega_{2}\)), and \(G_{0}\neq0\). Let X be defined by (4.6), where
At this time, X is also the canonical function which satisfies four properties from (1) to (4) in Sect. 4.1, and
 (5â€²):

\(X(z)\) 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
Under this case, two subcases arise:

(1)
When \(\kappa+m+1\geq0\), if and only if the condition of solvability
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0, \quad k=0,1,2,\ldots,\kappam1, $$(4.14)is fulfilled, \(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be expressed as
$$\begin{aligned} \Phi(z) =& \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)} \bigl[\zeta (tz)+\zeta(z) \zeta(tG_{0})\zeta(G_{0}) \bigr]\,\mathrm{d}t \\ &{}+X(z) \bigl[p_{\kappa +m}(z)p_{\kappa+m}(G_{0})\bigr], \end{aligned}$$(4.15)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).

(2)
When \(\kappa+m+1<0\), if and only if
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta(tG_{0}) \zeta (t)\bigr]\,\mathrm{d}t=0 $$(4.16)and
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0,\quad k=0,1,2,\ldots,\kappam2, $$(4.17)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).
In general, the freedom of solutions is \(\kappa+m\).
5 Riemann BVP for doublyperiodic bianalytic functions
In this section, we consider the following Riemann BVP for doublyperiodic bianalytic functions with the same factor: find a function \(V\in\mathit {DPH}_{2}(S^{+}\cup S^{};2\omega_{1},2\omega_{2})\) satisfying two Riemanntype boundary conditions and a growth condition
where the given boundary datum G and \(g_{j}\), \(j=1,2\), are HÃ¶ldercontinuous on every curve \(L_{k,\ell}\) and \(G(t)\neq0\), \(t\in L\). In addition, \(G(t+2\omega_{j})=G(t)\), \(g_{1}(t+2\omega_{j})=g(t)\), \(g_{2}(t+2\omega_{j})=g(t)\) for \(j=1,2\) and \(t\in L\). This problem is simply called \(\mathit {DBR}_{m}\) problem.
Since \(V\in\mathit{DPH}_{2}(S^{+}\cup S^{};2\omega_{1},2\omega_{2})\), by Theorem 2.1 or (2.12), one has the decomposition
where \(V_{j}\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega_{2})\) for \(j=1,2\).
Now, we will prove that \(\mathit{DBR}_{m}\) problem (5.1) can be transformed to two independent \(\mathit{DR}_{m+2}\) problem (5.3) and \(\mathit{DR}_{m+1}\) problem (5.4) as follows:
and
where
The solutions and conditions of solvability for those two problems have been presented in the preceding section.
Lemma 5.1
Let V, \(V_{1}\), \(V_{2}\) be given in (5.2). Then V is the solution of \(\mathit{DBR}_{m}\) problem (5.1) if and only if \(V_{1}\), \(V_{2}\) are respectively the solutions of \(\mathit{DR}_{m+2}\) problem (5.3) and \(\mathit{DR}_{m+1}\) problem (5.4) satisfying the relation
where \(\mathcal{L}_{j}\) is the operator defined by (3.9), and Î´ is given in (2.4).
Proof
Suppose that \(V_{1}\), \(V_{2}\) are respectively the solutions of \(\mathit{DR}_{m+2}\) problem (5.3) and \(\mathit{DR}_{m+1}\) problem (5.4) satisfying relation (5.6). By Theorem 3.1 and (5.6), \(\operatorname {Ord}(V_{1},0) \leq m+2\) and \(\operatorname{Ord}(V_{2},0) \leq m+1\) lead to
On the other hand, one has
and
Combining (5.7), (5.8) with (5.9), V is just a solution of \(\mathit{DBR}_{m}\) problem (5.1).
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.â€ƒâ–¡
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
If \(\kappa+m+1>0\), \(\mathit{DBR}_{m}\) problem (5.1) is solvable and its solution can be expressed as
with
where \(p_{\kappa+m+1}\in\Pi_{\kappa+m+1}(\zeta)\), \(q_{\kappa+m}\in\Pi _{\kappa+m}(\zeta)\) and
Proof
By Theorem 4.1, the solution of \(\mathit{DR}_{m+2}\) problem (5.3) can be expressed as
where \(\Pi_{\kappa+m+1}(\zeta)\) is defined by (4.9) and X is given by (4.6). By Theorem 4.1, the solution of \(\mathit{DR}_{m+1}\) problem (5.4) can be expressed as
According to Lemma 5.1, inserting (5.13) and (5.14) into (5.2), one gets
which leads to (5.10). At the same time, (5.6) is reduced to (5.11). This completes the proof.â€ƒâ–¡
Remark 5.1
Under this case, combining (5.10) with (5.11), the solution of \(\mathit{DBR}_{m}\) problem (5.1) can be rewritten as
where \(p_{\kappa+m1}\in\Pi_{\kappa+m1}(\zeta)\), \(q_{\kappa+m2}\in\Pi _{\kappa+m2}(\zeta)\), \(c_{\kappa+m}\in\mathbb{C}\) and \(c_{\kappa +m+1}\in\mathbb{C}\).
Theorem 5.2
If \(\kappa+m+1=0\), if and only if
\(\mathit{DBR}_{m}\) problem (5.1) is solvable and its solution can be represented as
with
where \(\{\widetilde{W}[g_{1},g_{2}]\}_{j}\) is jcomponent of \(\widetilde {W}[g_{1},g_{2}]\) defined by
Proof
By Theorem 4.1, the solution of \(\mathit{DR}_{m+2}\) problem (5.3) can be expressed as
By Theorem 4.1, if and only if the condition of solvability (5.15) is fulfilled, the solution of \(\mathit{DR}_{m+1}\) problem (5.4) can be expressed as
By Lemma 5.1, putting (5.19) and (5.20) into (5.2), one easily gets
with \(C_{1},C_{2}\in\mathbb{C}\). Also by Lemma 5.1, V given by (5.21) is the solution of \(\mathit{DBR}_{m}\) problem (5.1) if and only if \(C_{1}=\delta C_{2}\) and (5.17) are satisfied. And hence the proof of the theorem is completed.â€ƒâ–¡
Theorem 5.3
If \(\kappa+m+1<0\), if and only if
and
\(\mathit{DBR}_{m}\) problem (5.1) is solvable and its solution can be represented as
with
where \(\{\widehat{W}[g_{1},g_{2}]\}_{j}\) is jcomponent of \(\widehat {W}[g_{1},g_{2}]\) defined by
Proof
By Theorem 4.1, if and only if the conditions of solvability (5.22) are fulfilled, the solution of \(\mathit{DR}_{m+2}\) problem (5.3) can be expressed as
Analogously, by Theorem 4.1, if and only if the conditions of solvability (5.23) are satisfied, the solution of \(\mathit{DR}_{m+2}\) problem (5.4) can be written as
Therefore, by Lemma 5.1, if and only if the conditions of solvability (5.22) and (5.23) are fulfilled, the solution of \(\mathit{DBR}_{m}\) problem (5.1) can be expressed as
satisfying relation (5.25). This completes the proof.â€ƒâ–¡
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
If \(\kappa+m+2\geq0\), \(\mathit{DBR}_{m}\) problem (5.1) is solvable and its solution can be expressed as
with
and \(p_{\kappa+m+1}\in\Pi_{\kappa+m+1}(\zeta)\), \(q_{\kappa+m}\in\Pi _{\kappa+m}(\zeta)\), where X is the same as that in Sect. 4.2 and \(W[g_{1},g_{2}](z)\) is given in (5.12).
Proof
By Theorem 4.2, the solution of \(\mathit{DR}_{m+2}\) problem (5.3) can be expressed as
with \(p_{\kappa+m+2}\in\Pi_{\kappa+m+2}(\zeta)\), where \(\Pi_{\kappa+m+2}(\zeta)\) is defined by (4.9) and X is given by (4.6) with (4.13). By Theorem 4.2, the solution of \(\mathit{DR}_{m+1}\) problem (5.4) can be represented as
with \(p_{\kappa+m+1}\in\Pi_{\kappa+m+1}(\zeta)\). Observe
Inserting (5.31) and (5.32) into (5.2), one easily gets expression (5.29). By Lemma 5.1, (5.29) is the solution of \(\mathit{DBR}_{m}\) problem (5.1) if and only if (5.30) is satisfied.â€ƒâ–¡
Theorem 5.5
If \(\kappa+m+2=1\), if and only if
\(\mathit{DBR}_{m}\) problem (5.1) is solvable and its solution can be expressed as
with
where \(W[g_{1},g_{2}](z)\) is given by (5.12).
Proof
By Theorem 4.2, the solution of \(\mathit{DR}_{m+2}\) problem (5.3) can be expressed as
By Theorem 4.2, if and only if the condition of solvability (5.33) is fulfilled, the solution of \(\mathit{DR}_{m+1}\) problem (5.4) can be written as
By (5.33), expression (5.37) can be rewritten as
Thus, by Lemma 5.1, similar to the preceding discussion, the desired conclusion is obtained.â€ƒâ–¡
Theorem 5.6
If \(\kappa+m+2<1\), if and only if
and
\(\mathit{DBR}_{m}\) problem (5.1) is solvable and its solution can be expressed as
with
where \(W[g_{1},g_{2}](z)\) is given by (5.12).
Proof
By Theorem 4.2, if and only if conditions (5.38) and (5.40) are fulfilled, the solution of \(\mathit{DR}_{m+2}\) problem (5.3) can be expressed as
By Theorem 4.2, if and only if the conditions of solvability (5.39) and (5.41) are fulfilled, the solution of \(\mathit{DR}_{m+1}\) problem (5.4) can be written as
And hence, analogously to the preceding discussion, the desired conclusion is obtained.â€ƒâ–¡
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.
References
Balk, M.B.: Polyanalytic Functions. Akademie Verlag, Berlin (2001)
Gonchar, A.A., Havin, V.P., Nikolski, N.K. (eds.): Complex Analysis I: Entire and Meromorphic Functions, Polyanalytic Functions and Their Generalizations. Springer, Berlin (1997)
Begehr, H., Schmersau, D.: The Schwarz problem for polyanalytic functions. Z. Anal. Anwend. 24(2), 341â€“351 (2005)
Begehr, H., Hile, G.N.: A hierarchy of integral operators. Rocky Mt. J. Math. 27, 669â€“706 (1997)
Begehr, H., Kumar, A.: Boundary value problem for inhomogeneous polyanalytic equation I. Analysis 25, 55â€“71 (2005)
Begehr, H., Vanegas, C.J.: Iterated Neumann problem for higher order Poisson equation. Math. Nachr. 279, 38â€“57 (2006)
Begehr, H., Du, J.Y., Wang, Y.F.: A Dirichlet problem for polyharmonic functions. Ann. Mat. Pura Appl. 187(3), 435â€“457 (2008)
Du, Z.H., Qian, T., Wang, J.X.: \(L_{p}\) polyharmonic Dirichlet problems in regular domains II: the upper half plane. J. Differ. Equ. 252, 1789â€“1812 (2012)
Fatulaev, B.F.: The main Haseman type boundary value problem for metaanalytic function in the case of circular domains. Math. Model. Anal. 6(1), 68â€“76 (2001)
Wang, Y.F., Wang, Y.J.: On Riemann problems for singleperiodic polyanalytic functions. Math. Nachr. 287(16), 1886â€“1915 (2014)
Han, P.J., Wang, Y.F.: A note on Riemann problems for singleperiodic polyanalytic functions. Math. Nachr. 289(13), 1594â€“1605 (2016)
Wang, Y.F., Du, J.Y.: On Riemann boundary value problem of polyanalytic functions on the real axis. Acta Math. Sci. 24B(4), 663â€“671 (2004)
Du, J.Y., Wang, Y.F.: Riemann boundary value problems of polyanalytic functions and metaanalytic functions on the closed curves. Complex Var. Theory Appl. 50(7â€“11), 521â€“533 (2005)
Du, J.Y., Wang, Y.F.: On boundary value problems of polyanalytic function on the real axis. Complex Var. Theory Appl. 48(6), 527â€“542 (2003)
Wang, Y.F.: On modified Hilbert boundaryvalue problems of polyanalytic functions. Math. Methods Appl. Sci. 32, 1415â€“1427 (2009)
Wang, Y.F.: Schwarztype boundary value problems of polyanalytic equation on the upper half unit disk. Complex Var. Elliptic Equ. 57(9), 983â€“993 (2012)
Wang, Y.F.: On Hilberttype boundaryvalue problem of polyHardy class on the unit disc. Complex Var. Elliptic Equ. 58(4), 497â€“509 (2013)
Wang, Y.F., Wang, Y.J.: Schwarztype problem of nonhomogeneous Cauchyâ€“Riemann equation on a triangle. J. Math. Anal. Appl. 377, 557â€“570 (2011)
Wang, Y., Wang, Y.F.: Two boundaryvalue problems for the Cauchyâ€“Riemann equation in a sector. Complex Anal. Oper. Theory 6, 1121â€“1138 (2012)
Wang, Y.F., Han, P.J., Wang, Y.J.: On Riemann problem of automorphic polyanalytic functions connected with a rotation group. Complex Var. Elliptic Equ. 60(8), 1033â€“1057 (2015)
Begehr, H.: Complex Analytic Methods for Partial Differential Equation: An Introductory Text. World Scientific, Singapore (1994)
Kopperlman, K.: The Riemannâ€“Hilbert problem for finite Riemann surfaces. Commun. Pure Appl. Math. 12, 13â€“35 (1959)
Rodin, Y.L.: The Riemann Boundary Problem on Riemann Surfaces. Reidel, Dordrecht (1988)
Lu, J.K.: Boundary Value Problems for Analytic Functions. World Scientific, Singapore (1993)
Gakhov, F.D.: Boundary Value Problems. Pergamon, Oxford (1966)
Dang, P., Du, J.Y., Qian, T.: Boundary value problems for periodic analytic functions. Bound. Value Probl. 2015, 143 (2015)
Marin, M.: On weak solutions in elasticity of dipolar bodies with voids. J. Comput. Appl. Math. 82(1â€“2), 291â€“297 (1997)
Cai, H.T., Lu, J.K.: Mathematical Theory in Periodic Plane Elasticity. Gordon & Breach, Singapore (2000)
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 Chinaâ€™s Western Region (ZKZD2017009).
Author information
Authors and Affiliations
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Additional information
Publisherâ€™s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Han, H., Liu, H. & Wang, Y. Riemann boundaryvalue problem for doublyperiodic bianalytic functions. Bound Value Probl 2018, 88 (2018). https://doi.org/10.1186/s136610181005z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s136610181005z