Abstract
In this paper, we study Riemann boundary-value problem for doubly-periodic bianalytic functions. By the decomposition of doubly-periodic polyanalytic functions, the problem is transformed into two equivalent and independent Riemann boundary-value problems of doubly-periodic 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 doubly-periodic bianalytic functions.
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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 boundary-value 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].
Riemann BVP of single-periodic polyanalytic functions has been investigated [10, 11]. Analogously, a Riemann BVP of rotation-invariant polyanalytic functions has been discussed in [20]. Actually, a single-periodic polyanalytic function is defined by a translation-invariant group
which is generated by two elements \(\{\tau_{1}, \tau_{-1}\}\) with \(\tau_{\pm1}(z)=z\pm\omega\). Generally speaking, the single-periodic polyanalytic function is translation-invariant under the group \(\mathcal{T}\), and the rotation-invariant polyanalytic function is invariant under a rotation group
In general, single-periodic polyanalytic function and rotation-invariant polyanalytic function are automorphic.
In 1935, Natanzon first made use of doubly-periodic bianalytic function to deal with a problem on stresses deriving from a stretched plate. In 1957, Erwe also studied other classes of doubly-periodic polyanalytic functions. In 1982, Pokazeev further considered a general form of doubly-periodic polyanalytic functions. A concise history of investigation of doubly-periodic polyanalytic function has been introduced in the literature [2]. Doubly-periodic polyanalytic function is also automorphic and is determined by another translation-invariant 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 doubly-periodic 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 doubly-periodic polyanalytic function has not been well-posed and systematically investigated. In this article, our main objective is to set up the theory of doubly-periodic bianalytic function. The way to solve this problem is the conversion method used in [16]. Riemann BVP for doubly-periodic polyanalytic functions will be presented in the forthcoming paper.
This article is organized as follows. In Sect. 2, we give a decomposition of doubly-periodic polyanalytic functions, which will be used to solve BVPs of doubly-periodic 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 doubly-periodic polyanalytic functions at the origin is defined. To pose the reasonable BVPs of doubly-periodic bianalytic functions, the definition of growth order at the origin is needed. In the classical monographs [24, 25], the growth order of doubly-periodic functions is not explicitly defined. In Sect. 4, Riemann BVP of doubly-periodic 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 doubly-periodic bianalytic functions, the problem is transformed into two independent Riemann-type BVPs of doubly-periodic analytic functions. Finally, the solution is explicitly expressed as an integral representation.
2 Doubly-periodic 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 Weierstrass’s ζ-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 ϕ is a doubly-periodic bianalytic function.
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 doubly-periodic open set with periods \(2\omega_{1}\), \(2\omega_{2}\). Similar to the definition of single-periodic polyanalytic function in [10], we give the following definition.
Definition 2.1
Suppose f to be a polyanalytic function [1] of order n on Ω, where Ω is a doubly-periodic open set with periods \(2\omega_{1}\), \(2\omega _{2}\). If
then we say that f is a doubly-periodic polyanalytic function of order n with periods \(2\omega_{1}\), \(2\omega_{2}\) on Ω, or simply doubly-periodic polyanalytic function. The collection of all the doubly-periodic polyanalytic functions on Ω is denoted by \(\mathit{DPH}_{n}(\Omega)\).
By Definition 2.1, \(\mathit{DPH}_{1}(\Omega)\) is just a set of all the doubly-periodic analytic functions on the doubly-periodic open set Ω. The function \(f\in\mathit{DPH}_{1}(\Omega)\) is called doubly-periodic bianalytic function. \(\mathit{DPH}_{n}(\Omega)\) is a subset of the collection of polyanalytic functions on Ω 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 doubly-periodic polyanalytic functions, used to solve Riemann BVP of doubly-periodic bianalytic functions in the sequel.
Theorem 2.1
Let Ω be a doubly-periodic 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 \(j=0,1,\ldots,n-1\), where ζ is defined by (2.1), and λ, δ are given by (2.4).
Proof
We only need to verify the relation ⊆ 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 \(\partial_{\bar{z}}f\in\mathit{DPH}_{n-1}(\Omega;2\omega_{1},2\omega _{2})\). And hence, by the inductive hypothesis, there exist \(g_{j}(z)\in\mathit{DPH}_{1}(\Omega;2\omega_{1},2\omega _{2})\), \(j=0,1,\ldots,n-2\), 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. □
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 j-component of f with respect to the base \(\{[\overline{z}-\lambda z-\delta\zeta(z)]^{j}: j=0,1,2,\ldots ,n-1\}\). 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 doubly-periodic 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 Ω to be a doubly-periodic 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 +∞ 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 Ω to be a doubly-periodic 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 ζ 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 \(c_{j}\in \mathbb{C}\), \(j=1,2,\ldots \) .
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 Laurent’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. □
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 ζ is defined by (2.1), and \(c_{j}\in\mathbb{C}\), \(j=0,1,2,\ldots \) .
Proof
By Lemma 3.1, at the deleted neighbor of the origin, one has
where ζ 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 \(c_{j}\in \mathbb{C}\), \(j=1,2,\ldots \) . 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 Liouville’s theorem, \(c_{0}(z)\) is a constant function. □
Lemma 3.2
Suppose Ω to be a doubly-periodic 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 j-component 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 ϕ 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. □
In what follows, we need the operators
where \(d_{j}\) is the j-coefficient of Laurent’s expansion of the function f defined in (3.2).
Theorem 3.1
Suppose Ω to be a doubly-periodic 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 δ 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. □
4 Riemann BVP for doubly-periodic analytic functions
In this section, we will give the solutions and conditions of solvability for Riemann BVP for doubly-periodic 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 doubly-periodic 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 Hölder-continuous 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 μ 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 −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).
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
$$\begin{aligned} \Phi(z) =& \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta (t-z)+\zeta(z)\bigr]\, \mathrm{d}t \\ &{}+X(z)p_{\kappa+m-1}(z),\quad p_{\kappa+m-1}\in\Pi _{\kappa+m-1}( \zeta), \end{aligned}$$(4.10)where \(\Pi_{\kappa+m-1}(\zeta)\) is defined by (4.9) and X is given by (4.6).
-
(2)
When \(\kappa+m\leq0\), if and only if
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0,\quad k=-1,0,1,2,\ldots,-\kappa-m-1, $$(4.11)\(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be written as
$$ \Phi(z) = \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta (t-z)-\zeta(t)\bigr]\, \mathrm{d}t+CX(z),\quad C\in\Pi_{\kappa+m}(\zeta). $$(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
- (5′):
-
\(X(z)\) has a pole of order \(-\kappa-1\) at the origin, precisely \(\operatorname{Ord}(X,0)=-\kappa-1\).
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
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0, \quad k=0,1,2,\ldots,-\kappa-m-1, $$(4.14)is fulfilled, \(\mathit{DR}_{m}\) problem (4.1) is solvable and its solution can be expressed as
$$\begin{aligned} \Phi(z) =& \frac{X(z)}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)} \bigl[\zeta (t-z)+\zeta(z)- \zeta(t-G_{0})-\zeta(G_{0}) \bigr]\,\mathrm{d}t \\ &{}+X(z) \bigl[p_{\kappa +m}(z)-p_{\kappa+m}(G_{0})\bigr], \end{aligned}$$(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
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\bigl[\zeta(t-G_{0})- \zeta (t)\bigr]\,\mathrm{d}t=0 $$(4.16)and
$$ \frac{1}{2\pi i} \int_{L_{0}}\frac{g(t)}{X^{+}(t)}\zeta^{(k)}(t)\,\mathrm {d}t=0,\quad k=0,1,2,\ldots,-\kappa-m-2, $$(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 doubly-periodic bianalytic functions
In this section, we consider the following Riemann BVP for doubly-periodic 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 Riemann-type boundary conditions and a growth condition
where the given boundary datum G and \(g_{j}\), \(j=1,2\), are Hölder-continuous 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 δ 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. □
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. □
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 \(p_{\kappa+m-1}\in\Pi_{\kappa+m-1}(\zeta)\), \(q_{\kappa+m-2}\in\Pi _{\kappa+m-2}(\zeta)\), \(c_{\kappa+m}\in\mathbb{C}\) 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 j-component 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. □
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 j-component 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. □
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. □
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. □
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. □
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 doubly-periodic polyanalytic functions and growth order of doubly-periodic polyanalytic functions at the origin. Riemann BVP of doubly-periodic bianalytic functions is presented. The problem is transformed into two independent Riemann-type BVPs of doubly-periodic 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 doubly-periodic functions can be applied to the theory of planar elasticity. Furthermore, the number and the shape of cracks in the so-called 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 doubly-periodic functions.
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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.
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Han, H., Liu, H. & Wang, Y. Riemann boundary-value problem for doubly-periodic bianalytic functions. Bound Value Probl 2018, 88 (2018). https://doi.org/10.1186/s13661-018-1005-z
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DOI: https://doi.org/10.1186/s13661-018-1005-z