Schwarz boundary value problem for the Cauchy-Riemann equation in a rectangle

Ying Wang; Xuefang Zhao

doi:10.1186/s13661-016-0520-z

Research
Open Access

Schwarz boundary value problem for the Cauchy-Riemann equation in a rectangle

Ying Wang¹Email author View ORCID ID profile and
Xuefang Zhao²

Boundary Value Problems20162016:7

https://doi.org/10.1186/s13661-016-0520-z

Received: 5 October 2015
Accepted: 4 January 2016
Published: 12 January 2016

Abstract

In this paper, Schwarz problem for the inhomogeneous Cauchy-Riemann equation in a rectangle is investigated explicitly. By the parqueting reflection principle and the Cauchy-Pompeiu formula, a modified Schwarz-Poisson representation formula in a rectangle is constructed. In particular, the boundary behaviors for the related Schwarz-type and Pompeiu-type operators at the corner points are discussed in detail.

Keywords

Schwarz-Poisson formula
Cauchy-Riemann equation
Schwarz problem

MSC

30E25
35C15
35F45

1 Introduction

There are many investigations of boundary value problems for its theoretical significance and extensive applications in mathematical physics, such as elasticity theory, potential theory, medical imaging. Numerous results have been achieved for boundary value problems in different particular domains; see [1–10]. The basic boundary value problems are the Schwarz, Dirichlet, Robin and Neumann problems. In general, the well-known conformal invariance of the reflection across circles and lines can be used to find a Schwarz operator for simple domains like circles, half circles, rings, half planes, etc. But the explicit Schwarz-Poisson formula in a rectangle cannot be simply obtained from the classical Schwarz-Poisson formula on the unit disc or the half-plane by conformal mapping. In [11, 12], the authors introduced a method of plane parqueting, which can be used to determine a Schwarz operator, the harmonic Green and Neumann functions, and requires that reflecting the domain at all of these circular or straight curves produces a parqueting of whole complex plane with possible exception of singular points. In [7, 10, 13], the authors discussed Schwarz and Dirichlet problems in an isosceles orthogonal triangle and an equilateral triangle via different series. Also the Green and Neumann functions for a strip and a rectangle were established in detail in [6]. In the present paper, we construct a modified Schwarz-Poisson representation formula by the parqueting reflection principle and Cauchy-Pompeiu formula in a rectangle, discussing the related Schwarz problem explicitly.

Let Ω be a domain in the complex plane $\mathbb{C}$ defined by

$$ \Omega= \{z=x+iy: 0< x< a, 0< y< b \}, $$

(1)

where 0, a, $\mu=a+ib$, $\nu=ib$ are four corner points of the domain Ω. The boundary $\partial\Omega=[0,a]\cup[a,\mu]\cup[\mu,\nu]\cup[\nu,0]$ is oriented counter-clockwise. That is, the oriented line $[0,a]$ is parameterized by $t\mapsto t$, $t\in[0,a]$, the segment $[a,\mu]$ is parameterized by $t\mapsto a+it$, $t\in[0,b]$, the oriented segment $[\mu,\nu]$ is parameterized by $t\mapsto t+ib$, $t\in[a,0]$, and the oriented segment $[\nu,0]$ is parameterized by $t\mapsto it$, $t\in[b,0]$.

Lemma 1.1

([14–16])

If $w\in C^{1}(\Omega;\mathbb{C} )\cap C(\overline{\Omega};\mathbb{C})$, then

$$\frac{1}{2\pi i} \int_{\partial \Omega}w(\zeta)\frac{\mathrm{d}\zeta}{\zeta-z}- \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)\frac{\mathrm{d}\xi\, \mathrm{d}\eta}{\zeta-z}=\left \{ \textstyle\begin{array}{l@{\quad}l} w(z),&z\in\Omega, \\ 0,&z\notin\overline{\Omega} \end{array}\displaystyle \right . $$

and

$$- \frac{1}{2\pi i} \int_{\partial \Omega}w(\zeta)\frac{\mathrm{d}\overline{\zeta}}{\overline{\zeta-z}}- \frac{1}{\pi} \int_{ \Omega}w_{\zeta}(\zeta)\frac{\mathrm{d}\xi\, \mathrm{d}\eta}{\overline{\zeta-z}}=\left \{ \textstyle\begin{array}{l@{\quad}l} w(z),&z\in\Omega, \\ 0,&z\notin\overline{\Omega}, \end{array}\displaystyle \right . $$

where Ω is the rectangle defined by (1) and $\zeta=\xi +i\eta$, $\xi,\eta\in\mathbb{R}$.

Reflecting the point $z\in\Omega$ at $[a,\mu]$, the symmetric point is $z_{1}=2a-\overline{z}$ and the domain Ω is bijectively mapped onto a rectangle $\Omega_{1}=\{z=x+iy: a\leq x\leq2a, 0\leq y\leq b\}$. Furthermore, the symmetric point of $z_{1}$ at $[\mu,\nu]$ is $z_{2}=-z+2a+2bi$ and the domain $\Omega_{1}$ is bijectively mapped onto a rectangle $\Omega_{2}=\{z=x+iy: a\leq x\leq2a, b\leq y\leq2b\}$. Continuing the reflection at $[a,\mu]$, the symmetric point of $z_{2}$ is $z_{3}=\overline {z}+2bi$ and the domain $\Omega_{2}$ is bijectively mapped onto a rectangle $\Omega_{3}=\{z=x+iy: 0\leq x\leq a, b\leq y\leq2b\}$. Finally, reflecting $z_{3}$ at $[\mu,\nu]$, the symmetric point of $z_{3}$ is just point z. Therefore, let $\Omega_{0,0}=\Omega\cup\Omega_{1}\cup\Omega_{2}\cup\Omega _{3}$, we know that $\Omega_{0,0}$ can be a basic rectangle and the reflection along the horizontal and vertical direction is equivalent to extending the basic rectangle $\Omega_{0,0}$ by double periods $2bi$, 2a, respectively (see Figure 1).

Figure 1
$\pmb{\Omega_{0,0}=\Omega\cup\Omega_{1}\cup\Omega_{2}\cup \Omega_{3}}$ .

Denoting $\omega_{mn}=2(ma+nbi)$, $m,n\in\mathbb{Z}$, and

$$ \Omega_{m,n}=\{z+\omega_{mn}: z\in \Omega_{0,0}, m,n \in\mathbb{Z}\}, $$

(2)

then $\mathbb{C}=\bigcup^{+\infty}_{m,n=-\infty}\overline{\Omega} _{m,n}$. By the technique of plane parqueting, the complex plane is divided into infinitely rectangles, which are congruent to the domain Ω.

2 Schwarz-Poisson formula

Suppose $R_{M,N}=\{(k,j): |k|\leq M, |j|\leq N, k,j\in\mathbb{Z}\}$ for $M,N\in\mathbb{N}$ be a finite set of double series. If the limit $\lim_{(M,N)\rightarrow(\infty,\infty)}\sum_{(m,n)\in R_{M,N}}f_{m,n}(\zeta,z)$ exists, then the double series $\sum_{(m,n)\in\mathbb{Z}\times\mathbb{Z}}f_{m,n}(\zeta ,z)$, simply $\sum_{m,n}f_{m,n}(\zeta,z)$ is convergent along the rectangle, that is,

$$ \sum_{m,n}f_{m,n}(\zeta,z)= \lim_{(M,N)\rightarrow(\infty ,\infty)}\sum_{(m,n)\in R_{M,N}}f_{m,n}( \zeta,z). $$

(3)

Similarly in [10, 17], the following result is true.

Lemma 2.1

Suppose that $S,E,W\subset\mathbb{C}$ be three bounded sets. If $S\cap E_{m,n}=\emptyset$, $S\cap W_{m,n}=\emptyset$ for all $(m,n)\in\mathbb{Z}\times\mathbb{Z}$, then the double series

$$\sum_{m,n} \biggl[ \frac{1}{\zeta-z-\omega _{mn}}- \frac{1}{\zeta-w-\omega_{mn}} \biggr] $$

is uniformly convergent for $(\zeta,z,w)\in S\times E\times W$ with $E_{m,n}=\{z+\omega_{mn}: z\in E\}$ and $W_{m,n}=\{z+\omega_{mn}: z\in W\}$.

Let

$$\begin{aligned} q_{m,n}(\zeta,z)&= \frac{1}{\zeta-z-\omega _{mn}}+ \frac{1}{\zeta+z-\omega_{mn}} \\ &= \frac{1}{\zeta-z-2ma-2nbi}+ \frac{1}{\zeta +z-2ma-2nbi}, \end{aligned}$$

(4)

then we have the following result.

Theorem 2.1

Any $w\in C^{1}(\Omega;\mathbb{C} )\cap C(\overline{\Omega};\mathbb{C})$ can be represented as

$$ w(z)=w(\alpha)+ \frac{1}{2\pi i} \int_{\partial \Omega}w(\zeta)\Xi_{\alpha}(\zeta,z)\,\mathrm{d} { \zeta}- \frac {1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)\Xi_{\alpha}(\zeta,z)\, \mathrm{d} {\xi }\,\mathrm{d}\eta,\quad z\in\Omega, $$

(5)

and, for $z\in\Omega$,

$$\begin{aligned} w(z) =&w(\alpha)+ \frac{1}{\pi i} \int_{\partial \Omega}\operatorname{Re}w(\zeta)\Xi_{\alpha}(\zeta,z)\, \mathrm{d} {\zeta} \\ &{}- \frac{1}{\pi} \int_{ \Omega} \bigl[w_{\overline{\zeta}}(\zeta)\Xi_{\alpha}( \zeta,z) -\overline{w_{\overline{\zeta}}(\zeta)}\Xi_{\alpha}(\overline{\zeta },z) \bigr]\,\mathrm{d} {\xi}\,\mathrm{d}\eta, \end{aligned}$$

(6)

where α is a fixed point in Ω and

$$\Xi_{\alpha}(\zeta,z)=\sum_{m,n} \bigl[q_{m,n}(\zeta ,z)-q_{m,n}(\zeta,\alpha) \bigr]. $$

Proof

For $z\in\Omega$, we know $\pm\overline{z}+\omega_{mn}$, $-z+\omega_{mn}\notin\Omega$ for $(m,n)\in\mathbb{Z}\times\mathbb {Z}$, and $z+\omega_{mn}\notin\Omega$ for $(m,n)\neq(0,0)$. Then by Lemma 1.1, one has, for $z\in\Omega$,

$$\begin{aligned}& \frac{1}{2\pi i} \int_{\partial\Omega} \frac {w(\zeta)}{\zeta-z}\,\mathrm{d}\zeta - \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)\frac{\mathrm{d}\xi\,\mathrm{d}\eta}{\zeta-z}=w(z), \end{aligned}$$

(7)

$$\begin{aligned}& \frac{1}{2\pi i} \int_{\partial\Omega} \frac {w(\zeta)}{\zeta-z-\omega_{mn}}\,\mathrm{d}\zeta - \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)\frac{\mathrm{d}\xi\, \mathrm{d}\eta}{\zeta-z-\omega_{mn}}=0, \quad (m,n)\neq(0,0), \end{aligned}$$

(8)

$$\begin{aligned}& \frac{1}{2\pi i} \int_{\partial\Omega} \frac {w(\zeta)}{\zeta+z-\omega_{mn}}\,\mathrm{d}\zeta - \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)\frac{\mathrm{d}\xi\,\mathrm{d}\eta}{\zeta+z-\omega_{mn}}=0, \end{aligned}$$

(9)

$$\begin{aligned}& \frac{1}{2\pi i} \int_{\partial \Omega} \frac{w(\zeta)}{\zeta-\overline{z}-\omega_{mn}}\,\mathrm{d}\zeta - \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)\frac{\mathrm{d}\xi\,\mathrm{d}\eta}{\zeta-\overline{z}-\omega_{mn}}=0, \end{aligned}$$

(10)

$$\begin{aligned}& \frac{1}{2\pi i} \int_{\partial\Omega} \frac {w(\zeta)}{\zeta+\overline{z}-\omega_{mn}}\,\mathrm{d}\zeta - \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)\frac{\mathrm{d}\xi\,\mathrm{d}\eta}{\zeta+\overline{z}-\omega_{mn}}=0. \end{aligned}$$

(11)

Adding (7), (8) and (9), and taking the sum for all the indices $(m,n)\in R_{M,N}$, we obtain

$$ w(z)= \frac{1}{2\pi i} \int _{\partial\Omega} w(\zeta)Q_{M,N}(\zeta,z)\,\mathrm{d}\zeta - \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)Q_{M,N}(\zeta,z)\, \mathrm{d}\xi\,\mathrm{d}\eta, $$

(12)

which implies that

$$ w(\alpha)= \frac{1}{2\pi i} \int_{\partial\Omega } w(\zeta)Q_{M,N}(\zeta,\alpha)\,\mathrm{d} \zeta - \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta)Q_{M,N}(\zeta,\alpha) \,\mathrm{d}\xi \,\mathrm{d}\eta, $$

(13)

where

$$ Q_{M,N}(\zeta,z)=\sum_{(m,n)\in R_{M,N}}q_{m,n}( \zeta,z). $$

(14)

Subtracting (13) from (12), then

$$\begin{aligned} w(z) =&w(\alpha)+ \frac{1}{2\pi i} \int_{\partial \Omega}w(\zeta) \bigl[Q_{M,N}(\zeta,z)-Q_{M,N}( \zeta,\alpha) \bigr]\,\mathrm{d} {\zeta} \\ &{}- \frac{1}{\pi} \int_{ \Omega}w_{\overline{\zeta}}(\zeta) \bigl[Q_{M,N}( \zeta,z)-Q_{M,N}(\zeta ,\alpha) \bigr]\,\mathrm{d} {\xi}\,\mathrm{d}\eta. \end{aligned}$$

(15)

Putting $(M,N)\rightarrow(\infty,\infty)$ and from Lemma 2.1, we get (5).

Let

$$ g_{m,n}(\zeta,\overline{z})=\left \{ \textstyle\begin{array}{l@{\quad}l} q_{m,-n}(\zeta,\overline{z}),&\zeta\in[0,a], \\ q_{-m+1,n}(\zeta,\overline{z}),&\zeta\in[a,\mu], \\ q_{m,-n+1}(\zeta,\overline{z}),&\zeta\in[\mu,\nu], \\ q_{-m,n}(\zeta,\overline{z}),&\zeta\in[\nu,0], \end{array}\displaystyle \right . $$

(16)

then from the relationship

$$ \zeta=\left \{ \textstyle\begin{array}{l@{\quad}l} \overline{\zeta},& \zeta\in[0,a], \\ 2a-\overline{\zeta},&\zeta\in[a,\mu], \\ \overline{\zeta}+2bi,&\zeta\in[\mu, \nu], \\ -\overline{\zeta},&\zeta\in[\nu,0], \end{array}\displaystyle \right . $$

(17)

we obtain

$$ \overline{q_{m,n}(\zeta,z)}\,\mathrm{d}\overline{ \zeta}=g_{m,n}(\zeta ,\overline{z})\,\mathrm{d}\zeta, \quad \zeta\in \partial\Omega. $$

(18)

Furthermore, by (18) and taking the sum of conjugations of (10) and (11) for all the indices $(m,n)\in R_{M,N}$, we have

$$ \frac{1}{2\pi i} \int_{\partial \Omega} \overline{w(\zeta)}H_{M,N}(\zeta,z)\, \mathrm{d}\zeta + \frac{1}{\pi} \int_{ \Omega}\overline{w_{\overline{\zeta}}(\zeta)}H_{M,N}( \overline{\zeta },z)\,\mathrm{d}\xi\,\mathrm{d}\eta=0, $$

(19)

where

$$ H_{M,N}(\zeta,z)=\left \{ \textstyle\begin{array}{l@{\quad}l} \sum_{(m,n)\in R_{M,N}}q_{m,-n}(\zeta,z),&\zeta\in[0,a], \\ \sum_{(m,n)\in R_{M,N}}q_{-m+1,n}(\zeta,z),&\zeta\in[a,\mu], \\ \sum_{(m,n)\in R_{M,N}}q_{m,-n+1}(\zeta,z),&\zeta\in[\mu,\nu], \\ \sum_{(m,n)\in R_{M,N}}q_{-m,n}(\zeta,z),&\zeta\in[\nu,0], \\ \sum_{(m,n)\in R_{M,N}}q_{m,-n}(\zeta,z),&\zeta\in\Omega. \end{array}\displaystyle \right . $$

(20)

Obviously,

$$ \frac{1}{2\pi i} \int_{\partial \Omega} \overline{w(\zeta)}H_{M,N}(\zeta,\alpha)\, \mathrm{d}\zeta + \frac{1}{\pi} \int_{ \Omega}\overline{w_{\overline{\zeta}}(\zeta)}H_{M,N}( \overline{\zeta },\alpha)\,\mathrm{d}\xi\,\mathrm{d}\eta=0. $$

(21)

Subtracting (21) from the sum of (19) and (15), we know

$$\begin{aligned} w(z) =&w(\alpha)+ \frac{1}{2\pi i} \int_{\partial \Omega} \bigl[w(\zeta)A(\zeta,z) +\overline{w(\zeta)}B( \zeta,z) \bigr]\,\mathrm{d} {\zeta} \\ &{}- \frac{1}{\pi} \int_{ \Omega} \bigl[w_{\overline{\zeta}}(\zeta)A(\zeta,z)-\overline {w_{\overline{\zeta}}(\zeta)} B(\overline{\zeta},z) \bigr] \,\mathrm{d} {\xi}\, \mathrm{d}\eta \end{aligned}$$

(22)

with

$$A(\zeta,z)=Q_{M,N}(\zeta,z)-Q_{M,N}(\zeta,\alpha) $$

and

$$B(\zeta,z)=H_{M,N}(\zeta,z)-H_{M,N}(\zeta,\alpha). $$

From Lemma 2.1, (14) and (20), we obtain, for $z\in \Omega$,

$$\lim_{(M,N)\rightarrow(\infty,\infty)}A(\zeta,z)=\lim_{(M,N)\rightarrow(\infty,\infty)}B(\zeta,z) = \sum_{m,n} \bigl[q_{m,n}( \zeta,z)-q_{m,n}(\zeta,\alpha) \bigr] $$

and

$$\lim_{(M,N)\rightarrow(\infty,\infty)}B(\overline{\zeta},z) =\sum _{m,n} \bigl[q_{m,n}(\overline{\zeta},z)-q_{m,n}( \overline {\zeta},\alpha) \bigr]. $$

Letting $(M,N)\rightarrow(\infty,\infty)$ for (22), (6) is obviously true. □

3 Schwarz problem

The classical Schwarz kernel for the upper half-plane $\mathbb{C}^{+}$ is

$$\frac{z-\overline{z}}{|x-z|^{2}}, \quad (x,z)\in(-\infty ,+\infty)\times\mathbb{C}^{+} $$

and satisfies (see [14])

$$\begin{aligned}& \lim_{z\in\mathbb{C}^{+}, z\rightarrow t} \frac {1}{2\pi i} \int_{c}^{d}\rho(x) \biggl[ \frac{1}{x-z}- \frac {1}{x-\overline{z}} \biggr]\,\mathrm{d}x \\& \quad = \rho(t), \quad t\in(c,d)\subset(-\infty,+ \infty), \end{aligned}$$

(23)

with $\rho\in C([c,d],\mathbb{C})$, $c< d$.

Lemma 3.1

For $\rho\in C(\partial\Omega,\mathbb{C})$, $z\in\Omega$, we obtain

$$\begin{aligned}& \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[a,\mu ]}\rho(\xi) \biggl[ \frac{1}{\xi-z}- \frac{1}{\xi+\overline {z}-2a} \biggr]\,\mathrm{d}\xi= \rho(t), \quad t\in(a,\mu), \\& \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\mu,\nu ]}\rho(\xi) \biggl[ \frac{1}{\xi-z}- \frac{1}{\xi-\overline {z}-2bi} \biggr]\,\mathrm{d}\xi= \rho(t), \quad t\in(\mu,\nu), \\& \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\nu ,0]}\rho(\xi) \biggl[ \frac{1}{\xi-z}- \frac{1}{\xi+\overline {z}} \biggr]\,\mathrm{d}\xi= \rho(t), \quad t\in(\nu,0). \end{aligned}$$

Proof

When $\xi\in[a,\mu]$ and letting $\xi=a+iy$, then $y\in[0,b]$ and

$$\begin{aligned}& \lim_{z\rightarrow t, t\in(a,\mu )} \frac{1}{2\pi i} \int_{[a,\mu]}\rho(\xi) \biggl[ \frac{1}{\xi-z}- \frac{1}{\xi+\overline{z}-2a} \biggr]\,\mathrm{d}\xi \\& \quad =\lim_{z\rightarrow t, t\in(a,\mu )} \frac{1}{2\pi i} \int_{0}^{b}\rho(a+iy) \biggl[ \frac{1}{y-(a-z)i}- \frac{1}{y-(\overline{z}-a)i} \biggr]\,\mathrm{d}y. \end{aligned}$$

For $z\in\Omega$ and $z\rightarrow t\in(a,\mu)$, we obtain $(a-z)i\in \mathbb{C}^{+}$ and $\rightarrow(a-t)i\in(0,b)$. Thus, from the classical Schwarz kernel theory (23), the first limit in Lemma 3.1 equals $\rho(t)$.

For $\xi\in[\mu,\nu]$, we take $\xi=y+ib$, then $y\in[a,0]$ and the second equation is

$$\lim_{z\rightarrow t, t\in(\mu,\nu)} \frac{1}{2\pi i} \int_{0}^{a}\rho(y+ib) \biggl[ \frac{1}{y-(\overline{z}+i b)}- \frac{1}{y-(z-i b)} \biggr]\,\mathrm{d}y. $$

When $z\in\Omega$ and $z\rightarrow t\in(\mu,\nu)$, we obtain $\overline{z}+i b\in\mathbb{C}^{+}$ and $\rightarrow\overline{t}+i b\in(0,a)$. Hence, from (23), the second equation is true. In the same way, when $\xi\in[\nu,0]$, suppose $\xi=iy$, then $y\in[b,0]$ and the third limit equals

$$\lim_{z\rightarrow t\in(\nu,0)} \frac{1}{2\pi i} \int _{0}^{b}\rho(i y) \biggl[ \frac{1}{y-i\overline{z}}- \frac{1}{ y+iz} \biggr]\,\mathrm{d}y=\rho(t). $$

The proof is completed. □

Define the following Pompeiu-type operator:

$$\begin{aligned} \begin{aligned}[b] A_{\alpha}[f](z)={}&- \frac{1}{\pi} \int_{\Omega}\biggl\{ f(\zeta) \sum_{m,n} \biggl[q_{m,n}(\zeta ,z)- \frac{q_{m,n}(\zeta,\alpha)+q_{m,n}(\zeta,\overline {\alpha})}{2} \biggr] \\ &{}-\overline{f(\zeta)} \sum_{m,n} \biggl[q_{m,n}(\overline{\zeta},z)- \frac {q_{m,n}(\overline{\zeta},\alpha) +q_{m,n}(\overline{\zeta},\overline{\alpha})}{2} \biggr] \biggr\} \,\mathrm{d}\xi\,\mathrm{d}\eta, \end{aligned} \end{aligned}$$

(24)

for $f\in L_{p}(\Omega,\mathbb{C})$, $p>2$ with $q_{m,n}$ given by (4) and $\alpha\in\Omega$.

Lemma 3.2

If $f\in L_{p}(\Omega,\mathbb{C})$, $p>2$, then $A_{\alpha}[f](z)\in C(\overline{\Omega},\mathbb{C})$ and $\frac{\partial A_{\alpha}[f](z)}{\partial\overline{z}}=f(z)$ for $z\in\Omega$.

Proof

From the classical Pompeiu-type operator in [14],

$$T[f](z)=- \frac{1}{\pi} \int_{\Omega} \frac {f(\zeta)}{\zeta-z}\,\mathrm{d}\xi\,\mathrm{d}\eta\in C(\overline{\Omega },\mathbb{C}) $$

and $\frac{\partial T[f](z)}{\partial\overline{z}}=f(z)$ for $z\in\Omega$. When $z\in\Omega$, then $z+\omega_{mn}\notin\Omega$ for $(m,n)\neq(0,0)$ and $\pm\overline{z}+\omega_{mn}, -z+\omega_{mn}\notin\Omega$. Thus by (4) and (24), we have $A_{\alpha}[f](z)\in C(\overline {\Omega},\mathbb{C})$ and the integral in (24) is analytic for $z\in\Omega$ except for one term $T[f](z)$, therefore,

$$\frac{\partial A_{\alpha}[f](z)}{\partial\overline {z}}= \frac{\partial (- \frac{1}{\pi}\int _{\Omega} \frac{f(\zeta)}{\zeta-z}\,\mathrm{d}\xi\,\mathrm{d}\eta )}{\partial\overline{z}}=f(z). $$

□

Lemma 3.3

For $f\in L_{p}(\Omega,\mathbb{C})$, $p>2$,

$$\lim_{z\in\Omega, z\rightarrow t}\operatorname{Re}A_{\alpha}[f](z)=0, \quad t \in\partial\Omega. $$

Proof

By (24), we obtain

$$\operatorname{Re}A_{\alpha}[f](z)=- \frac{1}{2\pi} \int_{\Omega}\bigl\{ f(\zeta) \bigl[P(\zeta,z)+\Phi(\zeta,\alpha) \bigr] -\overline{f(\zeta)} \bigl[P(\overline{\zeta},z)+\Phi(\overline{\zeta }, \alpha) \bigr] \bigr\} \,\mathrm{d}\xi\,\mathrm{d}\eta $$

with

$$P(\zeta,z)=\sum_{m,n} \bigl[q_{m,n}( \zeta,z)-q_{m,-n}(\zeta ,\overline{z}) \bigr] $$

and

$$\Phi(\zeta,z)= \frac{1}{2}\sum_{m,n} \bigl[q_{m,-n}(\zeta,\overline{z})-q_{m,n}(\zeta,\overline{z}) +q_{m,-n}(\zeta,z)-q_{m,n}(\zeta,z) \bigr]. $$

Obviously, we know $\Phi(\zeta,\alpha)=\Phi(\overline{\zeta},\alpha)=0$ for $\zeta\in\Omega$. Therefore,

$$\begin{aligned} \operatorname{Re}A_{\alpha}[f](z) =&- \frac{1}{2\pi} \int_{\Omega}\biggl\{ f(\zeta)\sum_{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,-n}(\zeta,\overline {z}) \bigr] \\ &{}-\overline{f(\zeta)} \sum_{m,n} \bigl[q_{m,n}(\overline{\zeta},z)-q_{m,-n}(\overline{\zeta }, \overline{z}) \bigr] \biggr\} \,\mathrm{d}\xi\,\mathrm{d}\eta. \end{aligned}$$

Furthermore, when $z\in\partial\Omega$, it satisfies equation (17) by replacing ζ with z. For $\zeta\in\Omega$, $z\in[\mu ,\nu]$, we have $\overline{z}=z-2bi$, then

$$\sum_{m,n} \bigl[q_{m,n}( \zeta,z)-q_{m,-n}(\zeta,\overline{z}) \bigr]=\sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,-n}( \zeta,z-2bi) \bigr]=0. $$

Similarly,

$$\sum_{m,n} \bigl[q_{m,n}(\overline{ \zeta},z)-q_{m,-n}(\overline {\zeta},\overline{z}) \bigr]=0, \quad z\in[ \mu,\nu], \zeta\in\Omega. $$

Also, for $\zeta\in\Omega$ and $z\in\partial\Omega\setminus[\mu,\nu]$,

$$\sum_{m,n} \bigl[q_{m,n}( \zeta,z)-q_{m,-n}(\zeta,\overline{z}) \bigr] =\sum _{m,n} \bigl[q_{m,n}(\overline{\zeta},z)-q_{m,-n}( \overline {\zeta},\overline{z}) \bigr]=0, $$

thus the proof is completed. □

Consider the Schwarz-type operator

$$ S_{\alpha}[\gamma](z)= \frac{1}{\pi i} \int_{\partial\Omega }\gamma(\zeta) \sum_{m,n} \biggl[q_{m,n}(\zeta,z)- \frac {q_{m,n}(\zeta,\alpha)+q_{m,n}(\zeta,\overline{\alpha})}{2} \biggr] \,\mathrm{d}\zeta, $$

(25)

where $q_{m,n}$ is given by (4) and $\gamma\in C(\partial\Omega ,\mathbb{R})$. Then we have

$$\begin{aligned} \operatorname{Re}S_{\alpha}[\gamma](z) =& \frac{1}{2\pi i} \int _{\partial\Omega}\gamma(\zeta) \biggl\{ \sum _{m,n} \biggl[q_{m,n}(\zeta ,z)- \frac{q_{m,n}(\zeta,\alpha)+q_{m,n}(\zeta,\overline {\alpha})}{2} \biggr]\,\mathrm{d}\zeta \\ &{}- \sum_{m,n} \biggl[\overline {q_{m,n}( \zeta,z)}- \frac{\overline{q_{m,n}(\zeta,\alpha)} +\overline{q_{m,n}(\zeta,\overline{\alpha})}}{2} \biggr]\,\mathrm{d} {\overline{\zeta}} \biggr\} . \end{aligned}$$

By (18),

$$\begin{aligned} \operatorname{Re}S_{\alpha}[\gamma](z) =& \frac{1}{2\pi i} \int _{\partial\Omega}\gamma(\zeta) \biggl\{ \sum _{m,n} \bigl[q_{m,n}(\zeta,z)-g_{m,n}(\zeta , \overline{z}) \bigr] \\ &{}- \frac{1}{2} \sum_{m,n} \bigl[q_{m,n}(\zeta,\alpha)-g_{m,n}(\zeta,\overline{\alpha })+q_{m,n}(\zeta,\overline{\alpha})-g_{m,n}(\zeta,\alpha) \bigr] \biggr\} \,\mathrm{d} {\zeta}, \end{aligned}$$

where $g_{m,n}$ is given by (16). From (16) and Lemma 2.1, for $\zeta\in\partial\Omega$,

$$\sum_{m,n} \bigl[q_{m,n}(\zeta, \alpha)-g_{m,n}(\zeta ,\overline{\alpha})+q_{m,n}(\zeta, \overline{\alpha})-g_{m,n}(\zeta ,\alpha) \bigr]=0 $$

and

$$\sum_{m,n} \bigl[q_{m,n}( \zeta,z)-g_{m,n}(\zeta,\overline{z}) \bigr]=\sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta, \overline{z}) \bigr], $$

therefore,

$$ \operatorname{Re}S_{\alpha}[\gamma](z)= \frac{1}{2\pi i} \int _{\partial\Omega}\gamma(\zeta) \sum_{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta,\overline{z}) \bigr]\, \mathrm{d} {\zeta}. $$

(26)

Lemma 3.4

For $\gamma\in C(\partial\Omega,\mathbb{R})$, $S_{\alpha}[\gamma](z)$ is analytic in Ω, i.e.,

$$\frac{\partial S_{\alpha}[\gamma](z)}{\partial\overline{z}}=0. $$

Proof

From (25), the sum in integrand can be rewritten as

$$\frac{1}{2}\sum_{m,n} \bigl\{ \bigl[q_{m,n}(\zeta ,z)-q_{m,n}(\zeta,\alpha) \bigr] + \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta,\overline{\alpha}) \bigr] \bigr\} , $$

which is convergent for $\zeta\in\partial\Omega$, $z\in\Omega$ by Lemma 2.1. Obviously, from the expression of $q_{m,n}$ in (4), the integrand in $S_{\alpha}[\gamma](z)$ is analytic for $z\in\Omega$, hence the proof is completed. □

Lemma 3.5

For $z\in\Omega$, $\gamma\in C(\partial\Omega,\mathbb{R})$,

$$ \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{L} \bigl[\gamma(\zeta) -\gamma(t_{0}) \bigr] \sum_{m,n} \bigl[q_{m,n}( \zeta,z)-q_{m,n}(\zeta ,\overline{z}) \bigr]\,\mathrm{d} {\zeta}= \gamma(t)-\gamma(t_{0}), $$

(27)

where $t\in L^{0}\cup\{t_{0}\}$, $L^{0}$ is L except for two endpoints and

$$L=\left \{ \textstyle\begin{array}{l@{\quad}l} {[\mu,\nu]},& \textit{for } t_{0}=\mu \textit{ or } \nu, \\ {[\nu,0]},& \textit{for } t_{0}=0 \textit{ or } \nu, \\ {[0,a]},& \textit{for } t_{0}=0 \textit{ or } a, \\ {[a,\mu]},& \textit{for } t_{0}=a \textit{ or } \mu. \end{array}\displaystyle \right . $$

Furthermore, for $t\in\partial\Omega\setminus L^{0}$,

$$ \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{L} \bigl[\gamma(\zeta)-\gamma(t) \bigr] \sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta, \overline{z}) \bigr]\,\mathrm{d} {\zeta}=0. $$

(28)

Proof

When $L=[\mu,\nu]$, $t_{0}=\nu$ and $t\in L^{0}\cup\{t_{0}\}=(\mu,\nu]$,

$$\begin{aligned}& \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\mu,\nu ]} \bigl[\gamma(\zeta) -\gamma(\nu) \bigr]\sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta , \overline{z}) \bigr]\,\mathrm{d} {\zeta} \\& \quad = \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\mu,\nu ]} \bigl[\gamma(\zeta) -\gamma(\nu) \bigr] \biggl[ \frac{1}{\zeta-z}-\frac{1}{\zeta -\overline{z}-2bi}+\frac{1}{\zeta+z-2bi}-\frac{1}{\zeta+\overline {z}} \biggr]\,\mathrm{d} {\zeta} \\& \quad =\lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\mu,\nu ]\cup[\nu,-a+i b]}\Gamma(\zeta) \biggl[ \frac{1}{\zeta-z}- \frac {1}{\zeta-\overline{z}-2bi} \biggr]\,\mathrm{d} {\zeta}, \end{aligned}$$

where

$$\Gamma(\zeta)=\left \{ \textstyle\begin{array}{l@{\quad}l} \gamma(\zeta)-\gamma(\nu),&\zeta\in[\mu,\nu], \\ \gamma(\nu)-\gamma(2bi-\zeta),&\zeta\in[\nu,-a+i b]. \end{array}\displaystyle \right . $$

By $\Gamma\in C([\mu,\nu]\cup[\nu,-a+i b],\mathbb{R})$ and from the second equation in Lemma 3.1, the above limit is $\Gamma(t)=\gamma (t)-\gamma(\nu)$. That is,

$$\lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\mu,\nu ]} \bigl[\gamma(\zeta) -\gamma(\nu) \bigr]\sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta , \overline{z}) \bigr]\,\mathrm{d} {\zeta}=\gamma(t)-\gamma(\nu), \quad t\in (\mu, \nu]. $$

Similarly,

$$\lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\mu,\nu ]} \bigl[\gamma(\zeta) -\gamma(\mu) \bigr]\sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta , \overline{z}) \bigr]\,\mathrm{d} {\zeta}=\gamma(t)-\gamma(\mu), \quad t\in [\mu, \nu). $$

In the same way, when L and $t_{0}$ are in all other cases of (27), the result (27) is also true.

Furthermore, by (17), we obtain for $z\in[0,a]\cup[a,\mu)\cup (\nu,0]$ and $\zeta\in[\mu,\nu]$, $\sum_{m,n} [q_{m,n}(\zeta, z)-q_{m,n}(\zeta,\overline{z}) ]=0$, therefore, for $t\in[0,a]\cup[a,\mu)\cup(\nu,0]$,

$$ \lim_{z\rightarrow t} \frac{1}{2\pi i} \int_{[\mu,\nu ]} \bigl[\gamma(\zeta)-\gamma(t) \bigr] \sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta, \overline{z}) \bigr]\,\mathrm{d} {\zeta}=0. $$

(29)

Combining the result (27), we obtain (29) is also true for corner points $t=\mu,\nu$, hence (28) holds for $L=[\mu ,\nu]$ and $t\in\partial\Omega\setminus(\mu,\nu)$. Similarly, (28) is also true for L, t in the other cases. The proof is completed. □

Lemma 3.6

For $\gamma\in C(\partial\Omega,\mathbb{R})$ and $t\in\partial\Omega$,

$$\lim_{ z\in\Omega, z\rightarrow t}\operatorname{Re}S_{\alpha}[\gamma ](z)= \gamma(t). $$

Proof

From (26), we only need to prove

$$ \lim_{ z\in\Omega, z\rightarrow t} \frac{1}{2\pi i} \int_{\partial\Omega}\gamma(\zeta) \sum_{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta,\overline{z}) \bigr]\, \mathrm{d} {\zeta}=\gamma(t), \quad t\in\partial\Omega. $$

(30)

First of all, taking $w(z)\equiv1$ and by (12), (13), we know

$$\frac{1}{2\pi i} \int_{\partial\Omega} Q_{M,N}(\zeta ,z)\,\mathrm{d}\zeta=1, \quad z\in\Omega $$

and

$$\frac{1}{2\pi i} \int_{\partial\Omega} Q_{M,N}(\zeta,\alpha )\,\mathrm{d}\zeta=1. $$

In particular, for $\alpha\in\Omega$, $\pm\overline{\alpha}+\omega _{mn}$ do not belong to Ω, so we have

$$\frac{1}{2\pi i} \int_{\partial\Omega}Q_{M,N}(\zeta ,\overline{\alpha})\,\mathrm{d} \zeta=0. $$

Combining the above three equations and taking $(M,N)\rightarrow(\infty ,\infty)$, thus

$$S_{\alpha}[1](z)= \frac{1}{\pi i} \int_{\partial\Omega} \sum_{m,n} \biggl[q_{m,n}(\zeta,z)- \frac{q_{m,n}(\zeta ,\alpha)+q_{m,n}(\zeta,\overline{\alpha})}{2} \biggr] \,\mathrm{d}\zeta=1. $$

Then from (26),

$$ \operatorname{Re}S_{\alpha}[1](z)= \frac{1}{2\pi i} \int_{\partial\Omega} \sum_{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta,\overline{z}) \bigr]\, \mathrm{d} {\zeta}=1, \quad z\in\Omega, $$

(31)

hence,

$$\begin{aligned}& \lim_{ z\in\Omega, z\rightarrow t}\operatorname{Re}S_{\alpha}[ \gamma ](z) \\& \quad =\lim_{ z\in\Omega, z\rightarrow t} \frac{1}{2\pi i} \int_{\partial\Omega} \bigl[\gamma(\zeta)-\gamma(t) \bigr] \sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta, \overline{z}) \bigr]\,\mathrm{d} {\zeta}+\gamma(t). \end{aligned}$$

(32)

When $t\in(\mu,\nu)$, we rewrite $\operatorname{Re}S_{\alpha}[\gamma](z)$ as

$$\frac{1}{2\pi i} \int_{\partial\Omega} \bigl[\gamma(\zeta )-\gamma(\mu) \bigr] \sum _{m,n} \bigl[q_{m,n}(\zeta,z)-q_{m,n}(\zeta, \overline{z}) \bigr]\,\mathrm{d} {\zeta}+\gamma(\mu), $$

thus from the results for $L=[\mu,\nu]$, $t_{0}=\mu$ in (27) and for $L=[0,a]$, $[a,\mu]$, and $[\nu,0]$ in (28), we get $\lim_{ z\in\Omega, z\rightarrow t}\operatorname{Re}S_{\alpha}[\gamma ](z)=\gamma(t)$, $t\in(\mu,\nu)$. Furthermore, by (32) and Lemma 3.5, (30) obviously holds for $t=\mu,\nu$. Similarly, when $t\in \partial\Omega\setminus[\mu,\nu]$, the result is also true. Then the proof is completed. □

Theorem 3.1

The Schwarz problem

$$ \left \{ \textstyle\begin{array}{l}\partial_{\overline{z}}w=f\quad \textit{in } \Omega,\qquad \operatorname{Re}w=\gamma\quad \textit{on } \partial\Omega, \\ \operatorname{Im}w(\alpha)=c, \quad c\in \mathbb{R}, \end{array}\displaystyle \right . $$

(33)

for $f\in L_{p}(\Omega;\mathbb{C})$, $p>2$, $\gamma\in C(\partial\Omega;\mathbb{R}) $, and α is a fixed point in Ω, is uniquely solvable by

$$ w(z)=S_{\alpha}[\gamma](z)+ A_{\alpha}[f](z)+ic, $$

(34)

where $A_{\alpha}$, $S_{\alpha}$ are defined by (24) and (25), respectively.

Proof

By Lemmas 3.2-3.4 and Lemma 3.6, $\varphi(z)=S_{\alpha}[\gamma](z)+ A_{\alpha}[f](z)$ satisfies the first two conditions in (33). Suppose $\phi (z)=w(z)-\varphi(z)$, then $\phi(z)$ satisfies

$$\left \{ \textstyle\begin{array}{l} \partial_{\overline{z}}\phi=0 \quad \text{in } \Omega, \\ \operatorname{Re}\phi=0 \quad \text{on } \partial\Omega. \end{array}\displaystyle \right . $$

Then from Theorem 2.1, we know $\phi(z)=i\operatorname{Im}w(\alpha)=ic$, thus, the proof is completed. □

Declarations

Acknowledgements

This paper is supported by the Fundamental Research Funds for the Central Universities. The authors would like to thank the referees for their valuable suggestion and comments.

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

(1)

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430072, P.R. China

(2)

College of Mathematics and Econometrics, Hunan University, Changsha, 410082, P.R. China

References

Abdymanapov, SA, Begehr, H, Tungatarov, AB: Some Schwarz problems in a quarter plane. Eurasian Math. J. 3, 22-35 (2005) Google Scholar
Akel, M, Hussein, H: Two basic boundary-value problems for the inhomogeneous Cauchy-Riemann equation in an infinite sector. Adv. Pure Appl. Math. 3, 315-328 (2012) MathSciNetView ArticleMATHGoogle Scholar
Begehr, H, Schmersau, D: The Schwarz problem for polyanalytic functions. Z. Anal. Anwend. 24, 341-351 (2005) MathSciNetView ArticleMATHGoogle Scholar
Begehr, H, Gaertner, EA: Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane. Georgian Math. J. 14(1), 33-52 (2007) MathSciNetMATHGoogle Scholar
Begehr, H, Vaitekhovich, T: Schwarz problem in lens and lune. Complex Var. Elliptic Equ. 59(1), 76-84 (2014) MathSciNetView ArticleMATHGoogle Scholar
Begehr, H: Green function for a hyperbolic strip and a class of related plane domains. Appl. Anal. 93(11), 2370-2385 (2014) MathSciNetView ArticleMATHGoogle Scholar
Shupeyeva, B: Some basic boundary value problems for complex partial differential equations in quarter ring and half hexagon. Dissertation, FU, Berlin (2013). http://www.diss.fu-berlin.de/diss/receive/FUDISS-thesis-000000094596
Vaitekhovich, T: Boundary value problems for complex partial differential equations in a ring domain. Dissertation, FU, Berlin (2008). http://www.diss.fu-berlin.de/diss/receive/FUDISS-thesis-000000003859
Wang, Y, Wang, Y: Two boundary-value problems for the Cauchy-Riemann equation in a sector. Complex Anal. Oper. Theory 6, 1121-1138 (2012) MathSciNetView ArticleMATHGoogle Scholar
Wang, Y, Wang, Y: Schwarz-type problem of nonhomogeneous Cauchy-Riemann equation on a triangle. J. Math. Anal. Appl. 377, 557-570 (2011) MathSciNetView ArticleMATHGoogle Scholar
Begehr, H, Vaitekhovich, T: How to find harmonic Green functions in the plane. Complex Var. Elliptic Equ. 56(12), 1169-1181 (2011) MathSciNetView ArticleMATHGoogle Scholar
Begehr, H, Vaitekhovich, T: Green functions, reflections, and plane parqueting. Eurasian Math. J. 1, 17-31 (2010) MathSciNetMATHGoogle Scholar
Begehr, H, Vaitekhovich, T: Harmonic Dirichlet problem for some equilateral triangle. Complex Var. Elliptic Equ. 57(2-4), 185-196 (2012) MathSciNetView ArticleMATHGoogle Scholar
Begehr, H: Complex Analytic Methods for Partial Differential Equations: An Introductory Text. World Scientific, Singapore (1994) View ArticleMATHGoogle Scholar
Begehr, H: Boundary value problems in complex analysis I. Bol. Asoc. Mat. Venez. XII, 65-85 (2005) MathSciNetGoogle Scholar
Begehr, H: Boundary value problems in complex analysis II. Bol. Asoc. Mat. Venez. XII, 217-250 (2005) MathSciNetGoogle Scholar
Stein, EM, Shakarchi, R: Complex Analysis. Princeton University Press, Princeton (2003) MATHGoogle Scholar