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Schwarz boundary value problem for the Cauchy-Riemann equation in a rectangle
Boundary Value Problems volume 2016, Article number: 7 (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.
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
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
If \(w\in C^{1}(\Omega;\mathbb{C} )\cap C(\overline{\Omega};\mathbb{C})\), then
and
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).
Denoting \(\omega_{mn}=2(ma+nbi)\), \(m,n\in\mathbb{Z}\), and
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,
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
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
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
and, for \(z\in\Omega\),
where α is a fixed point in Ω and
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 \(z\in\Omega\),
Adding (7), (8) and (9), and taking the sum for all the indices \((m,n)\in R_{M,N}\), we obtain
which implies that
where
Subtracting (13) from (12), then
Putting \((M,N)\rightarrow(\infty,\infty)\) and from Lemma 2.1, we get (5).
Let
then from the relationship
we obtain
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
where
Obviously,
Subtracting (21) from the sum of (19) and (15), we know
with
and
From Lemma 2.1, (14) and (20), we obtain, for \(z\in \Omega\),
and
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
and satisfies (see [14])
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
Proof
When \(\xi\in[a,\mu]\) and letting \(\xi=a+iy\), then \(y\in[0,b]\) and
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
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
The proof is completed. □
Define the following Pompeiu-type operator:
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],
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,
 □
Lemma 3.3
For \(f\in L_{p}(\Omega,\mathbb{C})\), \(p>2\),
Proof
By (24), we obtain
with
and
Obviously, we know \(\Phi(\zeta,\alpha)=\Phi(\overline{\zeta},\alpha)=0\) for \(\zeta\in\Omega\). Therefore,
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
Similarly,
Also, for \(\zeta\in\Omega\) and \(z\in\partial\Omega\setminus[\mu,\nu]\),
thus the proof is completed. □
Consider the Schwarz-type operator
where \(q_{m,n}\) is given by (4) and \(\gamma\in C(\partial\Omega ,\mathbb{R})\). Then we have
By (18),
where \(g_{m,n}\) is given by (16). From (16) and Lemma 2.1, for \(\zeta\in\partial\Omega\),
and
therefore,
Lemma 3.4
For \(\gamma\in C(\partial\Omega,\mathbb{R})\), \(S_{\alpha}[\gamma](z)\) is analytic in Ω, i.e.,
Proof
From (25), the sum in integrand can be rewritten as
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 \(z\in\Omega\), \(\gamma\in C(\partial\Omega,\mathbb{R})\),
where \(t\in L^{0}\cup\{t_{0}\}\), \(L^{0}\) is L except for two endpoints and
Furthermore, for \(t\in\partial\Omega\setminus L^{0}\),
Proof
When \(L=[\mu,\nu]\), \(t_{0}=\nu\) and \(t\in L^{0}\cup\{t_{0}\}=(\mu,\nu]\),
where
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,
Similarly,
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 \(t\in[0,a]\cup[a,\mu)\cup(\nu,0]\),
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 \(t\in\partial\Omega\),
Proof
From (26), we only need to prove
First of all, taking \(w(z)\equiv1\) and by (12), (13), we know
and
In particular, for \(\alpha\in\Omega\), \(\pm\overline{\alpha}+\omega _{mn}\) do not belong to Ω, so we have
Combining the above three equations and taking \((M,N)\rightarrow(\infty ,\infty)\), thus
Then from (26),
hence,
When \(t\in(\mu,\nu)\), we rewrite \(\operatorname{Re}S_{\alpha}[\gamma](z)\) as
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
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
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
Then from Theorem 2.1, we know \(\phi(z)=i\operatorname{Im}w(\alpha)=ic\), thus, the proof is completed. □
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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.
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Wang, Y., Zhao, X. Schwarz boundary value problem for the Cauchy-Riemann equation in a rectangle. Bound Value Probl 2016, 7 (2016). https://doi.org/10.1186/s13661-016-0520-z
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DOI: https://doi.org/10.1186/s13661-016-0520-z