Dirichlet and Neumann boundary value problems for bi-polyanalytic functions on the bicylinder

1.   Introduction

Bi-analytic functions arise from the research on systems of some partial differential equations and play an important role in studying elasticity problems. In 1961, Sander ^[1] studied a system of partial differential equations of first order:

for $k\in \mathbb{R}, \; k\neq -1$, and introduced the concept of bi-analytic functions of type $k$. He extended the elementary properties of analytic functions to bi-analytic functions and extended the related problems of the plane strain, the generalized plane stress, and the flow of viscous fluids to the theory of bi-analytic functions.

Later, Lin and Wu ^[2] introduced a more extensive class of functions, i.e., bi-analytic functions of type $(\lambda, k)$, which are defined by the system of equations:

where $f(z) = u+iv$, $\lambda, k$ are real constants with $\lambda\neq0, 1, k^2$ and $0 < k < 1$. The complex form of the system of equations is

in which $\varphi(z) = k\vartheta-i\lambda\omega$ is analytic and is called the associated function of $f(z)$. Lin and Wu ^[2] obtained the general expression and some properties of bi-analytic functions of type $(\lambda, k)$, including the Cauchy integral theorem and formula, the Morera theorem, the Weierstrass theorem, and the power series expansions.

Hua et al. ^[3,4] investigated the systems of partial differential equations of second order, which have close association with $(\lambda, k)$ bi-analytic functions, and showed that $(\lambda, k)$ bi-analytic functions provide a powerful tool to deal with problems of plane elasticity.

Mu ^[5] obtained the Cauchy integral representation, the removable singularity theorem, the Weierstrass theorem, and the mean value theorem of bi-analytic functions of type $(\lambda, k)$ in the complex plane. Wen et al. ^[6,7] studied various types of boundary value problems for elliptic complex equations and systems, including boundary problems of $(\lambda, k)$ bi-analytic functions. Begehr and Lin ^[8] studied a mix-contact problem by applying $(\lambda, k)$ bi-analytic functions and singular integral operators. In addition, Lin and Zhao ^[9] reviewed the work on the applications of bi-analytic functions in elasticity, especially for solving the basic boundary value problems of plane elasticity (in isotropic or orthotropic cases). In ^[10], Lai investigated the properties of the vector-valued bi-analytic functions of type $(\lambda, k)$ in a complex couple Banach space and obtained the solution of the corresponding Dirichlet problem.

The special case of bi-analytic functions of type $(\lambda, k)$ for $k = 1$ is

which means

Thus, bi-analytic functions are different from bianalytic functions (which only satisfy $\frac{\partial^2f}{\partial\bar{z}^2} = 0$) in the sense of Bitsadze ^[11]. The generalization of a bianalytic function $f$ is $\frac{\partial^nf}{\partial\bar{z}^n} = 0\; (n\geq 1)$, which is called a polyanalytic function ^[12]. Obviously, bi-analytic functions can be similarly generalized.

In ^[13], Kumar extended bi-analytic functions on bounded simply connected domains. He considered the system of complex equations:

where $f$ is called bi-polyanalytic functions, and obtained the expressions of solutions to the corresponding boundary value problems on the unit disc. In ^[13], the systems

and the associated boundary problems were also discussed.

Begehr et al. ^[14,15] investigated boundary value problems for bi-polyanalytic functions in the upper half plane and on the unit disc. Similar researches can also be found in the reference ^[16].

In 2022, Lin and Xu ^[17] successfully obtained the solutions of Riemann problems of $(\lambda, k)$ bi-analytic functions. In 2023, Lin ^[18] discussed a type of inverse boundary value problems for $(\lambda, 1)$ bi-analytic functions. Nowadays, bi-analytic functions and polyanalytic functions have made great progress ^{[19,20,21,22]}.

The above conclusions are all in the complex plane. If these results can be extended to spaces of several complex variables, there will be more applications. There have been some conclusions about bi-analytic functions or polyanalytic functions with several complex variables. For example, in ^[23], Begehr and Kumar studied complex bi-analytic functions of $n$ variables. They successfully obtained the corresponding Cauchy integral formula, Taylor series and Poisson integral formula on the polydisc, and discussed the Dirichlet problem of the systems of partial differential equations on the polydisc. In ^[24], Kumar discussed a generalized Riemann boundary value problem by the Cauchy integral of bi-analytic functions with two complex variables. In 2023, Vasilevski ^[25] studied polyanalytic functions with several complex variables in detail.

However, there is relatively little research on bi-polyanalytic functions in spaces of several complex variables. Inspired by these, and on the basis of the previous works of the former researchers, we first investigate a kind of boundary value problem for polyanalytic functions with Schwarz conditions on the bicylinder, then we discuss boundary value problems with the Dirichlet, Neumann boundary conditions and mixed type boundary conditions for bi-polyanalytic functions on the bicylinder.

Throughout this paper, let the bicylinder $D^2 = D_1\times D_2 = \{(z_1, z_2):|z_1| < 1, \; |z_2| < 1\}$, whose characteristic boundary is denoted as $\partial_0D^2$, where $z_1$ and $z_2$ are on the complex plane. Let $C^m(G)$ represent the set of functions whose partial derivatives of order $m$ are all continuous within a bounded smooth domain $G$.

2.   Some lemmas

Lemma 2.1. ^[26] Let $G$ be a bounded smooth domain in the complex plane, $f\in L_1(G; \mathbb{C})$ and

Then, $\partial_{\bar{z}}Tf(z) = f(z)$.

Lemma 2.2. ^[26] Let $w\in C^1(G; \mathbb{C})\bigcap C(\overline{G}; \mathbb{C})$, where $m\geq 1$ and $G$ is a bounded smooth domain in the complex plane, then

where $d\sigma_\zeta = d\xi d\eta \; (\zeta = \xi+i\eta)$ and $d\sigma_z = dx dy \; (z = x+iy)$.

Lemma 2.3. ^[27] For $n\in \mathbb{N}$, $f\in L^p(D), p\geq 1$ with $D$ being the unit disc, let

and let $T_0f = f$. Then

Lemma 2.4. Let $g_{\mu\nu}\in C(\partial_0D^2;\mathbb{R})$ for $1\leq \mu, \nu\leq m-1$ ($m\geq2$), and let

where

in which

Then, $\widetilde{\phi}(z)$ is a specific solution to the problem

Proof. Obviously, $u_{\widetilde{v}_1\widetilde{v}_2}(z)$ is analytic on $D^2$, therefore, $\partial^{m}_{\bar{z}_1}\partial^{m}_{\bar{z}_2}\widetilde{\phi}(z) = 0$. In addition, from the proof of Theorem 2.1 in ^[28]: We obtain that $\Re\partial^{\mu}_{\bar{z}_1}\partial^{\nu}_{\bar{z}_2}\widetilde{\phi}(z) = g_{\mu\nu}(z)$ for $z\in \partial_0D^2$, and

for $z_1\in D_1, z_2\in D_2$. So, $\widetilde{\phi}(z)$ is a specific solution to the problem.

Lemma 2.5. ^[15] Let $D$ be the unit disc. For $1\leq k\leq n-1$, let $g_{k}\in C(\partial D)$ and $c_{k}\in\mathbb{C}$ be given. Then, there exists an analytic function $u_k$ on $D$ satisfying

if and only if for $\forall z\in D$

and $u_k$ is uniquely represented as

Lemma 2.6. ^[15] Let $f\in L_1(D)$, $\varphi\in C(\partial D)$ and $c\in\mathbb{C}$ be given; then the problem

has a unique solution

if and only if

Lemma 2.7. Let $u_0$ be an analytic function on $D^2$. Then

and

where $\partial_{\zeta_1}\partial_{\zeta_2}u_1(\zeta) = u_0(\zeta)$.

Proof. By Lemma 2.2,

So we obtain (2.2). Let $\partial_{\zeta_1}u_1(\zeta) = u_2(\zeta)$. Since $\partial_{\zeta_1}\partial_{\zeta_2}u_1(\zeta) = u_0(\zeta)$, then we have $\partial_{\zeta_2}u_2(\zeta) = u_0(\zeta)$, which follows $\partial_{\bar{\zeta}_2}\overline{u_2(\zeta)} = \overline{u_0(\zeta)}$. Therefore, by Lemma 2.2,

In addition, $\partial_{\zeta_1}u_1(\zeta) = u_2(\zeta)$ follows $\partial_{\bar{\zeta}_1}\overline{u_1(\zeta)} = \overline{u_2(\zeta)}$. Similarly, we have

Therefore, by (2.4) and (2.5), we obtain

Lemma 2.8. Let $\varphi\in C(\partial_0D^2;\mathbb{C})$ and $g_{\mu\nu}\in C(\partial_0D^2;\mathbb{R})$ for $1\leq \mu, \nu\leq m-1$ ($m\geq2$), and let $\lambda\in\mathbb{R}\setminus\{-1, 0, 1\}$. Let $W(z)$ and $u_0(z)$ be analytic functions on $D^2$ and

for $\tau\in\partial D^2$, where $\widetilde{\phi}$ is determined in Lemma 2.4 and $\partial_{\zeta_1}\partial_{\zeta_2}u_1(\zeta) = u_0(\zeta)$. Then

and

Proof. As $W(z)$ is analytic, applying the properties of the Poisson kernel on $D^2$, $W(z)$ can be expressed as

if and only if

(2.6) and (2.9) derive the expression of $W$:

which is due to

and

for $z\in D^2$.

Similarly, by (2.6) and (2.10), we obtain

Therefore,

which is in virtue of

and

Taking the partial derivative on both sides of Eq (2.11) with respect to $\bar{z}_1\bar{z}_2$ gives

Thus,

Particularly,

which follows

(2.13) and (2.14) derive

Plugging (2.15) into (2.12), we obtain the expression of $u_0(z)$, i.e., (2.8).

3.   Dirichlet boundary value problems

Theorem 3.1. Let $\varphi\in C(\partial_0D^2;\mathbb{C})$ and $g_{\mu\nu}\in C(\partial_0D^2;\mathbb{R})$ for $1\leq \mu, \nu\leq m-1$ ($m\geq2$), and let $\lambda\in\mathbb{R}\setminus\{-1, 0, 1\}$. Then the problem

with the conditions

for $1\leq \mu, \nu\leq m-1$ has a unique solution

where $\widetilde{\phi}$ is determined by (2.1) and

Proof. (1) Applying Lemma 2.1,

which means

is a special solution to

Therefore, by Lemma 2.4, the solution of the problem is

where $\widetilde{\phi}$ is determined in Lemma 2.4, $W$ and $u_0$ are analytic functions on $D^2$ to be determined, and $f = \varphi$.

Applying Lemma 2.7 and plugging (2.2) and (2.3) into (3.3), we obtain

which leads to

Considering the boundary condition $f = \varphi$ and applying Lemma 2.8, we obtain that $W(z)$ and $u_0(z)$ are determined by (2.7) and (2.8), respectively. Therefore, we have

and

In addition, (2.11) gives

in which

are used.

Plugging (2.7) and (3.8) into (3.4), $f(z)$ is determined as

where $u_0(z), u_0(0, \!z_2), u_0(z_1, \!0), u_0(0)$ are defined in (2.8), (2.15), (3.6), and (3.7), and the last equation is due to

(2) In the following, we verify that (3.1) is the solution to the problem.

(ⅰ) For $z\in \partial_0 D^2$ (i.e., $|z_1| = |z_2| = 1$), applying the properties of the Poisson kernel on $D^2$, (3.1) satisfies the boundary condition $f = \varphi$ obviously. In addition, applying Lemma 2.1, by (2.12) and (3.1), we obtain

(ⅱ) By Lemma 2.4, $\phi(z) = \widetilde{\phi}(z)+u_0(z)$ satisfies $\partial^{m}_{\bar{z}_1}\partial^{m}_{\bar{z}_2}\phi(z) = 0 \; (z\in D^2)$ and

for $1\leq \mu, \nu\leq m-1$.

From the above analysis in (1) and (2), and by the expression of $f$ in (3.1) and the uniqueness of $u_0 (z)$ in (3.2), it is shown that (3.1) is the unique solution of the problem.

Theorem 3.2. Let $\varphi\in C(\partial_0D^2;\mathbb{C})$ and $h_{\mu_1\nu_1}, g_{\mu_2\nu_2}\in C(\partial_0D^2;\mathbb{R})$ for $0\leq \mu_1, \nu_1\leq n\!-\!2$, $1\leq \mu_2, \nu_2\leq m\!-\!1$ ($m, n\geq2$), and let $\lambda\in\mathbb{R}\setminus\{-1, 0, 1\}$. Then

is the solution to the problem

with the conditions

where $F(z)$ is determined by (3.1) (in which $g_{\mu\nu}$ is replaced by $g_{\mu_2\nu_2}$), $\widetilde{\phi}(z)$ is determined by (2.1) (in which $g_{\mu\nu}$ is replaced by $h_{\mu_1\nu_1}$ and $m$ is replaced by $n-1$), and

with $T_{0, D_i}F = F$ and $n\in\mathbb{N}, \; i = 1, 2$.

Proof. Let $F(z) = \partial^{n-1}_{\bar{z}_1}\partial^{n-1}_{\bar{z}_2}f(z)$. Then, the problem is transformed to be

with the conditions

By Theorem 3.1, $F(z)$ determined by (3.1) (where $g_{\mu\nu}$ is replaced by $g_{\mu_2\nu_2}$) is the unique solution to

For $T_{n, D_i}F$ defined by (3.10); applying Lemma 2.3,

Therefore, $T_{n-1, D_1}(T_{n-1, D_2}F)$ is a special solution to $\partial^{n-1}_{\bar{z}_1}\partial^{n-1}_{\bar{z}_2}f = F$, and the solution to

is

where $\widetilde{\phi}(z)$ is a $n-1$-holomorphic function on $D^2$ satisfying

By Lemma 2.4, $\widetilde{\phi}(z)$ is determined by (2.1), in which $g_{\mu\nu}$ and $m$ are replaced by $h_{\mu_1\nu_1}$ and $n-1$, respectively.

4.   Neumann boundary value problems

In this section, we discuss systems of complex partial differential equations with Neumann boundary conditions on the bicylinder. Let $\partial \nu_i f(z) = z_i\partial z_if(z)+\bar{z_i}\partial\bar{z_i}f(z)$ denote the directional derivative of $f(z)$ in relation to the outer normal vector, where $i = 1, 2$.

Theorem 4.1. Let $g_{k_1 k_2}\in C(\partial_0D^2)$ and $c_{k_1 k_2}\in\mathbb{C}$. Let $b_{k_1 k_2}(z_1)$ and $d_{k_1 k_2}(z_2)$ be analytic functions about $z_1$ and $z_2$, respectively, with $b_{k_1 k_2}(0) = d_{k_1 k_2}(0) = c_{k_1 k_2}$ ($1\leq k_1, k_2\leq m-1$, $m\geq2$). Then, there exists an analytic function $u_{(m-k_1)(m-k_2)}(z)$ on $D^2$:

that satisfies

if and only if

for $\forall z_1\in D_1$ ($z_2\in\partial D_2$), and

for $\forall z_2\in D_2$, $z_1\in D_1$.

Proof. From (4.2), we obtain

for $z_1\in \partial D_1$ (at the same time $z_2\in \partial D_2$). Applying Lemma 2.5 on (4.5) for $z_1\in \partial D_1$, there exists an analytic function about $z_1$, i.e., $\partial_{\nu_2}\Big[\sum\limits^{k_2-1}_{l_2 = 0}\frac{\bar{z}_2^{l_2}}{l_2!} u_{(m-k_1)(l_2+m-k_2)}(z)\Big]$, satisfying (4.5) if and only if (4.3) is satisfied for $\forall z_1\in D_1$ ($z_2\in\partial D_2$). Besides that, $\partial_{\nu_2}\Big[\sum\limits^{k_2-1}_{l_2 = 0}\frac{\bar{z}_2^{l_2}}{l_2!} u_{(m-k_1)(l_2+m-k_2)}(z)\Big]$ is determined by

that is,

Further, applying Lemma 2.5 on (4.6) for $z_2\in \partial D_2$, there exists an analytic function about $z_2$, i.e., $u_{(m-k_1)(m-k_2)}(z)$, satisfying (4.6) and

if and only if (4.4) is satisfied for $\forall z_2\in D_2$ (at the same time $z_1\in D_1$). Moreover, $u_{(m-k_1)(m-k_2)}(z)$ is determined by

which leads to (4.1).

Since $\partial_{\nu_2}\Big[\sum\limits^{k_2-1}_{l_2 = 0}\frac{\bar{z}_2^{l_2}}{l_2!} u_{(m-k_1)(l_2+m-k_2)}(z)\Big]$ is analytic about $z_1$ and $u_{(m-k_1)(m-k_2)}(z)$ is analytic about $z_2$, then $u_{(m-k_1)(m-k_2)}(z)$ is analytic on $D^2$. Therefore, there exists an analytic function $u_{(m-k_1)(m-k_2)}(z)$ on $D^2$, determined by (4.1), satisfying (4.2) on the conditions of (4.3) and (4.4).

Theorem 4.2. Let $\varphi, g_{\mu\nu}\in C(\partial_0D^2)$ and $c_{\mu\nu}\in\mathbb{C}$. Let $b_{\mu\nu}(z_1)$ and $d_{\mu\nu}(z_2)$ be analytic functions about $z_1$ and $z_2$, respectively, with $b_{\mu\nu}(0) = d_{\mu\nu}(0) = c_{\mu\nu}$ ($1\leq\mu, \nu\leq m-1, \; \; m\geq2$). Let $\lambda\in\mathbb{R}\setminus\{-1, 0, 1\}$. Then the problem

with the conditions

for $1\leq \mu, \nu, k_1, k_2\leq m-1$, has a unique solution if and only if (4.3) and (4.4) are satisfied. The solution is given by (3.1), where $u_0$ is determined by (3.2), and

in which $u_{l_1l_2}(z)$ is determined by (4.1).

Proof. (4.8) follows that

is equivalent to (4.2). By Theorem 4.1 and similar to the proof of Theorem 3.1, we obtain the desired conclusion.

Theorem 4.3. Let $g\in L_1(D^2)$ and $\varphi_1, \varphi_2\in C(\partial D^2)$. Then the problem

with the compatibility condition

has a unique solution

if and only if

and

Proof. (4.9) follows $\partial_{\nu_1}(\partial_{\nu_2}f(z)) = \varphi(z)$, that is

which is equivalent to

Applying Lemma 2.6 on the problem

we get that the unique solution $\partial_{\bar{z}_2}f(z)$ is determined by

if and only if (4.11) is satisfied.

In addition, (4.13) is equivalent to

In view of

applying Lemma 2.6, we obtain the unique solution of the problem (4.14) with the conditions (4.15) is (4.10) if and only if (4.12) is satisfied. So we get the desired conclusion.

Theorem 4.4. Let $\varphi, g_{\mu\nu}\in C(\partial_0D^2)$ and $c_{\mu\nu}\in\mathbb{C}$. Let $b_{\mu\nu}(z_1)$ and $d_{\mu\nu}(z_2)$ be analytic functions about $z_1$ and $z_2$, respectively, with $b_{\mu\nu}(0) = d_{\mu\nu}(0) = c_{\mu\nu}$ ($1\leq\mu, \nu\leq m-1, \; \; m\geq2$). Let $\lambda\in\mathbb{R}\setminus\{-1, 0, 1\}$. Then the problem

under the conditions

with $1\leq \mu, \nu, k_1, k_2\leq m-1$ and

has a unique solution if and only if (4.3), (4.4) and

and

are satisfied. The solution is given by (4.10) where $g(\zeta)$ is replaced by $\frac{\lambda-1}{4\lambda}\phi(\zeta)+\frac{\lambda+1}{4\lambda}\bar{\phi}(\zeta)$, and

in which $u_{l_1l_2}(z)$ is determined by (4.1).

Proof. Applying Theorems 4.1 and 4.3, the problem (4.16) with conditions (4.17) has a unique solution (4.10) if and only if (4.3), (4.4), (4.11), and (4.12) are satisfied, where $g(\zeta)$ is replaced by $\frac{\lambda-1}{4\lambda}\phi(\zeta)+\frac{\lambda+1}{4\lambda}\bar{\phi}(\zeta)$. To obtain the specific representations of the solvable conditions, we need the following equations.

By the Gauss formula, we have

which follows

In addition,

follows

Similarly, we obtain

Using (4.21) and replacing $g(\zeta_1, z_2)$ in (4.11) by $\frac{\lambda-1}{4\lambda}\phi(\zeta_1, z_2)+\frac{\lambda+1}{4\lambda}\bar{\phi}(\zeta_1, z_2)$, we get (4.18). Using (4.20)–(4.22) and replacing $g(\zeta)$ in (4.12) by $\frac{\lambda-1}{4\lambda}\phi(\zeta)+\frac{\lambda+1}{4\lambda}\bar{\phi}(\zeta)$, we get

which leads to (4.19).

Remark 4.1. Dirichlet problems and Neumann problems are two typical types of boundary value problems. The conclusions obtained in this paper have enriched the research on boundary value problems for bi-polyanalytic functions. With the methods used in this paper, we can discuss other complex partial differential equation problems for bi-polyanalytic functions. For example, it would be interesting to discuss more complex mixed boundary value problems for bi-polyanalytic functions that simultaneously satisfy multiple boundary conditions, such as Schwarz boundary conditions, Riemann-Hilbert boundary conditions, Neumann boundary conditions and other boundary conditions. However, the corresponding boundary value problems of polyanalytic functions need to be investigated first. Besides that, with the methods used in this paper, we can also solve some complex partial differential equation problems (homogeneous or non-homogeneous) in higher-dimensional complex spaces.

5.   Conclusions

We first discuss a kind of boundary value problem for polyanalytic functions with Schwarz conditions on the bicylinder. On this basis, with the help of the properties of the singular integral operators as well as the Cauchy-Pompeiu formula on the unit disc, we investigate a type of boundary value problem with Dirichlet boundary conditions and a type of mixed boundary value problems of higher order for bi-polyanalytic functions on the bicylinder and obtain the specific representations of the solutions. In addition, we discuss a system of complex partial differential equations with respect to polyanalytic functions with Neumann boundary conditions. On this foundation, we obtain the solutions to Neumann boundary value problems for bi-polyanalytic functions on the bicylinder. The conclusions in this paper provide effective methods for discussing other boundary value problems of inhomogeneous complex partial differential equations of higher order in spaces of several complex variables.

Author Contributions

Yanyan Cui: Conceptualization, Project administration, Writing original draft, Writing–review and editing; Chaojun Wang: Investigation, Writing original draft, Writing–review and editing. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the NSF of China (No. 11601543), the NSF of Henan Province (No. 222300420397), and the Science and Technology Research Projects of Henan Provincial Education Department (No. 19B110016).

Conflicts of interest

The authors declare that they have no conflicts of interest.