Generalization of some overdetermined systems of complex partial differential equations

Theorem 2.1

The necessary and sufficient condition for the existence of a non-vanished solution of the system (2) is:

$$\frac{\partial A_{\mathit{js}}}{\partial {\overline{z}}_{\mathit{ks}}}=\frac{\partial A_{\mathit{ks}}}{\partial {\overline{z}}_{\mathit{js}}}\text{,}s\text{,}j\text{,}k=1\text{,}2\text{,}\dots \text{,}n$$

Proof

Let $W(z_{1s}\text{,}{z}_{2s}\text{,}\dots \text{,}{z}_{\mathit{ns}})=W(z_s)\in c^2(G)$ be a non-vanished solution of (2) then

$$\frac{{\partial }^{2}W}{\partial {\overline{z}}_{\mathit{js}}\partial {\overline{z}}_{\mathit{ks}}}=\frac{\partial A_{\mathit{js}}}{\partial {\overline{z}}_{\mathit{ks}}}W+A_{\mathit{js}}{A}_{\mathit{ks}}W$$ and

(5)

$$\frac{{\partial }^{2}W}{\partial {\overline{z}}_{\mathit{ks}}\partial {\overline{z}}_{\mathit{js}}}=\frac{\partial A_{\mathit{ks}}}{\partial {\overline{z}}_{\mathit{js}}}W+A_{\mathit{ks}}{A}_{\mathit{js}}W$$ Since $W\ne 0$ , by comparing (4) and (5) we have (3).

Now suppose that (3) holds. Consider the system:

$$\frac{\partial W}{\partial \overline{z_{\mathit{js}}}}=A_{\mathit{js}}(z_{1s}\text{,}{z}_{2s}\text{,}\dots \text{,}{z}_{\mathit{ns}})=A_{\mathit{js}}(z_s)s\text{,}j=1\text{,}2\text{,}\dots \text{,}n$$ in G. Since G is a domain of holomorphy, this system is solvable in $c^{\infty }(G)$ (see Tutschke, 1977). Take a fixed solution $w_0(z_s)\in c^{\infty }(G)\text{,}{z}_s=(z_{1s}\text{,}{z}_{2s}\text{,}\dots \text{,}{z}_{\mathit{ns}})$ of (6) and consider the function:

(7)

$$W(z_s)={\mathit{ce}}^{-w_0(z_s)}\text{;}c\ne 0\text{,}s=1\text{,}2\text{,}\dots \text{,}n$$ Then we get a solution $W(z_s)\ne 0$ for all $z_s\in G$ And we suppose that (6) always holds in sequel. Let $\Omega$ be an open subset of G and $W\in c^{\infty }(\Omega )$ be a solution of (2). Set:

(8)

$$\Phi (z_s)=W(z_s)e^{-w_0(z_s)}\text{,}s=1\text{,}2\text{,}\dots \text{,}n$$ where $w_0(z_s)$ is a solution of (6) as above. It may easily be verified that $\Phi$ is analytic in $\Omega$ . From (8) it follows:

(9)

$$W(z_s)=\Phi (z_s)e^{w_0(z_s)}\text{,}s=1\text{,}2\text{,}\dots \text{,}n$$ Conversely, let $\Phi$ be a given analytic function in $\Omega$ , then it follows immediately that the function $W(z_s)$ defined by formula (9) is a $c^{\infty }$ -solution of (2) in $\Omega$ . Thus we obtain. □

Theorem 2.2

There is a “1-1” correspondence between the set $\nu (\Omega )$ of all $c^{\infty }$ -solutions (of system (2)) and the set $\hslash (\Omega )$ of analytic functions in $\Omega$ . Now let $$D=D_1\times D_2\times \cdots \times D_n\subset G$$ be a polycylinder, where $D_j$ , are domains in $C(z_{\mathit{js}})$ with piece-smooth boundaries and $$W(z_s)\in \nu (D)\cap c(\bar{D})\text{,}s=1\text{,}2\text{,}\dots \text{,}n$$ By applying the Cauchy integral Formula we have

$$\begin{array}{cc}\hfill & \Phi (z_s)=\frac{1}{{(2\pi i)}^n}{\int }_{\partial D_1}{\int }_{\partial D_2}\cdots {\int }_{\partial D_n}\frac{\Phi ({\xi }_1\cdots {\xi }_n)}{({\xi }_1-z_{1s})\cdots ({\xi }_n-z_{\mathit{ns}})}d{\xi }_1\cdots d{\xi }_n\text{,}\hfill \\ \hfill & ss=1\text{,}2\text{,}\dots \text{,}n\hfill \end{array}$$ $$z_s=(z_{1s}\text{,}{z}_{2s}\text{,}\dots \text{,}{z}_{\mathit{ns}})\text{,}s=1\text{,}2\text{,}\dots \text{,}n$$ From (9) and (10) it follows

(11)

$$W(z_s)={\int }_{\Gamma (D)}F(\xi \text{,}{z}_s)W(\xi )d{\xi }_1\dots d{\xi }_n\text{,}s=1\text{,}2\text{,}\dots \text{,}n$$ For $z_s\in D$ where $$F(\xi \text{,}{z}_s)=\frac{1}{{(2\pi i)}^n}\frac{e^{w_0(z_s)-w_0(\xi )}}{({\xi }_1-z_{1s})\cdots ({\xi }_n-z_{\mathit{ns}})}$$ and $$\Gamma (D)=\partial D_1\times \partial D_2\times \cdots \times \partial D_n$$ Thus we obtain.

Theorem 2.3

If D is a polycylinder in G and $$W(z_s)\in \nu (D)\cap c(\bar{D})\text{,}s=1\text{,}\dots \text{,}n$$

Theorem 2.4

Theorem 2.4 Principle of Maximum Modulus Theorem

If $$W(z_s)\in \nu (D)\cap c(\bar{D})\text{,}s=1\text{,}\dots \text{,}n$$ Then $$|W(z_s)|⩽{\mathit{Mmax}}_{\xi \in \partial D}|W(\xi )|\text{,}s=1\text{,}\dots \text{,}n$$ For $z_s\in D$ where M is a positive constant depending only on the coefficient $A_{\mathit{js}}(z_s)$ of the system and on the domain D.

Proof

Let $W_0(z_s)\in c^{\infty }(G)$ be a solution of (4) then $W_0(z_s)\in c(\bar{D})$ . Set $$m_0={\mathit{min}}_{\xi \in \partial D}\mathit{Re}|w_0(\xi )|$$ and $$M_0={\mathit{max}}_{\xi \in \partial D}\mathit{Re}|w_0(\xi )|$$ From (8) and the Principle of maximum modulus for analytic functions it follows:

$$|\Phi (z_s)|=|W(z_s)|e^{-{\mathit{Rew}}_0(z_s)}\text{,}s=1\text{,}\dots \text{,}n$$ and

(13)

$$|\Phi (z_s)|⩽{\mathit{max}}_{\xi \in \partial D}|\Phi (\xi )|={\mathit{max}}_{\xi \in \partial D}|W(\xi )|e^{-{\mathit{Rew}}_0(\xi )}⩽e^{-m_0}{\mathit{max}}_{\xi \in \partial D}|W(\xi )|$$ By comparing (12) and (13) we have $$|W(z_s)|⩽e^{-m_0}{e}^{{\mathit{Rew}}_0(z_s)}{\mathit{max}}_{\xi \in \partial D}|W(\xi )|⩽e^{M_0-m_0}{\mathit{max}}_{\xi \in \partial D}|W(\xi )|=M·{\mathit{max}}_{\xi \in \partial D}|W(\xi )|$$ where $M=e^{M_0-m_0}$ . By definition, M depends only on $A_{\mathit{js}}(z_s)$ and on D. This completes the proof. □

Theorem 2.5

Theorem 2.5 Liouvllle’s Theorem

If a generalized analytic function $W(z_s)\text{,}s=1\text{,}2\text{,}\dots \text{,}n$ is continuous, bounded in $C^n$ and vanishes at a point $z_0\in c^n$ (in particular it may occur that $z_0=\infty$ ), then $W(z_s)=0$ everywhere.

Proof

From (6) it follows that $\Phi (z_s)$ is analytic, bounded in $C_n$ and vanishes at $z_0$ . By virtue of Liouvilie’s Theorem in Complex Analysis we have $\Phi (z_s)=0$ . Take the equality (9) into account we have $W(z_s)\equiv 0$ everywhere. If $W\in \nu (C_n)$ and bounded in $c^n$ , then because of (8) we have $\Phi (z_s)=c$ const. Hence it follows: □

Theorem 2.6

Every continuous and bounded generalized analytic function $W(z_s)$ in the whole of $C^n$ has the form: $$W(z_s)=c·e^{w_0(z_s)}\text{;}c=\mathit{const}\text{.}$$ where $w_0(z_s)$ is a solution of (6).