On an Unboundedness Property of Solutions of Elliptic Systems in the Plane

Abstract

The issue of the invariance of the unboundedness property of the solutions of the Carleman–Bers–Vekua system (generalized analytic functions) with respect to the transformation of the restriction is studied. The concept of the rating of an unbounded continuous function is introduced. A continuous unbounded function of zero rating is constructed, whose restriction to every strip of the plane is bounded. For entire and generalized entire functions of finite rating, rays are effectively constructed, along which the function is unbounded. It is shown that there exists an entire analytic generalized function of infinite rating that is bounded on every ray. The obtained results, in a somewhat modified form, allow for extension to sufficiently wide classes of elliptic systems on the complex plane.

1. Introduction

In this section, we will introduce all the necessary definitions and results from the theory of generalized analytic functions that we will use in the following sections. Generalized analytic functions, according to Vekua [1], and pseudoanalytic functions, according to Bers [2], emerged as a further extension of the theory of analytic functions in the middle of the last century, and in the following years, this theory played an important role in solving many problems in pure mathematics, in mechanics and in mathematical physics [3,4].

Suppose $F(x,y)$ and $G(x,y)$ is a pair of functions defined on the domain U of the complex plane $\mathbb{C},z=x+iy$ possessing Hölder continuous partial derivatives with respect to real variables x and $y.$ We call $F,G$ a generating pair if it satisfies the inequality

$$Im\left(\overline{F}G\right)>0$$

on $U,$ where $Im$ denotes imaginary part of $\overline{F}G$ and $\overline{F}$ is a complex conjugate of $F.$

The generating pair is independent in some sense if at any point $z_0\in U$, the value of any complex function w defined at $z_0$ can be represented as a real linear combination of F and G:

$$w\left(z_0\right)={\lambda }_{0}F\left(z_0\right)+{\mu }_{0}G\left(z_0\right).$$

(1)

We say that at the point $z_0\in U$, the function w possesses the $(F,G)$ -derivative $\dot{w}\left(z_0\right)$ if the finite limit

$$\dot{w}\left(z_0\right)=\underset{z\to z_0}{\lim }{\displaystyle \frac{w\left(z\right)-{\lambda }_{0}F\left(z\right)-{\mu }_{0}G\left(z\right)}{z-z_0}}$$

(2)

exists.

The functions

$$a_{(F,G)}=-{\displaystyle \frac{\overline{F}{G}_{\overline{z}}-F_{\overline{z}}\overline{G}}{F\overline{G}-\overline{F}G}},b_{(F,G)}={\displaystyle \frac{F{G}_{\overline{z}}-F_{\overline{z}}G}{F\overline{G}-\overline{F}G}}$$

(3)

are called the characteristic coefficients corresponding to the pair $(F,G).$ Here, $F_z,$$F_{\overline{z}},$$G_z,$$G_{\overline{z}}$ denote partial derivatives with respect to the variables z and $\overline{z}.$

If $\dot{w}\left(z_0\right)$ exists, then $w_z$ and $w_{\overline{z}}$ exist at $z_0$ and the equation

$$w_{\overline{z}}=a_{(F,G)}w+b_{(F,G)}\overline{w}$$

(4)

holds. On the contrary, if $w_z$ and $w_{\overline{z}}$ exist and are continuous in some neighborhood of $z_0$, and if (4) holds at $z_0,$ then $\dot{w}\left(z_0\right)$ exists.

Equation (4) is called a Carleman–Bers–Vekua equation. It is a complex form of a general elliptic equation on a complex plane. There is a one-to-one correspondence between the generating pairs and characteristic coefficients. For this reason, the Carlman–Bers–Vekua equation is often written without specifying the generating pairs:

$$w_{\overline{z}}=aw+b\overline{w}.$$

(5)

Moreover, if we take constants as a generating pair, e.g., 1 and i, we obtain the classical functions of a complex variable, and the condition of analyticity reduces to the Cauchy–Riemann equation, which is derived from Equation (5) when $a=0$ and $b=0.$

In what follows, we mean that the coefficients of the Equation (5) satisfy the regularity condition; that is, they belong to the special class $L_{p,2}\left(\mathbb{C}\right),$ for the number $p>2,$ i.e., the function $f\in L_{p,2}\left(\mathbb{C}\right),$ if the following two integrals satisfy the conditions

$$\int {\int }_G{\left|f\left(z\right)\right|}^{p}dz<\infty ,\int {\int }_G{\left|{\displaystyle \frac{1}{z}}f\left({\displaystyle \frac{1}{z}}\right)\right|}^{p}dz<\infty ,G=\left\{\right|z|\le 1\}.$$

(6)

The solution of Equation (5) is understood as a continuous generalized solution over the whole complex plane $\mathbb{C}$ (see [3,5]).

A complex function of the form (1) that satisfies condition (2) is called a pseudo-analytic function [2]. The solutions of Equation (5) are called generalized analytic functions [1]. A class of functions is pseudoanalytic if and only if it consists of generalized analytic functions. From this follows the equivalence of pseudoanalytic and generalized analytic functions.

Numerous properties of analytic functions (with some specific modifications) can be extended to generalized analytic functions (see [1,2,3]). For example, Liouville’s theorem for generalized analytic functions, which we will use below, claims that if a generalized analytic function is bounded and equal to zero at a fixed point, then this function is identically zero everywhere. In this paper, we find yet another property of analytic functions, which also extends to the class of generalized analytic functions. Furthermore, we show that this property is characteristic of analytic functions and not of continuous or differentiable functions. More exactly, we study the influence of a function’s local properties on its global properties, and vice versa.

2. Statement of the Problem: The Case of Continuous Function

Let $W=W\left(z\right)$ be a complex function defined on the entire complex plane $\mathbb{C},$ and let U be some part of the plane; let $W_U$ denote the restriction of the function W on the set U:

$$\left(W_U\right)\left(z\right)=W\left(z\right),z\in U.$$

We will study the influence of the properties of the function W on $W_U.$ We will answer the following question: does the boundedness of $W_U$ imply the boundedness of W?

Clearly, if U is a half-plane, its complement, or both, then $W_U$ is unbounded. For other types of subsets U of $\mathbb{C}$, the boundedness (unboundedness) problem of $W_U$ is non-trivial. Below, we provide an analysis of this problem for a special type of domain.

Consider any quadruple of real numbers

$$(A,B,D,K)$$

(7)

satisfying the condition

$$\left|A\right|+\left|B\right|\ne 0.$$

(8)

Let us construct the lines on the complex plane $\mathbb{C}$ using the quadruple of real numbers (7)

$$S_1=\{z=x+iy:Ax+By+K=0\},S_2=\{z=x+iy:Ax+By+D=0\}.$$

(9)

It is evident that the lines (9) are either parallel or coincident. Denote by

$$ST(A,B,D,K)$$

the part of the plane $\mathbb{C}$ that is enclosed between the lines (9), and we shall refer to this as a strip. When $D=K$, then $ST(A,B,D,K)$ degenerates into the line $S_1=S_2.$

Theorem 1.

There exists a continuous unbounded function $W_{\ast }=W_{\ast }\left(z\right)$ defined on the entire complex plane, which has no boundedness strip.

Proof of Theorem 1. 

Consider a system of points on complex plane $\mathbb{C}$

$$\{z_1,z_2,z_3,\dots \}$$

(10)

where

$$z_k=x_k+i{y}_k,x_k=k,y_k=k^2,k\in \mathbb{N}.$$

Using the system of points (10), we construct a system of circles

$$\{D_1,D_2,D_3,\dots \},$$

(11)

where $D_k=\{z:z\in \mathbb{C},|z-z_k| \le 1\},k\in \mathbb{N}$ and the function $W_{\ast }$ is defined in the following manner:

$$\left\{\begin{array}{c}{W}_{\ast }\left(z\right)=0,z\in \mathbb{C}\setminus {\cup }_{k=1}^{\infty }{D}_k,\hfill \\ W_{\ast }\left(z\right)=1-|z-z_1|,z\in D_1,\hfill \\ W_{\ast }\left(z\right)=2-2|z-z_2|,z\in D_2,\hfill \\ W_{\ast }\left(z\right)=3-3|z-z_3|,z\in D_3,\hfill \\ \dots .\hfill \end{array}\right.$$

(12)

It is clear that the constructed function $W_{\ast }=W_{\ast }\left(z\right)$ is continuous and unbounded on the entire complex plane $\mathbb{C}.$ It is also clear that any strip $ST(A,B,D,K)$ has a non-empty intersection with a finite number through circles in the system (11).

Based on all the above information, it follows that the restriction of the function $W_{\ast }$ constructed by Formulas (12) is bounded in each strip $ST(A,B,D,K).$ □

Remark 1.

In fact, we proved that for any strip $ST(A,B,D,K)$, the function $W_{\ast ST(A,B,D,K)}$ is bounded.

A consequence of Theorem 1 is as follows:

Corollary 1.

The local boundedness (boundedness in each strip of the plane) of a continuous complex function defined on the entire complex plane $\mathbb{C}$ does not imply the global boundedness of the function (boundedness over the entire plane).

The continuous function $W_{\ast }$ constructed in the proof of Theorem 1 is not smooth (differentiable) at any point of the subset of the complex plane $\mathbb{C}$:

$$\{z_1,z_2,z_3,\dots \}\cup \{z:|z-z_1|=1\}\cup \{z:|z-z_2|=2\}\cup \{z:|z-z_3|=3\}\cup \dots$$

It is clear that by “smoothing” this function we obtain a function with similar properties of the class $C^{\infty }$. Also, it is not difficult to prove that this type of function exists. As we will see below, the situation is completely different when we deal with generalized entire analytic functions (particularly classical entire functions).

3. The Case of Entire and Generalized Entire Functions

Below $\mathcal{A}$ is the class of all possible entire analytic functions. One of the current directions of contemporary mathematical analysis is a systematic study of this class of functions; it originates from the second half of the 19th century in the classical works of Weierstrass, Mittag-Leffleur, Picard, and others. For a modern overview of the fundamental properties of the meromorphic and entire functions and related current problems, see [6]. One of the most important results of the theory of entire functions is the classical Liouville theorem, according to which every entire function $f\left(z\right)$ is either constant or unbounded. Moreover, if the rate of growth of the entire function $f\left(z\right)$ in the neighborhood of infinity does not exceed the rate of growth of the function ${\left|z\right|}^N,$ i.e.,

$$f\left(z\right)=𝒪\left(\right|z{|}^N),z\to \infty ,$$

for some non-negative integer $N,$ then the function $f\left(z\right)$ is a polynomial whose order does not exceed $N.$

There are very few non-trivial properties that every member of this class obeys. Therefore, it is quite natural to consider subclasses (based on certain criteria) and to study these individually rather than to study the class $\mathcal{A}$ as a whole. Such criteria include the characteristic parameters of the growth in the value of the entire function $f\left(z\right)$ when the modulus of its argument $\left|z\right|\to \infty$ can be used. To obtain these parameters, fix a function $h\left(z\right)$, which is unbounded in the neighborhood of infinity and is simple in some sense, and take a subclass of functions of the class $\mathcal{A}$, with the rate of growth not exceeding the growth of $h\left(z\right)$. By virtue of the classical theorem of Liouville, function $h\left(z\right)={\left|z\right|}^N$, is of no interest from this point of view. Using this point of view, the function with exponential growth

$$h\left(z\right)=\exp \left\{\right|z{|}^{\delta }\}$$

is very interesting and important. Using $\delta \in \mathbb{R}$, the subclasses of the whole class $\mathcal{A}$ of the entire analytic functions $f\left(z\right)$ can be selected. These subclasses appeared in Hadamard’s work [7], most likely for the first time. A classification of all functions according to the exponential growth in the neighborhood of infinity is provided in the mentioned work. Hadamar’ s classification of all analytic functions is the basis of research on complex analysis. A similar classification of generalized analytic functions is provided in this work.

Suppose that $W=W\left(z\right),$ as above, is the continuous complex function of a complex variable defined on the whole complex plane $\mathbb{C}.$ Use the classical scheme (see e.g., [5,7]) and assign a non-negative real number $R\left(W\right)$ or symbol $+\infty$ to this function, as follows: if for any positive number $\delta \in \mathbb{R}$ there exists a set $U\subset \mathbb{C}$ such that

$$\sup \left\{\right|W\left(z\right)\left|\exp \right(-{\left|z\right|}^{\delta }\left)\right\}:z\in U\}<+\infty$$

(13)

then we write

$$R\left(W\right)=+\infty .$$

(14)

In such a case, we will say that the function $W=W\left(z\right)$ has an infinite exponential rating in the neighborhood of infinity (in short, it has an infinite rating). If (13) and (14) do not take place, then it is obvious that there exists a non-negative number $\delta \in \mathbb{R}$ for the function $W=W\left(z\right)$, fulfilling the following condition:

$$W\left(z\right)={𝒪\left(\exp \right\{\left|z\right|}^{\delta }\left\}\right),z\to \infty .$$

(15)

For function $W=W\left(z\right)$ with the finite rating, the number $\delta$ satisfying condition (15) is not uniquely defined; for every such number $\delta$, condition (15) is satisfied by any number greater than this number (and possibly by some numbers lower than this one).

Let us denote the set of all such possible numbers using ${\Delta }_W$, and let us call the non-negative real number

$$\rho =\inf {\Delta }_W$$

(16)

an exponential rating of a function in the neighborhood of infinity (i.e., the rating).

We call the sequence of the points of the plane $\mathbb{C}$

$$z_1,z_2,z_3,\dots ,z_n,\dots$$

(17)

the first type of canonical sequence if

$$\underset{n\to \infty }{lim}{z}_n=\infty ,\underset{n\to \infty }{lim}W\left(z_n\right)=\infty .$$

(18)

Sequence (18) is called the second type of canonical sequence if each of its terms is nonzero, ${lim}_{n\to \infty }{z}_n=\infty$ and for every nonnegative real number $\sigma \ge 0$,

$$\underset{n\to \infty }{lim}{\displaystyle \frac{W\left(z_n\right)}{|z_n{|}^{\sigma }}}=\infty .$$

Denote, using $\Gamma (z_0,\phi )$, a ray lying on the complex plane $\mathbb{C}$: the origin of this ray is at point $z_0\in \mathbb{C}$ and the direction is determined using the real number $\phi \in [0,2\pi )$

$${\Gamma }_{z_0,\phi }=\{z:z_0+r(cos\phi +isin\phi ),r\ge 0\}.$$

We call ray $\Gamma (z_0,\phi )$ the first-type canonical (the second-type canonical) ray of the continuous function $W=W\left(z\right).$ If the first-type canonical (the second-type canonical) sequence (18) of function W is lying on this ray, then

$$arg(z_n-z_0)=\phi ,n=1,2,3,\dots .$$

Obviously, every continuous unbounded function has the first type canonical sequence (and vice versa); studying the existence of the first- and especially the second-type canonical rays of such functions and their distribution on the plane is an interesting problem. In this paper we will not consider all possible classes of continuous functions, but we will limit ourselves to its most important subclass—the class of generalized entire analytic functions.

Let us denote the class of all possible solutions of system (5) using $A(a,b).$ Let $A_{\rho }(a,b)$ denote all elements of this class with the rating $\rho \in [0,+\infty ].$

The class of generalized analytic functions $A(a,b)$ can be represented as a disjunctive union:

$$A(a,b)={\cup }_{0\le \rho \le +\infty }{A}_{\rho }(a,b);$$

furthermore, for every non-negative number $\rho \in \mathbb{R}$, we have

$$\mathrm{card}{A}_{\rho }(a,b)=\mathrm{card}{A}_{\infty }(a,b)=\mathrm{card}\mathbb{C}.$$

Theorem 2.

Let function $W\in A_{\rho }(a,b),$ where the rating ρ from definition (16) satisfies the inequality $0<\rho <+\infty$ and the complex plane $\mathbb{C}$ is divided into equal angles by the system of the rays:

$${\Gamma }_{z_0,{\phi }_1},{\Gamma }_{z_0,{\phi }_2},\dots ,{\Gamma }_{z_0,{\phi }_n},$$

(19)

$z_0\in \mathbb{C},$$n\ne 2.$ Furthermore, assume

$$2\rho <n.$$

(20)

Then, in system (19), there is (at least one) of the second-type canonical ray of the function $W=W\left(z\right).$

Proof of Theorem 2. 

We represent the function in the form

$$W=\Phi exp(\Omega ),$$

(21)

where $\Phi$ is the entire analytic function and

$$\Omega \left(z\right)={\displaystyle \frac{1}{\pi }}\int {\int }_{\mathbb{C}}\left(a\left(\zeta \right)+b\left(\zeta \right){\displaystyle \frac{\overline{W\left(\zeta \right)}}{W\left(\zeta \right)}}\right){\displaystyle \frac{d_{\zeta }\mathbb{C}}{\zeta -z}}$$

(22)

(see [5]). The function $\Omega =\Omega \left(z\right)$ is continuous on the whole complex plane and ${lim}_{z\to \infty }\Omega \left(z\right)=0.$ Assume that none of the rays of the system (19) are the second-type canonical ray of function $W=W\left(z\right).$ Then, we find natural numbers $M_1,M_2$, and $N,$ for which

$$|W\left(z\right)=M_1{\left|z\right|}^N,$$

(23)

where $\left|z\right|>M_2,$$z\in {\bigcup }_{k=1}^n{\Gamma }_{z_0,{\phi }_k}.$

Due to the classical Phragmen–Lindelöf principle for analytic functions (see [5], p. 279), condition (23) implies that the function $W=W\left(z\right)$ is a generalized polynomial, which is impossible since the rating of this function is $\rho >0.$ □

Note that if condition (20) is not fulfilled, then the conclusion of Theorem 2 is generally not valid. Indeed, consider all of function

$${\Phi }_1\left(z\right)=sinz,{\Phi }_2\left(z\right)=cosz$$

and construct the corresponding (see Formula (21) generalized entire function $W_1=W_1\left(z\right)$ and $W_2=W_2\left(z\right).$ It is evident that

$$W_1,W_2\in A_1(a,b),$$

rays ${\Gamma }_{0,0}$ and ${\Gamma }_{0,\pi }$ (positive and negative parts of x-axis) are not the second canonical; moreover, they are not even the first-type canonical. Quite similarly, the generalized entire function $W_3$ corresponding to the function ${\Phi }_3=e^z$ is bounded on the ${\Gamma }_{0,\frac{\pi }{2}},$${\Gamma }_{0,\frac{3\pi }{2}}$ rays (the upper and lower rays of the y-axis).

The classes of generalized analytic functions

$$A_{\rho }(a,b),0<\rho <1/2$$

are very interesting. For any $W=W\left(z\right)$ function from these classes, every ray is the second-type canonical. In addition, if the function $W\in A_0(a,b)$ and, additionally, W has $\alpha$-canonical sequence (that is, it is unbounded), then every ray of the plane $\mathbb{C}$ will be the first-type canonical ray of $W=W\left(z\right)$.

In Theorem 2 and the results derived from it, we consider generalized entire functions with a finite rating (see (15)). As we have seen, their canonical rays are sufficiently dense (abundant) in the plane, and as can be seen from the next theorem, this property is essentially a property of functions with a finite rating. A generalized analytic function of infinite rating may not have the second-type canonical; furthermore, it may not have the first-type canonical ray.

As we saw above, the case of continuous (unbounded) functions completely differs from the case of generalized entire analytic functions. Function $W_{\ast }=W_{\ast }\left(z\right),$ constructed above, has no boundedness “strip” at all. However, $|W_{\ast }\left(z\right)|\le |z|$; therefore, its rating is $\rho =0.$

Theorem 3.

For every point $z_0$ of the plane $\mathbb{C}$, there exists $W_{\ast }\in A_{\infty }(a,b)$, such that for every number $\phi \in ,R$ we have

$$\underset{r\to +\infty }{lim}{W}_{\ast }(z_0+r{e}^{i\phi })=0.$$

Proof of Theorem 3. 

Consider the entire Mittag–Leffler function $E\left(z\right),$ which is obtained via analytic extension of the integral

$${\displaystyle \frac{1}{2\pi i}}{\int }_L{\displaystyle \frac{exp(exp(\zeta \left)\right)}{\zeta -z}}d\zeta$$

(see [8]).

Let us construct the entire function using the function $E=E\left(z\right)$ through formula $\Phi \left(z\right)=E\left(z\right)exp(-E(z\left)\right).$ This function satisfies the following condition:

$$\underset{r\to \infty }{lim}\Phi \left(r{e}^{i\phi }\right)=0,$$

This is satisfied via the well-known properties of the Mittag–Leffler entire function for any $\phi \in \mathbb{R}$ (see [9] Problem 184). Introduce the entire function

$${\Phi }_{\ast }\left(z\right)=\Phi (z-z_0)$$

and through this, using (21), construct the generalized entire function $W_{\ast }=W_{\ast }\left(z\right),$ which satisfies the condition of the theorem. □

The exotic nature of the generalized entire analytic function $W_{\ast }\left(z\right)$ mentioned in Theorem 3 is partially reflected by a fragment of the graph of a function

$$\zeta =|W_{\ast }\left(z\right)|,z_0=0$$

in the space ${\mathbb{R}}_{x,y,\zeta }^3$, given in Figure 1.

We should take into account the most important fact that the function $\zeta =|W_{\ast }\left(z\right)|$ is unbounded. The function $W_{\ast }\left(z\right)$ of the mentioned type “lives” only in the class $A_{\infty }(a,b).$ For any number $\rho \in [0,+\infty )$ the class of functions $A_{\rho }(a,b)$ does not contain a function of type $W_{\ast }\left(z\right).$

4. Discussion

The global properties of generalized analytic functions are used to analyse many problems in pure and applied mathematics. The unboundedness property of the pseudoanalytic functions discussed in this article can be used to determine the characteristics of global solutions to equations of mathematical physics. For regular equations, in the case of any functions a and b that satisfy condition (6), all spaces $A(a,b)$ are isomorphic, which is not the case for irregular equations [3]. In the future, we plan to study the property discussed in this article in relation to the solutions of such equations, which will broaden the scope of applications of irregular equations. These equations arise in various types of problems in mathematics and mathematical physics, including differential geometry and mechanics, as well as in the study of solutions to the Beltrami equation that degenerate at a point.