ON AN ELLIPTIC SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS WITH A SINGULAR POINT AT THE ORIGIN

The analytical theory of the Briot-Bouquet equation is well known:

$$\frac{dy}{dx} = f(x, y)$$

where $f(x, y)$ is a function holomorphic in the neighborhood of $x = y = 0$. The purpose of this note is to demonstrate that certain analogous results hold for the equation (here and hereafter, symbols with a bar denote complex conjugate quantities):

$$\bar{z} \frac{\partial u}{\partial \bar{z}} = f(z, \bar{z}, u, \bar{u}) \tag{1}$$

where $f(z, \bar{z}, u, \bar{u})$ is a holomorphic function in the neighborhood of $(0, 0, 0, 0)$ such that $f(0, 0, 0, 0) = 0$. According to \cite{1}, $\frac{\partial}{\partial \bar{z}}$ is the operator derivative, or according to \cite{2}, the generalized derivative. Therefore, equation (1) can be rewritten in the form:

$$\bar{z} \frac{\partial u}{\partial \bar{z}} = f(z, \bar{z}, u, \bar{u}) \tag{2}$$

Linear homogeneous equations of the form (2) have been previously investigated in \cite{3, 4, 5}. In this study, we first address the question of the existence of a solution to equation (2) that is holomorphic with respect to the powers of $z$ and $\bar{z}$ and vanishes at $z = \bar{z} = 0$. Let

$$f(z, \bar{z}, u, \bar{u}) = \sum f_{pqkl} z^p \bar{z}^q u^k \bar{u}^l$$

If the specified solution to equation (2) exists, it can be represented in the form:

$$u = \sum_{m+n=1}^{\infty} a_{mn} z^m \bar{z}^n \tag{3}$$

To determine the coefficients, we have the following equations:

$$\begin{aligned} (n - a) a_{mn} - b \bar{a}_{nm} &= \Phi_{mn} \\ -b a_{mn} + (m - a) \bar{a}_{nm} &= \Phi_{nm} \end{aligned}$$

The terms $\Phi_{mn}$ and $\Phi_{nm}$ are polynomials with respect to those coefficients $a_{m'n'}$ for which $m' + n' < m + n$, $m' \le m$, and $n' \le n$. These polynomials are obtained as the coefficients of the series resulting from the substitution of series (3) into the expression $F(z, \bar{z}, u, \bar{u}) - au - b\bar{u}$. We observe that if the determinants of the systems $\Delta_{mn} = (n - a)(m - a) - b^2$ are non-zero for all non-negative integers $m$ and $n$ satisfying the inequality $m + n > 0$, then a formal solution (3) exists and is unique.

We shall now prove the convergence of series (3) under the assumption that all $\Delta_{mn} \neq 0$. It is easy to see that there exists a positive constant $B$ such that we have

$$\frac{m}{|\Delta_{mn}|} < B, \quad \frac{n}{|\Delta_{mn}|} < B, \quad \frac{1}{|\Delta_{mn}|} < B$$

for all non-negative integers $m + n > 0$. Suppose there exists a majorant for the series $F(z, \bar{z}, u, \bar{u})$. Let us then consider the equation $U = B F(z, \bar{z}, U, \bar{U})$. It is easily seen that this equation possesses a holomorphic solution where all coefficients are positive. To determine these coefficients, we obtain the following equations:

$$A_{mn} = B \Phi_{mn} + B \Phi_{nm}$$

These are polynomials with respect to those indices for which $m' < m$ and $n' < n$. From the convergence of the majorizing series, the convergence of series (3) follows. Thus, we obtain the following:

Theorem

If $(m - a)(n - a) - b^2 \neq 0$ for all non-negative integers $m, n$ such that $m + n > 0$, the equation has a unique holomorphic solution in the neighborhood of the origin.

Suppose now that for certain values of $m$ and $n$, the expression vanishes. If $a$ is a real number and $b=0$, then $n-a$ can be zero for at most one value. If $a$ is a real irrational number, the expression can vanish for only one pair of numbers. Now, let $a$ be a rational number. Then the expression will vanish for an infinite number of pairs $(m, n)$ if $a$ is equal to a non-negative integer. In all other cases, it can vanish only for a finite set of values.

If the determinant vanishes at $(m_0, n_0)$ and is non-zero for all other values, and if the condition $(m_0 - a)a_{m_0 n_0} - b \bar{a}_{n_0 m_0} = \Phi_{m_0 n_0}$ is not satisfied, then equation (2) does not possess a solution of the specified form. If the condition is satisfied, the solution depends on an arbitrary constant.

Regarding the case where multiple pairs of indices cause the determinant to vanish, say $(m_1, n_1), \dots, (m_k, n_k)$, the equation will have a solution (3) if and only if certain consistency conditions are met. If these conditions hold identically with respect to the constants $C_1, \dots, C_k$, then the solution depends on these arbitrary constants.

Finally, let $b = 0$ and $a = s$, where $s$ is a non-negative integer. For the existence of a formal solution (3) to equation (2) in this case, it is necessary and sufficient that $\Phi_{mn} = 0$ for the corresponding indices. If these conditions are satisfied, the solution (3) will depend on an infinite number of arbitrary parameters, and the question of convergence depends on the choice of these parameters.

We now consider equation (2) under the assumption that $f(0, 0, 0, 0) = 0$. We seek a solution in the form:

$$u = \sum c_{mnjk} z^m u^n \bar{z}^j \bar{u}^k \tag{15}$$

To determine the coefficients, we obtain the corresponding equations:

$$\begin{aligned} (a - \alpha) p_{0001} - b p_{0010} &= 0 \\ -b p_{0001} + (\beta - \alpha) p_{0010} &= 0 \end{aligned}$$

If the determinant of this system is zero, we can take $p_{0010} = C_1$ and $p_{0001} = C_2$ as arbitrary constants. To determine the other coefficients $p_{mnjk}$, we obtain:

$$(n + j\beta + k\alpha - a) p_{mnjk} - b \bar{p}_{mkjn} = R_{mnjk}$$

If $j\beta + k\alpha - n\alpha \neq 0$ for all $m, n, j, k$ such that $m + n + j + k \geq 1$ and $j + k > 0$, then all coefficients $p_{mnjk}$ are uniquely determined. Under these conditions, the equation possesses a unique formal solution (15) that depends on two arbitrary constants. The convergence of this series for sufficiently small variables follows from the existence of a majorizing series.

Theorem

Let $\kappa_{mnjk}$ be defined such that the denominators in the coefficient formulas do not vanish. If $f(z, \bar{z}, 0, 0) \equiv 0$, equation (2) possesses a solution in the form of a convergent series (15) for sufficiently small $|z|$. The coefficients $p_{0010}$ and $p_{0001}$ are determined by the initial linear system.

In the case where $b = 0$, the equation under the condition $f(z, \bar{z}, 0, 0) \equiv 0$ possesses a convergent solution for sufficiently small values of the variables. This solution depends on the parameters $C_1, C_2$ as defined by the simplified linear relations.

For the general case with $q$ variables, the equation possesses a unique formal solution depending on $2q$ arbitrary constants $C_1, \dots, C_{2q}$. The convergence of this series for sufficiently small $|z_1|, \dots, |z_q|$ is guaranteed provided that the eigenvalues of the linear part satisfy the required non-resonance conditions.

Theorem

Suppose that there exists a constant $B > 0$ such that the small divisor conditions are satisfied for all non-negative integers $m, n, p, q, \dots$. Then, assuming $f(z, \bar{z}, 0, 0) = 0$, equation (2) possesses a convergent solution for sufficiently small $|z|$ and $|\bar{z}|$. The arbitrary constants can always be chosen such that the convergence inequalities are satisfied. This result generalizes the previous theorems to higher-dimensional systems.

References

1. Polozhii, G. N. Generalization of the Theory of Analytic Functions of a Complex Variable. Kiev University Press, 1965.
2. Vekua, I. N. Generalized Analytic Functions. Moscow, Fizmatgiz, 1959.
3. Mikhailov, L. G. A New Class of Singular Integral Equations and Its Application to Differential Equations with Singular Coefficients. Dushanbe, 1963.
4. Nazirov, G. Doklady Akademii Nauk Tadzhikskoi SSR, No. 3, 3–6, 1961.
5. Bliev, N. Author's abstract of Candidate's dissertation. Alma-Ata, Institute of Mathematics, Academy of Sciences.

Received by the editorial office on January 28, 1966.