V. S. VINOGRADOV

ON THE DIRICHLET PROBLEM FOR MULTIDIMENSIONAL ELLIPTIC SYSTEMS OF SECOND ORDER

(Presented by Academician I. N. Vekua on 23 V 1967)

It is known that the Dirichlet problem for a system of differential equations of second order of elliptic type, generally speaking, ceases to be Noetherian. The first examples of such systems of two equations in the plane were constructed by A. V. Bitsadze \((^1)\), and the first Noetherian condition was not satisfied—the homogeneous problem had an infinite set of linearly independent solutions. Later B. V. Boyarskii \((^2)\) established that, for systems of two second-order equations of elliptic type in spaces of dimensions \(n \geqslant 3\), the Dirichlet problem becomes Noetherian. However, for spaces of dimension \(n=4;\ 8\), Yu. T. Antokhin \((^3)\) and E. N. Kuzmin \((^4)\) constructed examples of systems again with violation of the first Noetherian condition. In the present paper examples of this kind of systems will be constructed in spaces of any even dimension.

Thus, consider in Euclidean space of dimension \(2n\) the following two systems of first order \((^5)\)

\[ \mathcal L_{\pm}\mathbf u = \sum_{\alpha=1}^{n} \left( \mathscr P_\alpha \frac{\partial}{\partial x_\alpha} \pm \mathscr Q_\alpha \frac{\partial}{\partial y_\alpha} \right)\mathbf u =0. \tag{1} \]

Here \(x_\alpha, y_\alpha\) are independent variables, and the matrices \(\mathscr P_\alpha\) and \(\mathscr Q_\alpha\) are constructed in the form of tensor products:

\[ \mathscr P_\alpha = 1' \times \cdots \times 1' \times \left\| \begin{matrix} 0 & 1\\ 1 & 0 \end{matrix} \right\| \times 1 \times \cdots \times 1, \]

\[ \mathscr Q_\alpha = 1' \times \cdots \times 1' \times \left\| \begin{matrix} 0 & i\\ -i & 0 \end{matrix} \right\| \times 1 \times \cdots \times 1, \tag{2} \]

\[ 1'= \left\| \begin{matrix} 1 & 0\\ 0 & -1 \end{matrix} \right\|, \qquad 1= \left\| \begin{matrix} 1 & 0\\ 0 & 1 \end{matrix} \right\|, \qquad \text{the matrices } P= \left\| \begin{matrix} 0 & 1\\ 1 & 0 \end{matrix} \right\| \text{ and } Q= \left\| \begin{matrix} 0 & i\\ -i & 0 \end{matrix} \right\| \]

stand in the \(\alpha\)-th place.

The matrices \(\mathscr P_\alpha\) and \(\mathscr Q_\alpha\) have order \(2^n\) and satisfy the relation

\[ \left[ \sum_{\alpha=1}^{n} \left( \mathscr P_\alpha \xi_\alpha \pm \mathscr Q_\alpha \eta_\alpha \right) \right]^2 = \left[ \sum_{\alpha=1}^{n} \left( \xi_\alpha^2+\eta_\alpha^2 \right) \right]\mathscr E, \tag{3} \]

where \(\mathscr E\) is the identity matrix (see \((^6)\), p. 363), from which the ellipticity of the systems (1) follows. We construct our elliptic system of second order in the form

\[ \Lambda \mathbf u=\mathcal L_{+}\mathcal L_{-}\mathbf u. \tag{4} \]

For it the following assertion is true.

Theorem. The homogeneous Dirichlet problem for the system (4) in the ball
\[ |z_1|^2+\cdots+|z_n|^2\leqslant R^2 \quad (z_\alpha=x_\alpha+i y_\alpha) \]
has as solutions vectors of the form
\[ \mathbf u_0=(w_0,0,\ldots,0), \qquad w_0= \left(R^2-|z_1|^2-\cdots-|z_n|^2\right) f(z_1^2+\cdots+z_n^2), \tag{5} \]
where \(f(\omega)\) is a function analytic with respect to \(\omega\) in the image of the ball
\[ |z_1|^2+\cdots+|z_n|^2\leqslant R^2 \]
under the mapping
\[ \omega=z_1^2+\cdots+z_n^2. \]

Proof. To verify that the vector \(\mathbf u_0\) satisfies the system (4), it is necessary to agree on the order of arrangement of the rows and columns—

rows of the matrices (2). As is known, the tensor product of the matrices
$A_1 \times A_2 \times \cdots \times A_n$, $A_\alpha=\{a^{(\alpha)}_{ij}\}$, is the matrix of the form
$\{a_{i_1\ldots i_n,\,j_1\ldots j_n}\}=\{a^{(1)}_{i_1j_1}\cdots a^{(n)}_{i_nj_n}\}$, composed of all possible products of elements of the factor matrices, where the set of numbers $i_1\ldots i_n$ characterizes a row, and the set $j_1\ldots j_n$ a column. We shall order these sets, and together with them the rows and columns of the matrices $\mathscr L_+$ and $\mathscr L_-$, in lexicographic order; it is precisely for this ordering rule that our solution $\mathbf u_0$ was chosen. Any other ordering of rows (columns) will cause a certain permutation of rows (columns), the same for all our matrices (2); therefore, to obtain a solution it will be necessary to perform the very same permutation of the components of the vector $\mathbf u_0$.

Let us find the differential operators of the first column of the matrix $\Lambda$; they are obtained by multiplying the matrix $\mathscr L_+$ by the first column of the matrix $\mathscr L_-$.

In the first column of $\mathscr L_-$, in the rows $(1,\ldots,2,\ldots,1)$, $\alpha=1,\ldots,n$, there stand the operators $2\partial/\partial \overline z_\alpha=(\partial/\partial x_\alpha+i\partial/\partial y_\alpha)$, while in the remaining rows there are operators identically equal to zero. This is easy to see from the following notation for our operators:

\[ \begin{gathered} \mathscr L_+=2\sum_{\alpha=1}^n \left\|\begin{matrix}1&0\\[2pt]0&-1\end{matrix}\right\| \times\cdots\times \left\|\begin{matrix}0&\partial/\partial \overline z_\alpha\\[2pt]\partial/\partial z_\alpha&0\end{matrix}\right\| \times \left\|\begin{matrix}1&0\\[2pt]0&1\end{matrix}\right\| \times\cdots\times \left\|\begin{matrix}1&0\\[2pt]0&1\end{matrix}\right\|,\\ \mathscr L_-=2\sum_{\alpha=1}^n \left\|\begin{matrix}1&0\\[2pt]0&-1\end{matrix}\right\| \times\cdots\times \left\|\begin{matrix}1&\partial/\partial z_\alpha\\[2pt]\partial/\partial \overline z_\alpha&0\end{matrix}\right\| \times \left\|\begin{matrix}1&0\\[2pt]0&1\end{matrix}\right\| \times\cdots\times \left\|\begin{matrix}1&0\\[2pt]0&1\end{matrix}\right\|, \end{gathered} \tag{6} \]

\[ \partial/\partial z_\alpha=\tfrac12(\partial/\partial x_\alpha-i\partial/\partial y_\alpha). \]

The first row of $\mathscr L_+$ consists of the operators $2\partial/\partial \overline z_\alpha$, standing in the columns $(1,\ldots,2,\ldots,1)$, $\alpha=1,\ldots,n$, and identically equal to zero in the others. Hence the operator obtained by multiplying the first row of $\mathscr L_+$ by the first column of $\mathscr L_-$ will be

\[ 4(\partial^2/\partial z_1^2+\cdots+\partial^2/\partial z_n^2). \tag{7} \]

Consider an arbitrary row $(a_1\ldots a_n)$ of the matrix, different from the first. When it is multiplied by the first column of $\mathscr L_-$, a nonzero operator can be obtained only in the case when among the $a_i$ there are exactly two twos. Indeed, the operators of the row under consideration are nonzero in the columns $(c a_1,a_2,\ldots,a_n),\ldots,(a_1,\ldots,c a_n)$; $c a_i=1$ for $a_i=2$ and $c a_i=2$ for $a_i=1$; therefore, if the number of twos among the $a_i$ is not equal to two, the numbers of the columns with operators identically not equal to zero do not coincide with the numbers of those rows of the first column where there also stand operators different from identically zero. Suppose that among the $a_i$ there are two twos,

\[ (a_1\ldots a_n)=(1\ldots \underset{(k)}{2}\ldots \underset{(l)}{2}\ldots 1), \]

then after multiplication one obtains the operator

\[ 4(-\partial^2/\partial z_l\partial \overline z_k+\partial^2/\partial z_k\partial \overline z_l),\qquad k,l=1,\ldots,n. \tag{8} \]

This is obtained by a simple computation from (6).

Now it is easy to verify that the function $w_0$ is a zero of the operators (7) and (8), and consequently the vector $\mathbf u_0$ satisfies our system (4).

It is clear that among the solutions of the form (5) of the homogeneous Dirichlet problem there is an infinite set of linearly independent solutions, although they may not exhaust all its solutions. It follows that the Dirichlet problem for our system is not Noetherian.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
16 V 1967

CITED LITERATURE

1. A. V. Bitsadze, UMN, 3, no. 6 (28), 241 (1948).
2. B. V. Boyarskii, Bull. Polish Acad. Sci., ser. mat., astr. i fiz. nauk, 8, no. 1 (1960).
3. Yu. T. Antonin, Differential Equations, 2, no. 4, 525 (1966).
4. E. N. Kuzmin, Differential Equations, 3, no. 1, 155 (1967).
5. V. S. Vinogradov, DAN, 178, no. 2 (1968).
6. H. Weyl, The Classical Groups, Their Invariants and Representations, 1947.