UDC 517.946.8

ON THE DIRICHLET PROBLEM FOR ELLIPTIC SYSTEMS IN SPACE

E. N. Kuz’min

As early as 1948 A. V. Bitsadze [1] observed that the Dirichlet problem for elliptic, in the sense of Petrovskii, systems of differential equations in the plane may fail to be Noetherian even in the case of systems with constant coefficients—in this is manifested a striking difference between elliptic systems and a single elliptic equation. An analogous situation, apparently, should also occur in spaces of higher dimension. Nevertheless, a number of mathematicians, in particular at the Soviet-American Symposium on Partial Differential Equations in 1963, expressed the opinion that the ellipticity condition in multidimensional space is stronger, and that the Dirichlet problem for elliptic systems should be Noetherian when the number of independent variables is \(n \geqslant 3\).

We shall give examples of elliptic systems of second order with four and eight variables, analogous to the examples of [1] and possessing analogous properties. These examples were found by the author and reported to the participants of A. V. Bitsadze’s seminar at the Institute of Mathematics of the Siberian Branch of the Academy of Sciences of the USSR in 1964. Similar examples were also indicated by Yu. T. Antokhin [2]. Like the author, Yu. T. Antokhin uses quaternions \((n=4)\) and Cayley numbers \((n=8)\) to construct his examples. We note that one of the examples of [2] (with the matrix \(C_8(\lambda)\)) is erroneous, since the corresponding system is not elliptic. The example becomes correct if the matrix \(A_8(\lambda)\) used in its construction is replaced by a matrix in which the minor \((4 \times 4)\) situated in the upper right corner and the minor \((3 \times 3)\) situated in the lower right corner are transposed with respect to the corresponding minors of the matrix \(A_8(\lambda)\).

Let the system of equations be given in matrix form

\[ L_n(p)u = 0 \qquad (n=4,8), \tag{1} \]

where \(u\) is a column with components \(u_1, u_2, \ldots, u_n\); \(p=\left(\dfrac{\partial}{\partial x_1}, \dfrac{\partial}{\partial x_2}, \ldots \right.\)

\[ \left. \ldots, \dfrac{\partial}{\partial x_n}\right), \]

and the characteristic matrix \(L_n(\lambda)\) is represented in the form of a product of block matrices

\[ L_n(\lambda)= \begin{pmatrix} A_m & B_m\\ -B_m' & A_m' \end{pmatrix} \begin{pmatrix} A_m & B_m'\\ -B_m & A_m' \end{pmatrix}; \]

the sign \(({}')\) denotes transposition, \(m=\dfrac{n}{2}\);

\[ A_2=\begin{pmatrix} \lambda_1 & \lambda_2\\ -\lambda_2 & \lambda_1 \end{pmatrix}, \qquad B_2=\begin{pmatrix} \lambda_3 & \lambda_4\\ -\lambda_4 & \lambda_3 \end{pmatrix}, \]

\[ A_4= \begin{pmatrix} \lambda_1 & \lambda_2 & \lambda_3 & \lambda_4\\ -\lambda_2 & \lambda_1 & -\lambda_4 & \lambda_3\\ -\lambda_3 & \lambda_4 & \lambda_1 & -\lambda_2\\ -\lambda_4 & -\lambda_3 & \lambda_2 & \lambda_1 \end{pmatrix}, \]

\[ B_4= \begin{pmatrix} \lambda_5 & \lambda_6 & \lambda_7 & \lambda_8\\ -\lambda_6 & \lambda_5 & \lambda_8 & -\lambda_7\\ -\lambda_7 & -\lambda_8 & \lambda_5 & \lambda_6\\ -\lambda_8 & \lambda_7 & -\lambda_6 & \lambda_5 \end{pmatrix}. \]

It can be verified that
\[ \det L_n(\lambda)=\bigl(\lambda_1^2+\lambda_2^2+\cdots+\lambda_n^2\bigr)^n \quad (n=4,8), \]
so that the corresponding systems of equations (1) are elliptic in the sense of Petrovsky. We shall show that the homogeneous Dirichlet problem for systems (1) in the ball
\[ x_1^2+x_2^2+\cdots+x_n^2\le R^2 \]
has infinitely many linearly independent solutions for any \(R\); hence it follows that the Dirichlet problem for these systems in the ball is not Noetherian.

Let us pass in equations (1) to complex variables, setting
\[ z_k=x_{2k-1}+ix_{2k}, \qquad \bar z_k=x_{2k-1}-ix_{2k}, \]

\[ w_k=u_{2k-1}+iu_{2k}\qquad \left(k=1,2,\ldots,\frac n2\right); \]
then, for \(n=4\), we obtain
\[ \begin{pmatrix} \dfrac{\partial}{\partial z_1} & \dfrac{\partial}{\partial z_2}\\[6pt] -\dfrac{\partial}{\partial \bar z_2} & \dfrac{\partial}{\partial \bar z_1} \end{pmatrix} \begin{pmatrix} \dfrac{\partial}{\partial \bar z_1} & \dfrac{\partial}{\partial \bar z_2}\\[6pt] -\dfrac{\partial}{\partial z_2} & \dfrac{\partial}{\partial z_1} \end{pmatrix} \begin{pmatrix} w_1\\ w_2 \end{pmatrix} =0. \tag{2} \]

It is easy to see that the formulas
\[ w_1=(R^2-z_1\bar z_1-z_2\bar z_2)\,f_1(z_1^2-z_2^2), \]

\[ w_2=(R^2-z_1\bar z_1-z_2\bar z_2)\,f_2(z_1^2-z_2^2), \]
where \(f_1(\zeta), f_2(\zeta)\) are arbitrary functions holomorphic in the disk \(|\zeta|<R^2\) and continuous in the closed disk \(|\zeta|\le R^2\), give solutions of system (2), regular in the domain
\[ |z_1|^2+|z_2|^2\le R^2 \]
and vanishing on the boundary of this domain. Similarly, for \(n=8\) one can indicate solutions of the form
\[ w_1=(R^2-r^2)f_1(z_2\cdot \bar z_4), \qquad w_2=(R^2-r^2)f_2(\bar z_2\cdot z_4), \]

\[ w_3=(R^2-r^2)f_3(z_2\cdot \bar z_4), \qquad w_4=(R^2-r^2)f_4(\bar z_2\cdot z_4), \]
where
\[ r^2=z_1\bar z_1+z_2\bar z_2+z_3\bar z_3+z_4\bar z_4, \]
and \(f_i(\zeta)\) are arbitrary functions holomorphic in the disk \(|\zeta|<R^2\).

As was already mentioned above, in constructing system (1) quaternions and the Cayley numbers were used—well-known examples of division algebras of finite rank over the field of real numbers. Their application in the present case is based on the fact that to each division algebra one can associate a certain system of elliptic type. Indeed, let \(A\) be a division algebra over the field of real numbers, and let \(e_1, e_2, \ldots, e_n\) be a basis of \(A\). To an arbitrary element \(e=\lambda_1 e_1+\lambda_2 e_2+\ldots+\lambda_n e_n \in A\) we associate the operator of right multiplication \(R_e:x\to xe\), acting in the linear space \(A\). Denote by \(P(\lambda)\) the matrix of the operator \(R_e\) in the chosen basis; the entries of the matrix \(P(\lambda)\) are linear forms in the variables \(\lambda_1,\lambda_2,\ldots,\lambda_n\). Since \(xe\ne 0\) for \(x,e\ne 0\), it follows that \(\det P(\lambda)\ne 0\) for all \(\lambda\ne 0\). Hence it follows that the system of linear equations whose characteristic matrix coincides with \(P(\lambda)\) is elliptic. Various classes of division algebras over the field of real numbers were constructed, for example, in [4]. It is known, however, that division algebras exist only for dimensions \(n=1,2,4,8\) [3], and therefore the indicated method is not applicable to spaces of other dimensions.

References

1. Bitsadze A. V. UMN, 3, issue 6 (28), 241—242, 1948.
2. Antokhin Yu. T. Differential Equations, 2, No. 4, 525—532, 1966.
3. Adams J. F. On the non-existence of mappings with Hopf invariant one, equal to one. Matematika, 5, issue 4, 3—86, 1961.
4. Kuzmin E. N. Algebra and logic. Seminar, 5, issue 2, 57—102, 1966.

Received by the editors
March 1, 1966

Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences