UDC 517.946

MATHEMATICS

A. DZHURAEV

ON AN ELLIPTIC SYSTEM OF SECOND-ORDER EQUATIONS IN THE PLANE

(Presented by Academician I. N. Vekua on October 3, 1966)

The second-order differential equation

\[ D_\lambda(w)\equiv \partial_z^2 w+A(z)\partial_{\bar z}w+\lambda B(z)w=h(z) \tag{1} \]

\[ (2\partial_{\bar z}=\partial_x+i\partial_y,\quad \partial_z^2\equiv \partial_z(\partial_{\bar z})), \]

in which \(A(z)\), \(B(z)\), \(h(z)\) are given complex-valued functions; \(\lambda\) is a certain parameter; \(w(z)=u(x,y)+iv(x,y)\) is the unknown function, is equivalent to a certain real linear elliptic system of two partial differential equations of the second order. In particular, when \(A(z)\equiv0\), \(\lambda B(z)\equiv0\), this equation is equivalent to the well-known system of equations of A. V. Bitsadze \((^{2-4})\). In the present note, we study some properties of equation (1) connected with the formulation of boundary-value problems for it.

1. General representation of solutions. Let \(G\) be an open bounded domain in the complex plane \(z=x+iy\), bounded by a closed Lyapunov curve \(\Gamma\), \(\bar G=G+\Gamma\). We shall assume that \(A(z)\in C_\nu^1(\bar G)\), \(B(z)\in C_\nu(G)\), \(h(z)\in C_\nu(\bar G)\), \(0<\nu<1\). Every solution of equation (1) belonging to the class \(C^2(G)\) will be called a regular solution. Let \(\lambda_1,\lambda_2,\ldots\) \((0<|\lambda_1|<|\lambda_2|<\ldots)\) be the eigenvalues of the homogeneous Fredholm-type integral equation

\[ w(z)+\lambda T(w)\equiv \]

\[ \equiv w(z)+\frac{\lambda}{\pi}\iint_G \frac{w^*(\zeta)-w^*(z)}{\zeta-z}e^{w(\zeta)}B(\zeta)w(\zeta)\,dG_\zeta=0, \tag{2} \]

where

\[ w^*(z)=\frac{1}{\pi}\iint_G\frac{\exp[-w(\zeta)]}{\zeta-z}\,dG_\zeta,\qquad w(z)=-\frac{1}{\pi}\iint_G\frac{A(\zeta)}{\zeta-z}\,dG_\zeta. \]

If \(\lambda\ne\lambda_k\), then there exist functions \(w_1(z)\), \(w_2(z)\) which are regular in \(G\) solutions of the homogeneous equation (1) \((h(z)\equiv0)\), continuous in the whole plane together with \(\partial_{\bar z}w_1\), \(\partial_{\bar z}w_2\), such that \(w_1(z)\), \(w_2(z)\), \(\partial_{\bar z}w_1\), \(\partial_{\bar z}w_2\) are holomorphic outside \(G+\Gamma\) and \(w_1(\infty)=1\), \(w_2(\infty)=0\), \(\partial_{\bar z}w_1|_{z=\infty}=0\), \(\partial_{\bar z}w_2|_{z=\infty}=1\). These functions can be obtained by solving the integral equations \(w(z)+\lambda T(w)=1\), \(w(z)+\lambda T(w)=w^*(z)\). It is not difficult to verify that

\[ w_1(z)\cdot\partial_{\bar z}w_2-w_2(z)\cdot\partial_{\bar z}w_1\equiv \exp[-w(z)]. \]

Let \(w(z)\) be an arbitrary regular in \(G\) solution of the homogeneous equation (1). Introducing into consideration the functions

\[ v_1(z)=-\partial_{\bar z}w_1\cdot w(z)+w_1(z)\cdot\partial_{\bar z}w,\qquad v_2(z)=\partial_{\bar z}w_2\cdot w(z)-w_2(z)\cdot\partial_{\bar z}w, \]

we have

\[ \partial_z v_1 + A(z)v_1 = 0,\qquad \partial_z v_2 + A(z)v_2 = 0. \]

Hence it follows that

\[ \partial_z w_2\cdot w(z)-w_2(z)\cdot \partial_z w = \varphi(z)\exp[-\omega(z)], \]

\[ -\partial_z w_1\cdot w(z)+w_1(z)\cdot \partial_z w = \psi(z)\exp[-\omega(z)], \]

where \(\varphi(z)\), \(\psi(z)\) are arbitrary functions holomorphic in \(G\). Solving the latter relations with respect to \(w(z)\), we obtain the representation

\[ w(z)=w_1(z)\varphi(z)+w_2(z)\psi(z). \tag{3} \]

Formula (3) gives the general representation of all regular solutions of the homogeneous equation (1) for \(\lambda\ne \lambda_k\) in terms of two arbitrary holomorphic functions \(\varphi(z)\), \(\psi(z)\). Adding to (3) a particular solution

\[ \frac{1}{\pi}\iint_G \frac{w_1(z)w_2(\zeta)-w_2(z)w_1(\zeta)}{\zeta-z}\, h(\zeta)\exp\omega(\zeta)\,dG_\zeta \]

of the nonhomogeneous equation (1), we obtain the formula for the general representation of all regular solutions of equation (1):

\[ w(z)=w_1(z)\varphi(z)+w_2(z)\psi(z) +\frac{1}{\pi}\iint_G \frac{w_1(z)w_2(\zeta)-w_2(z)w_1(\zeta)}{\zeta-z}\, h(\zeta)e^{\omega(\zeta)}\,dG_\zeta. \tag{4} \]

Formula (4) is valid for \(\lambda\ne\lambda_k\). In particular, it is valid for sufficiently small values of \(|\lambda|\) \((0<|\lambda|<|\lambda_1|)\).

From representation (3) the following assertion follows easily:

Theorem 1. If \(\lambda\ne\lambda_k\) and the angular boundary values of a regular solution \(w(z)\) of the homogeneous equation (1) vanish together with \(\partial_z w\) on an arc \(\gamma\subset\Gamma\), then \(w(z)\equiv 0\).

2. The Dirichlet problem. Find regular solutions of equation (1), continuous in \(\overline{G}+\Gamma\), satisfying the condition

\[ w|_{\Gamma}=0. \tag{D} \]

The problem of determining a solution regular in \(G\) of the equation

\[ D_\lambda^*(v)\equiv \partial_z^2 v-\partial_z(Av)+\lambda B(z)v=0, \tag{5} \]

continuous in \(\overline{G}\), satisfying condition (D), will be called problem \((D_0^*)\).

From Green’s identity

\[ \iint_G\{vD_\lambda(w)-wD_\lambda^*(v)\}\,dx\,dy = \frac{1}{2i}\int_\Gamma \{v(z)(\partial_z w+A(z)w)-w\partial_z v\}\,dz \]

it follows that, for the solvability of the Dirichlet problem, it is necessary that the function \(h(z)\) satisfy the equality

\[ \iint_G h(z)v(z)\,dx\,dy=0 \tag{6} \]

for every regular solution \(v(z)\) of problem \((D_0^*)\). However, this condition, generally speaking, is not sufficient for solvability of the Dirichlet problem. Below we shall indicate one criterion for the sufficiency of condition (6) in terms of the functions \(w_1(z)\), \(w_2(z)\).

Using representation (4), we reduce the boundary value problem (D) to the following problem of determining holomorphic functions \(\varphi(z)\), \(\psi(z)\):

\[ w_1(t_0)\varphi(t_0)+w_2(t_0)\psi(t_0)=f(t_0),\qquad t_0\in\Gamma, \tag{7} \]

where

\[ f(t_0)=-\frac{1}{\pi}\iint_G \frac{w_1(t_0)w_2(\xi)-w_2(t_0)w_1(\xi)}{\xi-t_0}\, h(\xi)\exp\omega(\xi)\,dG_\xi . \]

It follows immediately from condition (7) that the Dirichlet problem cannot be nontrivial. Let \(|\lambda|\) be sufficiently small \((0<|\lambda|<|\lambda_1|)\). Then \(w_1(z)\ne 0\), and condition (7) can be written as follows:

\[ \varphi(t_0)+\widetilde w(t_0)\psi(t_0)=\widetilde f(t_0), \tag{8} \]

where \(\widetilde w(z)=w_2(z)/w_1(z)\), \(\widetilde f(t_0)=f(t_0)/w_1(t_0)\). Using the fact that \(\varphi(t_0)\) is the boundary value of a function holomorphic in \(G\), condition (8) can also be written as

\[ \widetilde w(t_0)\psi(t_0)-\frac{1}{\pi i}\int_\Gamma \frac{\widetilde w(t)\psi(t)\,dt}{t-t_0} =F(t_0)=\widetilde f(t_0)-\frac{1}{\pi i}\int_\Gamma \frac{\widetilde f(t)\,dt}{t-t_0}. \tag{9} \]

Representing now the function \(\psi(z)\) in the form of a Cauchy-type integral with density \(\mu(t)\), determined up to the boundary value of an arbitrary function holomorphic outside \(G+\Gamma\) and vanishing at infinity, and using the formula for interchanging singular integrals, we obtain, for determining \(\mu(t)\), the integral equation

\[ \frac{1}{2\pi i}\int_\Gamma \frac{\widetilde w(t_0)-\widetilde w(t)}{t-t_0}\,\mu(t)\,dt + \frac{1}{2\pi^2} \int_\Gamma \left\{\int_\Gamma \frac{\widetilde w(t)\,dt}{(t-t_0)(\tau-t)} \right\}\mu(\tau)\,d\tau =F(t_0). \tag{10} \]

The continuous function \(\widetilde w(t)\) is the boundary value of the function \(w_2(z)/w_1(z)\), holomorphic outside \(G+\Gamma\), and vanishing at infinity. Therefore

\[ \int_\Gamma \frac{\widetilde w(t)\,dt}{(t-t_0)(\tau-t)} = \frac{1}{\tau-t_0} \left\{ \int_\Gamma\frac{\widetilde w(t)\,dt}{t-t_0} - \int_\Gamma\frac{\widetilde w(t)\,dt}{t-\tau} \right\} = \frac{\pi}{i}\bigl(\widetilde w(t_0)-\widetilde w(\tau)\bigr), \]

and equation (10) takes the form

\[ K(\mu)\equiv \frac{1}{\pi i}\int_\Gamma \frac{w_1(t)w_2(t_0)-w_2(t)w_1(t_0)} {w_1(t)(t-t_0)}\,\mu(t)\,dt = H(t_0)=w_1(t_0)F(t_0). \tag{11} \]

In view of the Hölder continuity of the functions \(w_1(t)\) and \(w_2(t)\), equation (11) is a Fredholm integral equation of the first kind. From the very method of deriving equation (11), it follows that it is equivalent to the Dirichlet problem, while the corresponding homogeneous equation \(K(\mu)=0\) has an infinite set of linearly independent solutions.

The condition for normal solvability of equation (11) has the form

\[ \int_\Gamma H(t)\nu(t)\,dt=0, \tag{12} \]

where \(\nu(t)\) is any solution of the homogeneous equation \(K'(\nu)=0\) adjoint to it. It is easy to see that if \(\nu(t)\) is a solution of the equation \(K'_1(\nu)=0\), then \(\nu_1(t)=w_1(t)\nu(t)\) is a solution of the equation \(K(\nu)=0\). Substituting the value of the function \(H(t)\) into condition (12), we easily obtain a condition of the form (6), in which the function \(v(z)\) has the form

\[ v(z)=- \frac{\exp\omega(z)}{\pi} \int_\Gamma \frac{w_1(t)w_2(z)-w_2(t)w_1(z)} {w_1(t)(z-t)}\,\widetilde\nu(t)\,dt, \tag{13} \]

where

\[ \widetilde\nu(t)=\nu_1(t)+\frac{1}{\pi i}\int_\Gamma \frac{\nu_1(\tau)\,d\tau}{\tau-t}. \]

But it is easy to verify that \(\tilde v(t)\) satisfies the homogeneous equation \(K(v)=0\). Hence it follows that

\[ \lim_{z\to t_0} v(z)=v(t_0)=-i\exp\omega(t_0)K(\tilde v)=0, \]

i.e., the function (13) satisfies condition (D). On the other hand, direct application of the operation \(\partial_{\bar z}K\) to (13) gives \(\mathscr D_\lambda^*(v)\equiv 0\) in \(G\). Thus, the condition of normal solvability of the integral equation (11) is equivalent to the condition of normal solvability of the Dirichlet problem (6):

Theorem 2. If \(|\lambda|\) is sufficiently small, then the Dirichlet problem is not a Noether problem. It is normally solvable in the sense of Hausdorff if and only if the integral equation (11) is normally solvable in the sense of Hausdorff.

3. Noether problems. For the equation \(\mathscr D_\lambda(w)=h(z)\) consider the boundary-value problem:

\[ \operatorname{Re}\{a(t)w(t)+b(t)\partial_{\bar t}w\}=0,\qquad \operatorname{Re}\{c(t)w(t)+d(t)\partial_{\bar t}w\}=0. \tag{14} \]

If the Hölder-continuous functions \(a(t), b(t), c(t), d(t)\) are such that
\(\theta(t)=a(t)d(t)-b(t)c(t)\ne 0\) for \(t\in \Gamma\), then for an arbitrary \((m+1)\)-connected domain \(G\) the following result holds:

Theorem 3. For solvability of problem (14) it is necessary and sufficient that the equality
\[ \operatorname{Re}\iint h(z)v(z)\,dx\,dy=0 \]
hold for every solution \(v(z)\) of the homogeneous adjoint problem to (14), and the index of this problem is equal to \(2\varkappa-4(m-1)\), where \(\varkappa=(2\pi)^{-1}\{\arg\theta(t)\}_{\Gamma}\). Here the homogeneous adjoint problem to (14) has the form:

\[ \mathscr D_\lambda^*(v)=0\quad \text{in }G, \]

\[ \operatorname{Re}\frac{t'(s)}{\theta(t)} \{[A(t)d(t)-c(t)]v(t)-d(t)\partial_{\bar t}v\}=0, \]

\[ \operatorname{Re}\frac{t'(s)}{\theta(t)} \{[a(t)-A(t)b(t)]v(t)+b(t)\partial_{\bar t}v\}=0. \]

If \(G\) is the unit disk \(|z|<1\), then in the case \(b(t)\equiv c(t)\equiv 0,\ a(t)=d(t)\equiv 1\), problem (14), for all values of \(\lambda\), except perhaps a discrete set \(\{\lambda_i\}\), is always solvable, and the corresponding homogeneous problem has exactly two linearly independent solutions.

Physical-Technical Institute
named after S. U. Umarov
Academy of Sciences of the Tajik SSR

Received
22 IX 1966

CITED LITERATURE

1. I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
2. A. V. Bitsadze, Uspekhi Mat. Nauk, 3, 6 (28) (1948).
3. A. V. Bitsadze, Equations of Mixed Type, Moscow, 1959.
4. A. V. Bitsadze, Dokl. Akad. Nauk SSSR, 164, No. 6 (1965).
5. Nguyen Tkha Khop, Dokl. Akad. Nauk SSSR, 166, No. 4 (1966).
6. Nguyen Tkha Khop, Differential Equations, 2, 2 (1966).