UDC 517.944

MATHEMATICS

A. DZHURAEV

ON A SYSTEM OF BELTRAMI EQUATIONS DEGENERATING ON A LINE

(Presented by Academician I. N. Vekua on 14 VIII 1968)

1. Consider the equation

\[ \partial w/\partial \bar z-q(x)\partial w/\partial z=0, \tag{1} \]

where \(z=x+iy\), \(\partial/\partial \bar z=\tfrac12(\partial/\partial x+i\partial/\partial y)\), \(\partial/\partial z=\tfrac12(\partial/\partial x-i\partial/\partial y)\), \(q(z)\) is a given, and \(w(z)\) the sought, complex-valued function. This equation is equivalent to a system of two real first-order equations, known as the system of Beltrami equations in the case of its uniform ellipticity; moreover, the condition of uniform ellipticity for (1) has the form

\[ |q(z)|\leqslant \mathrm{const}<1. \tag{2} \]

In the present note we shall consider equation (1) when condition (2) is violated along a certain line. Let \(|q(z)|\) be an analytic function of the variables \(x,y\) in some domain \(D\), and let the set of points of the domain \(D\) at which \(|q(z)|=1\) \((q(z)\ne 1)\) be a simple analytic curve \(\gamma\).

If, along \(\gamma\), all partial derivatives of the function \(|q(z)|^2-1\) up to order \(n-1\) (inclusive) are equal to zero, while among the derivatives of order \(n\) at least one is different from zero, then in some neighborhood \(\Delta\subset D\) of an arc of \(\gamma\) there is the representation

\[ |q(z)|^2-1=\eta^n(x,y)a(x,y), \tag{3} \]

where \(\eta(x,y)=0\) is the equation of the curve \(\gamma\), and \(a(x,y)\ne 0\); moreover the functions \(\partial \xi/\partial x\) and \(\partial \xi/\partial y\) are not simultaneously equal to zero in \(\Delta\).

2. Suppose that the direction of the characteristics of equation (1) at the points of the arc \(\gamma\) does not coincide with the direction of the tangent to the arc \(\gamma\), i.e. that along \(\gamma\) the inequality

\[ (1-\operatorname{Re} q(z))\partial\eta/\partial x-\operatorname{Im} q(z)\partial\eta/\partial y\ne 0 \tag{4} \]

holds.

Let the function \(\xi(x,y)\) be a solution of the equation

\[ (1-\operatorname{Re} q(z))\partial\xi/\partial x-\operatorname{Im} q(z)\partial\xi/\partial y=0. \tag{5} \]

It is evident that one can find a subdomain \(\delta\) of the domain \(\Delta\), containing inside it an arc of \(\gamma\), in which the function \(b(x,y)\), satisfying the equalities

\[ \partial\xi/\partial x=b(x,y)\operatorname{Im} q(z),\qquad \partial\xi/\partial y=b(x,y)(1-\operatorname{Re} q(z)), \tag{6} \]

is different from zero. Since

\[ I=\frac{\partial \xi}{\partial x}\frac{\partial \eta}{\partial y} -\frac{\partial \eta}{\partial x}\frac{\partial \xi}{\partial y} =-b(x,y)\{(1-\operatorname{Re} q(z))\partial\eta/\partial x-\operatorname{Im} q(z)\partial\eta/\partial y\}\ne 0 \]

in \(\delta\), the mapping \(\xi=\xi(x,y)\), \(\eta=\eta(x,y)\) is a homeomorphism of the neighborhood \(\delta\) onto some domain \(\tilde\delta\) in the plane of the variables \(\xi,\eta\). As a result of this mapping, equation (1) is transformed to the form:

\[ (\partial w/\partial \bar \zeta+\partial w/\partial \zeta)(\partial \xi/\partial \bar z-q(z)\partial\xi/\partial z)- \]

\[ -i(\partial w/\partial \bar\zeta-\partial w/\partial\zeta)(\partial\eta/\partial\bar z-q(z)\partial\eta/\partial z)=0, \tag{7} \]

where \(\zeta=\xi+i\eta\), \(\partial/\partial\bar\zeta=\tfrac12(\partial/\partial\xi+i\partial/\partial\eta)\), \(\partial/\partial\zeta=\tfrac12(\partial/\partial\xi-i\partial/\partial\eta)\).

If we now take into account that, by virtue of equalities (6) and (3),

\[ \frac{\partial \xi}{\partial \bar z}-q(z)\frac{\partial \xi}{\partial z} =-\frac{i b(x,y)}{2}\left(|q(z)|^2-1\right) =-\frac{i a(x,y)b(x,y)}{2}\eta^n, \]

and, by virtue of inequality (4), \(\partial \eta/\partial \bar z-q(z)\partial \eta/\partial z\ne 0\) in \(\delta\), then, dividing (7) by \(\eta^*(z)\), we shall have

\[ (1+\eta^n c(\xi))\partial w/\partial \bar \xi -(1-\eta^n c(\xi))\partial w/\partial \xi=0, \tag{8} \]

where

\[ c(\xi)=a[x(\xi,\eta),y(\xi,\eta)]b[x(\xi,\eta),y(\xi,\eta)]/\eta^*[z(\xi)], \quad \eta^*(x)=\partial\eta/\partial\bar z-q(z)\partial\eta/\partial z. \]

Let us note that, by virtue of inequality (4), the function \(\operatorname{Re} c(\xi)\) is different from zero in the domain \(\delta\), since

\[ \operatorname{Re} c(\xi)= \frac{ab}{|\eta^*|^2} \left\{(1-\operatorname{Re} q)\frac{\partial\eta}{\partial x} -\operatorname{Im} q\cdot\frac{\partial\eta}{\partial y}\right\}\ne 0 \quad\text{in }\delta. \]

1. Let us now consider the case when along \(\gamma\) the equality

\[ (1-\operatorname{Re} q(z))\partial\eta/\partial x-\operatorname{Im} q(z)\partial\eta/\partial y=0 \tag{9} \]

holds. In this case one can find functions \(\mu(x,y)\) and \(\nu(x,y)\) for which the inequality

\[ (1-\operatorname{Re} q(z))\mu(x,y)-\operatorname{Im} q(z)\cdot\nu(x,y)\ne 0 \tag{10} \]

holds.

Let \(\xi(x,y)\) be a solution of equation (5), and let \(\widetilde{\eta}(x,y)\) satisfy the equality

\[ \nu(x,y)\partial\widetilde{\eta}/\partial x-\mu(x,y)\partial\widetilde{\eta}/\partial y=0. \tag{11} \]

Obviously, one can find such an arc \(\gamma_1\) of the arc \(\gamma\) and a domain \(\delta\), containing within itself the arc \(\gamma_1\), in which inequality (10) holds and the functions \(b_1(x,y)\), \(b_2(x,y)\), distinct from zero, satisfy the relations

\[ \begin{aligned} \partial\xi/\partial x&=b_1(x,y)\operatorname{Im}q(z),& \partial\xi/\partial y&=b_1(x,y)(1-\operatorname{Re}q(z)),\\ \partial\widetilde{\eta}/\partial x&=b_2(x,y)\mu(x,y),& \partial\widetilde{\eta}/\partial y&=b_2(x,y)\nu(x,y). \end{aligned} \tag{12} \]

From these relations and inequalities (11) it follows that the Jacobian of the transformation \(\xi=\xi(x,y)\), \(\widetilde{\eta}=\widetilde{\eta}(x,y)\) is different from zero in the neighborhood \(\delta\):

\[ J=b_1b_2\bigl(\operatorname{Im}q(z)\cdot\nu(x,y)-(1-\operatorname{Re}q(z))\mu(x,y)\bigr)\ne 0. \]

Let \(\eta^0(\xi,\widetilde{\eta})=\eta[x(\xi,\widetilde{\eta}),y(\xi,\widetilde{\eta})]\). Since along \(\gamma_1\) the directions of the tangent curves \(\eta(x,y)=\mathrm{const}\), \(\xi(x,y)=\mathrm{const}\), by virtue of (5) and (9), coincide, it follows that \(\eta^0(0,\widetilde{\eta})=0\). But it is easy to see that

\[ \frac{\partial\eta^0}{\partial\xi} = \frac{b_2}{J}\left(\nu(x,y)\frac{\partial\eta}{\partial x} -\mu(x,y)\frac{\partial\eta}{\partial y}\right)\ne 0. \]

Therefore, in some neighborhood \(\delta_1\subset\delta\) of the arc \(\gamma_1\), the function \(\eta^0(\xi,\widetilde{\eta})\) has the form

\[ \eta^0(\xi,\widetilde{\eta})=\xi\eta_0(\xi,\widetilde{\eta}), \qquad \eta_0(\xi,\widetilde{\eta})\ne 0, \tag{13} \]

and since

\[ \partial\xi/\partial\bar z-q(z)\partial\xi/\partial z =-{}^{1}/_{2}i b_1\left(|q(z)|^2-1\right) =[\eta^0(\xi,\widetilde{\eta})]^n a[x(\xi,\eta),y(\xi,\eta)], \]

then, taking (13) into account, we are convinced that, as a result of the nondegenerate transformation \(\xi=\xi(x,y)\), \(\widetilde{\eta}=\widetilde{\eta}(x,y)\), equation (1) takes the form \((\zeta=\xi+i\widetilde{\eta})\)

\[ (1+\xi^n c_0(\zeta))\partial w/\partial\bar\zeta -(1-\xi^n c_0(\zeta))\partial w/\partial\zeta=0, \tag{14} \]

where

\[ c_0(\zeta)=a_1b_1\eta_0(\xi,\widetilde{\eta})/2(\partial\widetilde{\eta}/\partial\bar z-q(z)\partial\widetilde{\eta}/\partial z)_{z=z(\zeta)}. \]

By virtue of inequality (10) it is easy to see that the function \(\operatorname{Re} c_0(\zeta)\) is different from zero in the domain \(\tilde\delta_1\)—the image of the domain \(\delta_1\) under the mapping \(\xi=\xi(x,y)\), \(\eta=\eta(x,y)\). Thus, in a neighborhood of the line of degeneration \(\gamma\), equation (1) can always be reduced either to the form (8) or to the form (14).

1. Let \(G\) be a domain in the \((\xi,\eta)\)-plane, bounded by some arc \(\sigma\), situated in the half-plane \(\eta>0\) (or \(\eta<0\)) and adjacent to the axis \(\eta=0\), and also by a segment \(AB\) of the real axis \(\eta=0\). In the domain \(G\) consider an equation of the form (8), and we shall assume that \(c(\zeta)\) is a function of class \(C_\alpha'\) in the domain \(G+\sigma+AB\) and that \(\operatorname{Re} c(\zeta)>0\) (if \(n\) is an even number, then, obviously, the latter condition does not restrict generality). Let \(w(\zeta)\) be a solution of equation (8), regular in the domain \(G\), continuous in \(G+\sigma+AB\). Making the change of variable
   \[ \zeta_*(\zeta)=\xi_*+i\eta_*=\xi+\frac{i}{n+1}\eta^{n+1}, \]
   which maps the domain \(G\) onto the domain \(G_*\), situated in the half-plane \(\eta^*>0\) (or \(\eta^*<0\)), bounded by the arc \(\sigma\) and the segment \(AB\) of the axis \(\eta^*=0\), instead of equation (8) we obtain the Beltrami equation
   \[ \frac{\partial w}{\partial \overline{\zeta_*}}-\frac{1-c(\zeta_*)}{1+c(\zeta_*)}\frac{\partial w}{\partial \zeta_*}=0. \tag{15} \]

Since \(\operatorname{Re} c(\zeta)>0\) in \(\overline G\), the function \(\operatorname{Re} c(\zeta_*)\) is positive in the domain \(\overline{G_*}\), and, consequently, the modulus of the function \(\bigl(1-c(\zeta_*)\bigr)/\bigl(1+c(\zeta_*)\bigr)\) is bounded in \(G_*\) from above by unity; in this case, as is known \((^1)\), all regular solutions of equation (15) have the form
\[ w=\Phi[\chi(\zeta_*)], \tag{16} \]
where \(\chi(\zeta_*)\) is a univalent solution of equation (15), and \(\Phi(\chi)\) is an arbitrary holomorphic function. Consequently, all solutions of equation (8) regular in \(G\), continuous in the domain \(G+\sigma+AB\), are represented by the formula
\[ w(\zeta)=\Phi\left[\chi\left(\xi+\frac{i}{n+1}\eta^{n+1}\right)\right]. \tag{17} \]

If one uses the known representations for holomorphic functions in the form of Cauchy-type integrals with real density, then, using formula (17), it is easy to study the following boundary-value problem: find regular solutions of equation (8), continuous in \(G+\sigma+AB\), and satisfying the condition
\[ \operatorname{Re}[\lambda(\zeta)w(\zeta)]=h(\zeta),\qquad \zeta\in\sigma+AB. \]
Let \(\varkappa\) be the integer equal to the increment of the function \(\arg\lambda(\zeta)\), obtained when the contour \(\sigma-AB\) is traversed once counterclockwise, divided by \(2\pi\). Then, with respect to this problem, the following assertions hold: if \(\varkappa\ge0\), then the homogeneous problem has exactly \(2\varkappa+1\) linearly independent solutions, and the nonhomogeneous problem is solvable; but if \(\varkappa<0\), then the homogeneous problem has no nontrivial solution, and the nonhomogeneous problem is solvable provided that \(-2\varkappa-1\) conditions are satisfied by the right-hand side \(h(\zeta)\). In particular, the Dirichlet problem
\[ \operatorname{Re} w(\zeta)=h(\zeta),\qquad \zeta\in\sigma+AB \]
for equation (8) is always solvable uniquely up to an imaginary constant. An analogous problem can also be considered for equations of the form (14).

Physicotechnical Institute
Academy of Sciences of the Tajik SSR

Received
22 VII 1968

CITED LITERATURE

1. I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
2. A. V. Bitsadze, Equations of Mixed Type, Moscow, 1959.