G. KARATOPRAKLIEV

ON A GENERALIZATION OF PROBLEM T FOR THE EQUATION

\[ u_{xx}+\operatorname{sign} y\,u_{yy}=0 \]

(Presented by Academician M. A. Lavrent'ev, 27 IV 1963)

In the present article one boundary-value problem is considered for the Lavrent'ev—Bitsadze equation
\[ u_{xx}+\operatorname{sign} y\,u_{yy}=0, \tag{1} \]
which is a generalization of the Tricomi problem T, posed and investigated in the work of M. A. Lavrent'ev and A. V. Bitsadze \((^1)\) and in the works of A. V. Bitsadze \((^2,{}^3)\).

Let \(D\) be a simply connected finite domain of the \(xy\)-plane, bounded by a Jordan line \(\sigma\) with endpoints at the points \(A(-1,0)\), \(B(1,0)\), lying in the upper half-plane \(y>0\), and by the characteristics \(AC: y=-x-1\) and \(BC: y=x-1\), issuing from the point \(C(0,-1)\). Denote by \(D_1\) and \(D_2\), respectively, the elliptic and hyperbolic parts of the mixed domain \(D\).

Problem \(T_\alpha\). It is required to determine a function \(u(x,y)\) with the following properties: 1) \(u(x,y)\) is a solution of equation (1) in the domain \(D\) for \(y\ne0\); 2) \(u(x,y)\) is continuous in the closed domain \(D\) for \(y\ne0\); 3) the partial derivatives \(u_x\) and \(u_y\) are continuous in the domain \(D\) for \(y\ne0\), and near the points \(A\) and \(B\) they may become infinite of order less than one; 4) on the segment \(AB\) of the real axis the functions \(u(x,y)\) and \(u_y(x,y)\) satisfy the gluing conditions
\[ u(x,+0)=\alpha(x)u(x,-0), \tag{2} \]
\[ u_y(x,+0)=\beta(x)u_y(x,-0), \tag{3} \]
where \(\alpha(x)\) and \(\beta(x)\) are given functions which do not vanish anywhere on the segment \(AB\), \(\beta(x)\) is once and \(\alpha(x)\) twice differentiable, and the functions \(\beta'(x)\) and \(\alpha''(x)\) belong to the class \(\bar H\) on the segment \(^{*}AB\), with
\[ \alpha'(x)\beta(x)\le 0,\qquad -1<x<1; \tag{4} \]
5) \(u(x,y)\) assumes the prescribed values
\[ u=\varphi \qquad \text{on } \sigma, \tag{5} \]
\[ u=\psi(x) \qquad \text{on } AC, \tag{6} \]
where \(\varphi\) is continuous, and \(\psi(x)\) is a twice differentiable function whose second derivative belongs to the class \(\bar H\) on the segment \([-1,0]\), and \(\varphi(-1)=\alpha(-1)\psi(-1)\).

For \(\alpha(x)=\beta(x)=1\), problem \(T_\alpha\) coincides with problem T.

In the domain \(D_2\), by virtue of (2) and (3), the solution \(u(x,y)\) of equation (1) has the form
\[ 2u(x,y)=\frac{\tau(x+y)}{\alpha(x+y)}+\frac{\tau(x-y)}{\alpha(x-y)}+\int_{x-y}^{x+y}\frac{\nu(t)}{\beta(t)}\,dt, \tag{7} \]
where \(\tau(x)=u(x,+0)\), \(\nu(x)=u_y(x,+0)\), \(-1\le x\le1\).

By virtue of (6), from (7) we obtain
\[ \frac{\tau(x)}{\alpha(x)}+\frac{\tau(-1)}{\alpha(-1)}-\int_{-1}^{x}\frac{\nu(t)}{\beta(t)}\,dt =2\psi\!\left(\frac{x-1}{2}\right),\qquad -1\le x\le1, \]

* We use the terminology adopted in \((^6)\).

or, after differentiation,

\[ \left[\frac{\tau(x)}{\alpha(x)}\right]' - \frac{\nu(x)}{\beta(x)} = 2\frac{d}{dx}\psi\left(\frac{x-1}{2}\right), \qquad -1 < x < 1. \tag{8} \]

Hence, taking (4) into account, just as in problem \(T\) \(\left({}^{3}\right)\), we conclude that if \(\psi(x)\equiv 0\), the solution \(u(x,y)\) of problem \(T_\alpha\) in the closed domain \(\overline{D}_1\) attains a nonzero extremum on the arc \(\sigma\) (the extremum principle). From this principle the uniqueness of the solution of problem \(T_\alpha\) follows directly.

The existence of a solution of problem \(T_\alpha\) will be proved if the function \(\nu(x)\) can be determined. Let us note that, without loss of generality, one may assume that \(\varphi=0\). We shall additionally assume that \(\sigma\) is a smooth arc satisfying the Lyapunov condition, and that \(u_x\) and \(u_y\) are continuous in the closed domain \(\overline{D}_1\) everywhere except, possibly, at the points \(A\) and \(B\). By a conformal mapping one can arrange that \(\sigma\) coincide with the semicircle \(\sigma_0\) with endpoints at the points \(A\) and \(B\) \(\left({}^{3}\right)\). We shall assume that \(\sigma\) coincides with \(\sigma_0\).

The relation between the functions \(\tau(x)\) and \(\nu(x)\) from the elliptic part \(D_1\) of the mixed domain \(D\) has the form \(\left({}^{3}\right)\)

\[ \tau(x)-\frac{1}{\pi}\int_{-1}^{1}[\ln|t-x|-\ln(1-tx)]\nu(t)\,dt=0, \qquad -1\le x\le 1. \tag{9} \]

We shall seek the function \(\nu(x)\) in the class \(H^*\) on the interval \([-1,1]\). Eliminating \(\tau(x)\) from (8) and (9), to determine \(\nu(x)\) we obtain the singular integral equation

\[ \alpha(x)\nu(x)+\frac{\beta(x)}{\pi}\int_{-1}^{1} \left(\frac{1}{t-x}-\frac{t}{1-tx}\right)\nu(t)\,dt+ \]

\[ +\alpha'(x)\beta(x)\int_{-1}^{1}\frac{\omega(x,t)}{\beta(t)}\nu(t)\,dt = f(x), \tag{10} \]

where

\[ f(x)=-2\beta(x)\left[\alpha(x)\psi\left(\frac{x-1}{2}\right)\right]', \]

and \(\omega(x,t)=1\) for \(t\in[-1,x]\), \(\omega(x,t)=0\) for \(t\notin[-1,x]\).

Equation (10) can very simply be reduced to a Fredholm equation. For this purpose we rewrite it in the form

\[ \alpha(x)\nu(x)+\frac{\beta(x)}{\pi}\int_{-1}^{1} \left(\frac{1}{t-x}-\frac{t}{1-tx}\right)\nu(t)\,dt = g(x), \tag{11} \]

where

\[ g(x)=f(x)-\alpha'(x)\beta(x)\int_{-1}^{1} \frac{\omega(x,t)}{\beta(t)}\nu(t)\,dt, \tag{12} \]

and we shall regard \(g(x)\) (belonging to the class \(H\)) as a known function.

Equation (11) is associated with the cyclic group \(\mathcal W_0(z)=z\), \(\mathcal W_1(z)=1/z\), and therefore is solvable explicitly \(\left({}^{3,4}\right)\). We note only the following: since the contour of integration is open and the transformation \(\mathcal W_1(z)=1/z\), taking the interval \([-1,1]\) into the remaining part of the real axis, leaves the endpoints fixed, then, as is easy to see, one cannot seek a solution of class \(h_0\) of equation (11). The index of this equation is equal to zero.

For the function \(v(x)\) we obtain the expression

\[ v(x)=a(x)g(x)-\frac{b(x)Z(x)}{\pi}\int_{-1}^{1}\frac{g(t)}{Z(t)} \left(\frac{1}{t-x}-\frac{t}{1-tx}\right)\,dt, \tag{13} \]

where

\[ a(x)=\frac{\alpha(x)}{\alpha^2(x)+\beta^2(x)},\qquad b(x)=\frac{\beta(x)}{\alpha^2(x)+\beta^2(x)},\qquad Z(x)=\sqrt{\alpha^2(x)+\beta^2(x)}\,e^{\Gamma(x)}, \]

\[ \Gamma(x)=\int_{-1}^{1}\theta(t)\left[\frac{1}{t-x}-\frac{1}{t(1-tx)}\right]\,dt,\qquad \theta(x)=-\frac{1}{\pi}\operatorname{arc\,tg}_{(-\pi/2,\ \pi/2)}\frac{\beta(x)}{\alpha(x)}. \]

Taking into account (12), for \(v(x)\) we obtain a Fredholm integral equation equivalent to equation (10):

\[ v(x)+\int_{-1}^{1}K(x,t)v(t)\,dt=h(x), \tag{14} \]

where

\[ K(x,t)=\frac{a(x)\alpha'(x)\beta(x)\omega(x,t)}{\beta(t)} -\frac{b(x)Z(x)}{\pi\beta(t)} \int_{-1}^{1}\frac{\alpha'(t_1)\beta(t_1)\omega(t_1,t)}{Z(t_1)} \left(\frac{1}{t_1-x}-\frac{t}{1-t_1x}\right)\,dt_1, \]

\[ h(x)=a(x)f(x)-\frac{b(x)Z(x)}{\pi}\int_{-1}^{1}\frac{f(t)}{Z(t)} \left(\frac{1}{t-x}-\frac{t}{1-tx}\right)\,dt. \]

All the basic Fredholm theorems are applicable to equation (14) \({}^{(5,6)}\). From the uniqueness of the solution of problem \(T_\alpha\) it follows that equation (14) is solvable. It is not hard to see that the function \(v(x)\), which is the solution of equation (14), belongs to the class \(H^*\) on the interval \([-1,1]\) and is differentiable in the interval \((-1,1)\).

It is interesting to note that if \(\beta(x)/\alpha(x)<0\), \(-1\le x\le 1\), the solution \(v(x)\) tends to infinity of order less than one at \(x=-1\); if \(\beta(x)/\alpha(x)>0\), \(-1\le x\le 1\), then, as is to be expected, \(v(x)\) tends to infinity of order less than one at \(x=1\).

Corollary. If \(\alpha(x)=\mathrm{const}\), condition (4) is satisfied, \(K(x,t)=0\), and the solution \(v(x)\) of equation (10) is found explicitly:

\[ v(x)=\frac{\alpha(x)f(x)}{\alpha^2(x)+\beta^2(x)} -\frac{\beta(x)e^{\Gamma(x)}}{\pi\sqrt{\alpha^2(x)+\beta^2(x)}} \int_{-1}^{1} \frac{f(t)e^{-\Gamma(t)}}{\sqrt{\alpha^2(x)+\beta^2(x)}} \left(\frac{1}{t-x}-\frac{t}{1-tx}\right)\,dt. \tag{15} \]

Let us note that in this case, in order to prove the existence of a solution of problem \(T_\alpha\), one can use the results of \({}^{(7)}\), if equation (1) is reduced to a system of mixed type.

For \(\alpha(x)=\beta(x)=1\), from formula (15) we obtain the function \(v(x)\) for problem \(T\).

Mathematical Institute with Computing Center
of the Bulgarian Academy of Sciences

Received
18 IV 1963

REFERENCES

\({}^{1}\) M. A. Lavrent’ev, A. V. Bitsadze, DAN, 70, No. 3, 373 (1950).
\({}^{2}\) A. V. Bitsadze, DAN, 70, No. 4, 561 (1950).
\({}^{3}\) A. V. Bitsadze, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 41 (1953).
\({}^{4}\) F. D. Gakhov, L. I. Chibrikova, Mat. sborn., 35, No. 3, 395 (1954).
\({}^{5}\) F. Goursat, Cours d’analyse mathématique, 3, 5 éd., Paris, 1956.
\({}^{6}\) N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
\({}^{7}\) F. I. Karamyshev, Sibirsk. matem. zhurn., 2, No. 4, 537 (1961).