UDC 517.946

MATHEMATICS

M. M. ZAINULABIDOV

ON ONE BOUNDARY-VALUE PROBLEM FOR A MODEL EQUATION OF MIXED TYPE WITH TWO PERPENDICULAR LINES OF DEGENERATION

(Presented by Academician M. A. Lavrent'ev on 4 III 1969)

In the finite simply connected domain \(\Omega\), bounded by the arcs \(AB: x^2+y^2=1,\ x\geqslant 0,\ y\geqslant 0;\ B^*A^*: x^2+y^2=1,\ x\leqslant 0,\ y\leqslant 0\) and the segments \(BB^*: y-x=1,\ y\geqslant 0;\ A^*A: x-y=1,\ y\leqslant 0\), consider the equation of mixed type

\[ u_{xx}+\operatorname{sgn}(xy)u_{yy}=0. \tag{1} \]

Let \(CO(OD)\) be the segment \(-\tfrac12\leqslant x\leqslant 0\) \((0\leqslant x\leqslant \tfrac12)\) of the characteristic \(x+y=0\) of equation (1); \(BA^*(AB^*)\) the segment of the straight line \(x=0\) \((y=0)\); \(\Omega_1(\Omega_1^*)\) and \(\Omega_2(\Omega_2^*)\) the hyperbolic parts of the mixed domain \(\Omega\), where \(x>0,\ x+y>0\) \((x>0,\ x+y<0)\) and \(x<0,\ x+y>0\) \((x<0,\ x+y<0)\), respectively; \(\Omega_3(\Omega_3^*)\) the elliptic part of \(\Omega\), where \(x>0,\ y>0\) \((x<0,\ y<0)\);

\[ \Delta=\bigcup_{i=1}^{3}\Omega_i\cup OA\cup OB, \qquad \Delta^*=\bigcup_{i=1}^{3}\Omega_i^*\cup OA^*\cup OB^*. \]

Problem A. It is required to determine a function \(u(x,y)\) with the following properties: 1) \(u(x,y)\in C(\overline{\Omega})\cap C^{(1,0)}(\Omega\setminus CD)\); 2) \(u_x(x,0),\ u_x(0,y),\ u_y(x,0),\ u_y(0,y)\) may tend to \(\infty\) of order less than unity at the points \(A,\ B,\ A^*,\ B^*\) and \(O\); 3) \(u(x,y)\) is a regular solution in \(\Omega\setminus(BA^*\cup AB^*\cup CD)\) of equation (1), satisfying the boundary conditions

\[ u\big|_{AB}=\varphi_1(\theta),\quad 0\leqslant\theta\leqslant\pi/2; \qquad u\big|_{B^*A^*}=\varphi_2(\theta),\quad \pi\leqslant\theta\leqslant 3\pi/2; \tag{2} \]

\[ u\big|_{BC}=\psi_1(y),\quad \tfrac12\leqslant y\leqslant 1; \qquad u\big|_{A^*D}=\psi_2(y),\quad -1\leqslant y\leqslant -\tfrac12. \]

With respect to the prescribed functions it is assumed that

\[ \varphi_i(\theta)=\sin^2 2\theta\,\overline{\varphi}_i(\theta),\quad i=1,2; \qquad \overline{\varphi}_1(\theta)\in C(0\leqslant\theta\leqslant\pi/2), \]

\[ \overline{\varphi}_2(\theta)\in C(\pi\leqslant\theta\leqslant 3\pi/2); \tag{3} \]

\[ \psi_1(y)\in C(\tfrac12\leqslant y\leqslant 1)\cap C^{(2,\alpha)}(\tfrac12<y<1), \]

\[ \psi_2(y)\in C(-1\leqslant y\leqslant-\tfrac12)\cap C^{(2,\alpha)}(-1<y<-\tfrac12), \tag{4} \]

where \(\psi_1'(y)\) as \(y\to\tfrac12,\ y\to 1\) and \(\psi_2'(y)\) as \(y\to -1,\ y\to-\tfrac12\) may tend to \(\infty\) of order lower than 1; \(\varphi_1(\pi/2)=\psi_1(1),\ \varphi_2(3\pi/2)=\psi_2(-1)\).

We shall show that the solution of boundary-value problem A exists, is unique, and can be written in explicit form.

\[ u(x,y)=u_i(x,y),\quad (x,y)\in\Omega_i;\qquad u(x,y)=u_i^*(x,y),\quad (x,y)\in\Omega_i^*;\quad i=1,2; \tag{5} \]

\[ u_y(x,0)= \begin{cases} \nu_1(x),&0<x<1,\\ \nu_1^*(x),&-1<x<0; \end{cases} \qquad u_x(0,y)= \begin{cases} \nu_2(y),&0<y<1,\\ \nu_2^*(y),&-1<y<0; \end{cases} \tag{6} \]

\[ u(x,x-1)=X_1(x),\qquad \tfrac12\leqslant x\leqslant 1; \]

\[ u(x,x+1)=X_2(x),\qquad -1\leqslant x\leqslant-\tfrac12, \tag{7} \]

where we shall assume that \(X_1(x)\in C^{(2,\alpha)}(1/2<x<1)\), \(X_2(x)\in C^{(2,\alpha)}(-1<x<-1/2)\), while \(X_1'(x)\) at \(x=1/2,\ x=1\) and \(X_2'(x)\) at \(x=-1,\ x=-1/2\) may have singularities of order less than 1 (below it will be seen that \(X_1(x)\) and \(X_2(x)\) possess these properties).

The solution \(u(x,y)\) of problem A in the domain \(\Delta\) coincides with the solution of the following Gellerstedt boundary-value problem: find a function \(u(x,y)\) having the properties: 1) \(u(x,y)\in C(\overline{\Delta})\cap C^{(1,0)}(\Delta)\); 2) \(u_x(x,0)\), \(u_x(0,y)\), \(u_y(x,0)\), \(u_y(0,y)\) may have singularities of order less than 1 at the points \(A,B\), and \(O\); 3) \(u(x,y)\) is a regular solution of equation (1) in \(\Delta\setminus(OA\cup OB)\), satisfying the boundary conditions \(u|_{AB}=\varphi_1(\theta)\), \(u|_{BC}=\psi_1(y)\), \(u|_{DA}=X_1(x)\).

In paper \((^2)\) it was proved that there exists a unique solution of the formulated boundary-value problem, which can be written in explicit form, and that the functions (6) and (7) are connected by the relations:

\[ \nu_i(x)=\frac{F_i(x)}{2}+\frac{1}{\pi}\int_0^1 N_1(x,t)F_i(t)\,dt-\frac{1}{\pi}\int_0^1 N_2(x,t)F_j(t)\,dt, \tag{8} \]

\[ \nu_i^*(-x)=\frac{F_i^*(-x)}{2}+\frac{1}{\pi}\int_0^1 N_1(x,t)F_i^*(-t)\,dt-\frac{1}{\pi}\int_0^1 N_2(x,t)F_j^*(-t)\,dt, \tag{9} \]

where

\[ N_k(x,t)=t\left(\frac{1-x^4}{1-t^4}\right)^{1/2} \left[\frac{1}{t^2+\beta x^2}-\frac{t^2}{1+\beta t^2x^2}\right], \qquad \beta= \begin{cases} -1, & k=1,\\ 1, & k=2; \end{cases} \tag{10} \]

\[ F_1(x)=2\frac{d}{dx}\left[X_1\!\left(\frac{x+1}{2}\right)-g_1(x)\right], \qquad F_2(x)=2\frac{d}{dx}\left[\psi_1\!\left(\frac{x+1}{2}\right)-g_2(x)\right]; \tag{11} \]

\[ F_1^*(-x)=2\frac{d}{dx}\left[g_1^*(-x)-X_2\!\left(-\frac{x+1}{2}\right)\right], \]

\[ F_2^*(-x)=2\frac{d}{dx}\left[g_2^*(-x)-\psi_2\!\left(-\frac{x+1}{2}\right)\right]; \tag{12} \]

\[ g_i(x)=\frac{1}{2\pi}\int_0^{\pi/2}\varphi_1(\theta) \left.\frac{\partial G_i}{\partial n}\right|_{|\zeta|=1}\,d\theta, \qquad g_i^*(-x)=\frac{1}{2\pi}\int_{\pi}^{3\pi/2}\varphi_2(\theta) \left.\frac{\partial G_i}{\partial n}\right|_{|\zeta|=1}\,d\theta; \tag{13} \]

\[ G_1(x;\xi,\eta)=\ln|1-\zeta^2x^2|-\ln|\zeta^2-x^2|, \qquad G_2(x;\xi,\eta)=\ln|1+\zeta^2x^2|-\ln|\zeta^2+x^2|, \]
\[ \zeta=\xi+i\eta;\quad 0<x<1,\quad i,j=1,2,\quad i\ne j;\quad n\text{ is the inner normal.} \]

Relations (9) are obtained in the same way as (8)—by solving the Gellerstedt problem in the domain \(\Delta^*\) with the data \(\varphi_2(\theta)\), \(\psi_2(y)\), and \(X_2(x)\) on \(B^*A^*\), \(A^*D\), and \(B^*C\), respectively.

By virtue of the continuity of the solution \(u(x,y)\) of problem A in the closed domain \(\overline{\Omega}\), we have

\[ u_1(x,-x)=u_1^*(x,-x),\qquad u_2(x,-x)=u_2^*(x,-x),\qquad u_1(0,0)=u_2^*(0,0); \tag{14} \]

\[ X_1(1/2)=\psi_2(-1/2),\qquad X_2(-1/2)=\psi_1(1/2). \tag{15} \]

From d’Alembert’s formula it is obvious that the functions \(u_1\) and \(u_2\) (\(u_1^*\) and \(u_2^*\)) can be written explicitly in terms of the functions \(X_1,\nu_1,\psi_1,\nu_2\) (respectively \(\psi_2,\nu_2^*\), \(X_2,\nu_1^*\)). Hence, on the basis of equalities (14) and (15), we conclude that

\[ \nu_1(x)+\nu_2^*(-x)+\frac{d}{dx}\left[\psi_2\!\left(-\frac{x+1}{2}\right)-X_1\!\left(\frac{x+1}{2}\right)\right]=0, \]

\[ \nu_2(x)+\nu_1^*(-x)+\frac{d}{dx}\left[X_2\!\left(-\frac{x+1}{2}\right)-\psi_1\!\left(\frac{x+1}{2}\right)\right]=0, \tag{16} \]

\[ 2\,[\psi_2(-1/2)-\psi_1(1/2)]+X_2(-1)-X_1(1)+ \int_0^1[\nu_1(t)+\nu_1^*(-t)]\,dt=0. \]

Eliminating from (8), (9), and (16) \(v_i(x)\), \(v_i^*(-x)\), \(i=1,2\), and taking into account (11), (12), we obtain

\[ \frac{2}{\pi}\int_0^1 N_1(x,t)\frac{d}{dt}\left[X_1\left(\frac{t+1}{2}\right)\right]dt + \frac{2}{\pi}\int_0^1 N_2(x,t)\frac{d}{dt}X_2\left(-\frac{t+1}{2}\right)dt = P_1(x), \tag{17} \]

\[ \frac{2}{\pi}\int_0^1 N_1(x,t)\frac{d}{dt}X_2\left(-\frac{t+1}{2}\right)dt + \frac{2}{\pi}\int_0^1 N_2(x,t)\frac{d}{dt}X_1\left(\frac{t+1}{2}\right)dt = P_2(x), \tag{18} \]

\[ \frac{2}{\pi}\int_0^1 ds\int_0^1 N_1(s,t)\frac{d}{dt} \left[ X_1\left(\frac{t+1}{2}\right)-X_2\left(-\frac{t+1}{2}\right) \right]dt = P, \tag{19} \]

where

\[ P_1(x)=\frac{d}{dx}\left[g_1(x)-g_2^*(-x)\right] +\frac{1}{\pi}\int_0^1 N_1(x,t)h_1(t)dt +\frac{1}{\pi}\int_0^1 N_2(x,t)h_2(t)dt, \tag{20} \]

\[ P_2(x)=\frac{d}{dx}\left[g_1^*(-x)-g_2(x)\right] +\frac{1}{\pi}\int_0^1 N_1(x,t)h_2(t)dt +\frac{1}{\pi}\int_0^1 N_2(x,t)h_1(t)dt, \tag{21} \]

\[ h_1(x)=2\frac{d}{dx}g_1(x)-F_2^*(-x),\qquad h_2(x)=2\frac{d}{dx}g_1^*(-x)+F_2(x), \tag{22} \]

and \(P\) is a particular number which depends only on the prescribed functions \(\psi_1\), \(\psi_2\), \(\varphi_1\), and \(\varphi_2\).

Consequently, problem A has been reduced to finding functions
\(X_1\left(\frac{x+1}{2}\right)\), \(X_2\left(-\frac{x+1}{2}\right)\in C^{(2,\alpha)}(0<x<1)\), satisfying the system of equalities (17), (18), (19) and the conditions (15), where the first derivatives of \(X_1\) and \(X_2\) may become infinite of order less than unity at \(x=0\) and \(x=1\).

It is clear that the functions \(X_1\) and \(X_2\) satisfying equalities (17) and (18) can be obtained from the solution of the equations

\[ \frac{2}{\pi}\int_0^1 [N_1(x,t)\pm N_2(x,t)]\mu_i(t)dt = M_i(x),\qquad i=1,2, \tag{23} \]

where

\[ \mu_i(x)=\frac{d}{dx} \left[ X_1\left(\frac{x+1}{2}\right) \pm X_2\left(-\frac{x+1}{2}\right) \right], \qquad M_i(x)=P_1(x)\pm P_2(x), \tag{24} \]

with the plus sign taken for \(i=1\), and the minus sign for \(i=2\).

Substituting the values of \(N_1(x,t)\), \(N_2(x,t)\) from (10) into (23), after the change of variables

\[ t=\tau^{1/4}\left(1+\sqrt{1-\tau^2}\right)^{-1/4},\qquad x=y^{1/4}\left(1+\sqrt{1-y^2}\right)^{-1/4}, \tag{25} \]

equations (23) take the form \({}^{2}\)

\[ \frac{1}{\pi}\int_0^1 \frac{\rho_i(\tau)}{\tau-y}\,d\tau = R_i(y),\qquad i=1,2, \tag{26} \]

where

\[ \rho_i(y)=\delta_i(x)\mu_i(x),\qquad R_i(y)=\delta_i(x)M_i(x); \tag{27} \]

\[ \delta_1(x)=(1+x^8)(1+x^4)^{-1}(1-x^4)^{-1/2},\qquad \delta_2(x)=(1+x^8)x^{-2}(1-x^4)^{-1/2}, \tag{28} \]

and \(x\) is related to \(y\) by equality (25).

It follows from (13) that \(g_i(x)\) and \(g_i^*(-x)\), for \(0<x<1\), are analytic functions.

Taking (3) into account, we ascertain that
\(\frac{d}{dx}g_i(x)\), \(\frac{d}{dx}g_i^*(-x)\) have finite limits as \(x\to 0\), \(x\to 1\). Hence, by virtue of the assumptions on the given functions \(\psi_i(x)\) (see (4)), the equalities (11), (12), (20), (21), (22), (24), and the properties of integrals of Cauchy type [1], we obtain that
\(M_i(x)\in C^{(1,\alpha)}(0<x<1)\), \(i=1,2\); moreover, at \(x=0\) and \(x=1\) these functions may tend to \(\infty\) with order less than 1.

On the basis of equalities (25), (27), and (28) we now conclude that the functions
\(R_i(y)\in C^{(1,\alpha)}(0<y<1)\) and may tend to \(\infty\) with order less than \(1/4\) \((3/4)\) in the case \(i=1\), and less than \(3/4\) \((3/4)\) in the case \(i=2\), when \(y\to 0\) \((y\to 1)\). From the same relations (24), (27), and (28) it follows that the solutions of equations (26), \(\rho_i(y)\), on the interval \(0\le y\le 1\) must possess the same properties as \(R_i(y)\). It is known that such solutions of equations (26) exist and can be written in quadratures [1].

Solving (26) and returning to the functions \(X_1, X_2\), by virtue of equality (19) and conditions (15), we ascertain that \(X_1\) and \(X_2\) are determined uniquely. Consequently, problem A is uniquely solvable.

I express my deep gratitude to A. V. Bitsadze, who proposed studying boundary value problems of type A for equations of mixed type with two intersecting lines of degeneration, and to A. M. Nakhushev for valuable advice and attention.

CITED LITERATURE

1. N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
2. M. M. Zainulabidov, Differential Equations, 5, No. 1 (1969).