MATHEMATICS

G. KARATOPRAKLIEV

ON A GENERALIZATION OF THE TRICOMI PROBLEM

(Presented by Academician M. A. Lavrent’ev, 23 V 1964)

In the present note we consider a boundary-value problem for the Tricomi equation

\[ y u_{xx}+u_{yy}=0, \tag{1} \]

which is a generalization of the Tricomi problem (1) in the case when the unknown function \(u(x,y)\) and its partial derivative \(u_y(x,y)\) have discontinuities of the first kind on the line of parabolic degeneration.

Let \(D\) be a simply connected finite domain of the plane \(xOy\), bounded by a Jordan curve \(\sigma\) with endpoints at the points \(A(-1,0)\), \(B(1,0)\), situated in the upper half-plane \(y>0\), and by the segments \(AC\) and \(BC\) of the characteristics \(x+1=(2/3)(-y)^{3/2}\) and \(1-x=(2/3)(-y)^{3/2}\) of equation (1), issuing from the point \(C[0,-(3/2)^{2/3}]\). 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 \(y\ne 0\); 2) \(u(x,y)\) is continuous in the closed domain \(\overline D\) for \(y\ne 0\); 3) the partial derivatives \(u_x\) and \(u_y\) are continuous in the domain \(D\) for \(y\ne 0\), and at the points \(A\) and \(B\) they may tend to infinity of order lower than \(2/3\); 4) on the line of parabolic degeneration the functions \(u(x,y)\) and \(u_y(x,y)\) satisfy the gluing conditions

\[ u(x,-0)=\alpha(x)u(x,+0)+\gamma(x),\qquad -1\le x\le 1, \]

\[ u_y(x,-0)=\beta(x)u_y(x,+0)+\delta(x),\qquad -1<x<1, \tag{2} \]

where \(\alpha(x)\), \(\beta(x)\), \(\gamma(x)\), and \(\delta(x)\) are given functions, with \(\beta(x)\) and \(\gamma(x)\) differentiable and \(\beta'(x)\), \(\gamma'(x)\) satisfying a Hölder condition for \(-1\le x\le 1\), while \(\delta(x)\) and \(\alpha(x)\) are, respectively, twice and three times continuously differentiable for \(-1\le x\le 1\); \(\beta(x)\ne 0\) for \(-1\le x\le 1\) and

\[ \alpha(x)\beta(x)\ge 0,\qquad \beta(x)\int_{-1}^{x}\frac{\alpha'(t)}{(x-t)^{1/3}}\,dt\ge 0,\qquad -1<x<1; \tag{3} \]

5) \(u(x,y)\) assumes the prescribed values

\[ u=\varphi \quad \text{on } \sigma, \tag{4} \]

\[ u=\psi(x) \quad \text{on } AC, \tag{5} \]

where \(\varphi\) and \(\psi\) are twice continuously differentiable functions, \(\psi''(x)\) satisfies a Hölder condition for \(-1\le x\le 0\), and \(\varphi(A)=\alpha(A)\psi(A)+\gamma(A)\).

For \(\alpha(x)=\beta(x)=1\), \(\gamma(x)=\delta(x)=0\), the problem \(T_\alpha\) coincides with the Tricomi problem.

In the domain \(D_2\), by virtue of (2), the solution \(u(x,y)\) has the form

\[ u(x,y)=2^{2/3}k_1\int_{-1}^{1}[\alpha(v)\tau(v)+\gamma(v)](1-t^2)^{-5/6}\,dt+ \]

\[ +\left(\frac{2}{3}\right)^{2/3}k_2y\int_{-1}^{1}[\beta(v)\nu(v)+\delta(v)](1-t^2)^{-1/6}\,dt, \tag{6} \]

where \(\tau(x)=u(x,+0)\), \(\nu(x)=u_y(x,+0)\), \(-1\le x\le 1\), and \(v=x+\frac{2}{3}(-y)^{3/2}t\), \(k_1=\Gamma(1/3)/\Gamma^2(1/6)\), \(k_2=(3/4)^{2/3}\Gamma(5/3)/\Gamma^2(5/6)\).

By virtue of (5), from (6) we obtain

\[ \alpha(x)\tau(x)+\gamma(x)=\psi_1(x)+k\int_{-1}^{x}\frac{\beta(t)\nu(t)+\delta(t)}{(x-t)^{1/3}}\,dt, \tag{7} \]

where \(k=3^{2/3}\Gamma^3(1/3)/4\pi^2\) and

\[ \psi_1(x)=\frac{(x+1)^{5/6}}{2k_1\pi}\frac{d}{dx} \int_{-1}^{1} \frac{\psi\left(\dfrac{t-1}{2}\right)} {(x-t)^{1/6}(t+1)^{2/3}}\,dt . \]

For \(\psi(x)=\gamma(x)=\delta(x)\equiv 0\), from (7), applying Abel’s inversion formula, we obtain

\[ \beta(x)\nu(x)=\frac{\sqrt{3}}{2k\pi}\frac{d}{dx} \int_{-1}^{x}\frac{\alpha(t)\tau(t)}{(x-t)^{2/3}}\,dt . \]

Hence, taking (3) into account, just as in the Tricomi problem \((^2)\), we conclude that the solution \(u(x,y)\) attains a positive maximum and a negative minimum in the closed domain \(\overline{D}_1\) on the curve \(\sigma\) (the extremum principle). From this principle the uniqueness of the solution of problem \(T_\alpha\) follows immediately.

The existence of a solution of problem \(T_\alpha\) will be proved if the function \(\nu(x)\) can be determined (1). We shall assume that \(\sigma\) coincides with the normal contour \(\sigma_0: x^2+\frac{4}{9}y^3=1\) and that \(\varphi(A)=\psi(A)=\varphi(B)=0\), \(\varphi'(A)=\psi'(A)=\varphi'(B)=0\), where the derivatives are taken in the direction tangent to the contour \(\sigma_0+AC\).

The relation between \(\tau(x)\) and \(\nu(x)\), brought from the elliptic part of the domain \(D\), has the form

\[ \tau(x)+k\int_{-1}^{1}\left[|t-x|^{-1/3}-(1-tx)^{-1/3}\right]\nu(t)\,dt=F^*(x), \tag{8} \]

where

\[ F^*(x)=\frac{k}{2^{1/3}3^{7/3}}(1-x^2) \int_{-1}^{1}(1-2tx+x^2)^{-7/6}(1-t^2)^{-1/3}\varphi(t)\,dt . \]

We shall seek the function \(\nu(x)\) in the class \(H^*\) on the segment \([-1,1]\) (the definition of the class \(H^*\) is given in \((^4)\)). Let \(-1<x<1\). Eliminating \(\tau(x)\) from (7) and (8), to determine \(\nu(x)\) we obtain the singular integral equation

\[ [\alpha(x)+2\beta(x)]\nu(x)+\frac{\sqrt{3}\,\alpha(-1)}{\pi} \int_{-1}^{1} \left(\frac{t+1}{x+1}\right)^{2/3} \left(\frac{1}{t-x}-\frac{1}{1-tx}\right)\nu(t)\,dt+ \]

\[ +\int_{-1}^{1}\frac{k_1(x,t)\nu(t)}{t-x}\,dt+ \int_{-1}^{1}\frac{k_2(x,t)\nu(t)}{1-tx}\,dt+ \int_{-1}^{1}k_3(x,t)\nu(t)\,dt=F(x), \tag{9} \]

where

\[ f(x)=\frac{\sqrt{3}}{\pi}\frac{d}{dx}\int_{-1}^{x}\frac{\varphi^{*}(t)\,dt}{(x-t)^{2/3}}, \]

\[ \varphi^{*}(x)=\frac{1}{k}\,[\alpha(x)F^{*}(x)-\psi_{1}(x)+\gamma(x)]-\int_{-1}^{x}\frac{\delta(t)\,dt}{(x-t)^{1/3}}, \]

\[ k_{1}(x,t)=\frac{\sqrt{3}}{\pi}\int_{-1}^{x}\left(\frac{t-\xi}{x-\xi}\right)^{2/3}\alpha'(\xi)\,d\xi,\qquad k_{2}(x,t)=-\frac{\sqrt{3}}{\pi}\int_{-1}^{x}\left(\frac{1-t\xi}{x-\xi}\right)^{2/3}\alpha'(\xi)\,d\xi, \]

\[ k_{3}(x,t)=\frac{2\sqrt{3}\,\omega(x,t)}{\pi}\frac{1}{x-t}\int_{t}^{x}\left(\frac{t-\xi}{x-\xi}\right)^{2/3}\alpha'(\xi)\,d\xi, \]

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

In addition, we shall assume that

\[ \alpha(-1)=\alpha'(1)=0. \tag{10} \]

In this case equation (9) takes the form

\[ a(x)v(x)+\frac{b(x)}{\pi}\int_{-1}^{1}\left(\frac{1}{t-x}-\frac{1}{1-tx}\right)v(t)\,dt +\int_{-1}^{1}k(x,t)v(t)\,dt=F(x), \tag{11} \]

where \(a(x)=\alpha(x)+2\beta(x)\), \(b(x)=\sqrt{3}\alpha(x)\), and

\[ k(x,t)=\frac{2}{\pi\sqrt{3}}\int_{-1}^{x} \frac{\alpha'(\xi)\,d\xi}{(x-\xi)^{2/3}[x+\theta(t-x)-\xi]^{1/3}} \]
\[ -\frac{\sqrt{3}}{\pi}\frac{1}{1-tx}\int_{-1}^{x} \left[\left(\frac{1-t\xi}{x-\xi}\right)^{2/3}-1\right]\alpha'(\xi)\,d\xi +\frac{2\sqrt{3}\,\omega(x,t)}{\pi}\frac{1}{x-t}\int_{t}^{x} \left(\frac{t-\xi}{x-\xi}\right)^{2/3}\alpha'(\xi)\,d\xi, \]

\[ 0<\theta<1. \]

To equation (11) one can apply the well-known method of regularization \((^{2})\). For this purpose we rewrite it in the form

\[ a(x)v(x)\frac{b(x)}{\pi}\int_{-1}^{1}\left(\frac{1}{1-x}-\frac{1}{1-tx}\right)v(t)\,dt=g(x), \tag{12} \]

\[ g(x)=F(x)-\int_{-1}^{1}k(x,t)v(t)\,dt. \tag{13} \]

It is not difficult to see that \(g'(x)\) satisfies a Hölder condition for \(-1<x<1\).

The solution of equation (12) is given by the formula \((^{3})\)

\[ v(x)=A(x)g(x)-\frac{B(x)Z(x)}{\pi}\int_{-1}^{1}\frac{g(t)}{Z(t)} \left(\frac{1}{t-x}-\frac{1}{1-tx}\right)\,dt, \tag{14} \]

\[ A(x)=\frac{a(x)}{a^{2}(x)+b^{2}(x)},\qquad B(x)=\frac{b(x)}{a^{2}(x)+b^{2}(x)},\qquad Z(x)=\sqrt{a^{2}(x)+b^{2}(x)}\,e^{\Gamma^{*}(x)}. \]

\[ \Gamma^{*}(x)=\int_{-1}^{1}\theta(t)\left[\frac{1}{t-x}-\frac{1}{t(1-tx)}\right]dt,\qquad \theta(x)=-\frac{1}{\pi}\underset{(-\pi/2,\ \pi/2)}{\operatorname{arctg}}\frac{b(x)}{a(x)}. \]

Taking (13) into account, for \(\nu(x)\) we obtain the Fredholm integral equation equivalent to equation (11)

\[ \nu(x)+\int_{-1}^{1} H(x,t)\,\nu(t)\,dt=h(x), \tag{15} \]

where

\[ H(x,t)=-A(x)k(x,t)=\frac{B(x)Z(x)}{\pi}\int_{-1}^{1}\frac{k(t_1,t)}{Z(t_1)} \left(\frac{1}{t_1-x}-\frac{1}{1-t_1x}\right)\,dt_1, \]

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

The kernel \(H(x,t)\) has a moving singularity of order \(1/3\) for \(t=x\) and a fixed singularity of order less than \(2/3\) for \(x=1\). Consequently, the Fredholm theorems are applicable to equation (15). From the uniqueness of the solution of problem \(T_\alpha\) it follows that equation (15) is solvable. From (14) it follows directly that the solution \(\nu(x)\) belongs to the class \(H^*\) on the interval \([-1,1]\) and is differentiable in the interval \((-1,1)\).

Remark 1. If the function \(\alpha(x)\) is linear, i.e. \(\alpha(x)=px+q\), the existence of a solution of the problem \(T_\alpha\) can be proved without additional assumptions of the type (10). In the case where \(\alpha(x)=C=\mathrm{const}\), the solution \(\nu(x)\) of equation (9) is found explicitly:

\[ \nu(x)=A(x)F(x)-\frac{B(x)Z(x)}{\pi}\int_{-1}^{1}\frac{F(t)}{Z(t)} \left(\frac{t+1}{x+1}\right)^{2/3} \left(\frac{1}{t-x}-\frac{1}{1-tx}\right)\,dt. \]

From this formula, for \(\alpha(x)=\beta(x)=1,\ \gamma(x)=\delta(x)=0\), one obtains the function \(\nu(x)\) for the Tricomi problem.

Remark 2. The case where the curve \(\sigma\) has continuous curvature and ends in arcs \(AA'\) and \(BB'\) of an arbitrarily small length of a normal curve is investigated in the same way as in the Tricomi problem \(({}^{1,2})\).

The author expresses gratitude to A. V. Bitsadze for helpful suggestions and attention to this work.

Mathematical Institute with Computing Center
of the Bulgarian Academy of Sciences
Sofia, Bulgaria

Received
21 V 1964

REFERENCES

1. F. Tricomi, Mem. Lincei, ser. 5, 14, fasc. 7, 133 (1923).
2. A. V. Bitsadze, Equations of Mixed Type, Moscow, 1959.
3. F. D. Gakhov, L. I. Chibrikova, Mat. sbornik, 35, No. 3, 395 (1954).
4. N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.