MATHEMATICS

G. KARATOPRAKLIEV

ON SOME BOUNDARY-VALUE PROBLEMS FOR THE EQUATION

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

(Presented by Academician M. A. Lavrent’ev on 22 XII 1962)

In the present paper two boundary-value problems are considered for the Lavrent’ev–Bitsadze equation

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

of type of problem \(T_1\) \((^{1-3})\).

Let \(D\) be a simply connected domain of the plane \(xy\), bounded by a Jordan curve \(\sigma\) with endpoints at \(A(-1,0)\), \(B(1,0)\), situated in the upper half-plane \(y>0\), and by the characteristics \(AC:\ y=-x-1\) and \(BC:\ y=x-1\), issuing from the point \(C(0,-1)\). Let \(E_k(a_k,0)\), \(k=1,\ldots,n,\ -1<a_1<\cdots<a_n<1\), be prescribed points of the segment \(AB\). The points \(A_k[\,\tfrac12(a_k-1),-\tfrac12(a_k+1)\,]\) and \(B_k[\,\tfrac12(a_k+1),\tfrac12(a_k-1)\,]\), \(k=0,1,\ldots,n+1\) \((a_0=-1,\ a_{n+1}=1)\), lie respectively on the characteristics \(AC\) and \(BC\). Denote by \(E_{ik}[\,\tfrac12(a_i+a_k),\tfrac12(a_i-a_k)\,]\), \(i\le k,\ i=0,1,\ldots,n;\ k=1,\ldots,n+1\), the point of intersection of the characteristics \(E_iB_i\) and \(E_kA_k\) \((E_0=A,\ E_{n+1}=B,\ E_{0k}=A_k,\ E_{k,n+1}=B_k)\). Denote by \(D_1\) and \(D_2\), respectively, the elliptic and hyperbolic parts of the mixed domain \(D\).

Problem \(T_1^1\). 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 \(D\) everywhere except for the points of the segment \(AB\), the real axis, and the characteristics \(E_kA_k, E_kB_k\); 2) \(u(x,y)\) is continuous in the closed domain \(\overline D\); 3) the partial derivatives \(u_x\) and \(u_y\) are continuously matched at all points of the segment \(AB\), except, possibly, for the points \(E_k,\ k=0,1,\ldots,n+1\), at which \(u_x\) and \(u_y\) may tend to infinity of order less than one; 4) \(u(x,y)\) assumes the prescribed values

\[ u=\varphi \quad \text{on } \sigma; \tag{2} \]

\[ u=\psi_k \text{ on } E_kE_{k-1},\ k \text{ odd}; \qquad u=\psi_k \text{ on } E_{k-1}E_{k-1},\ k \text{ even}, \tag{3} \]

where \(\varphi\) is continuous, while \(\psi_k(x)\), \(k=1,\ldots,n+1\), are twice differentiable functions whose second derivatives satisfy a Hölder condition, and moreover \(\psi_{2k-1}(a_{2k-1})=\psi_{2k}(a_{2k-1})\), \(k=1,2,\ldots\) (for \(n=2m\) the condition \(\psi_{n+1}(1)=\varphi(1)\) must also be satisfied).

Problem \(T_1^1\) in the case \(n=1,\ a_1=0\) was investigated in the work of T. D. Dzhuraev \((^4)\)*. A problem of type \(T_1^1\) in the case \(n=1\) was first posed and investigated in the work of Gellerstedt \((^5)\) for the equation \(y^m z_{xx}+z_{yy}=0\).

Let \(n=2m\). The case \(n=2m-1\) is investigated analogously. In the domain \(D_2\) the solution \(u(x,y)\) of equation (1) has the form

\[ u(x,y)=\frac{\tau(x+y)+\tau(x-y)}{2} +\frac12\int_{x-y}^{x+y}\nu(t)\,dt, \tag{4} \]

* Arbitrary constants \(c_1\) and \(c_2\) cannot be determined by the method proposed in \((^4)\). It is easy to see that the function \(F(z)\) satisfies the condition \(F(1/\overline z)=-F(z)\), whence it follows that \(\operatorname{Re}F(z)=0\) for \(z=-1\) and \(z=1\) for any \(c_1\) and \(c_2\). These constants may be determined from the conditions \(F(-1)=0\) and \(\operatorname{Re}F(0)=\psi_1(0)-\omega_2(0)-W(0,0)\).

where \(\tau(x)=u(x,0)\), \(-1\leq x\leq 1\), and \(\nu(x)=u_y(x,0)\), \(-1<x<1\).

By virtue of (3), from (4) we obtain

\[ u_x-\lambda(x)u_y=f(x),\quad y=0,\quad a_k<x<a_{k+1},\quad k=0,1,\ldots,2m, \tag{5} \]

where \(\lambda(x)=-1\) on \(L_1\); \(\lambda(x)=1\) on \(L_2\); \(f(x)=\psi'_{2k-1}\bigl[\tfrac12(x+a_{2k-1})\bigr]\) on \(L_1\); \(f(x)=\psi'_{2k}\bigl[\tfrac12(x+a_{2k-1})\bigr]\) on \(L_2\); \(L_1\) and \(L_2\) are, respectively, the union of the intervals \((a_{2k-2},a_{2k-1})\), \(k=1,\ldots,m+1\), and \((a_{2k-1},a_{2k})\), \(k=1,\ldots,m\).

Hence, analogously to problem \(T_1\) for \(n=2m-1\), we conclude that if \(\psi_k(x)\equiv 0\), \(k=1,\ldots,2m+1\), then the solution \(u(x,y)\) of problem \(T_1^{\,1}\) 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 \(T_1^{\,1}\) follows immediately.

Without loss of generality one may assume that \(\varphi\equiv 0\) \((^6)\). We shall additionally assume that \(\sigma\) is a smooth arc satisfying Lyapunov’s condition, and that \(u_x\) and \(u_y\) are continuous in the closed domain \(\overline D_1\) everywhere except, possibly, at the points \(E_k\), \(k=0,1,\ldots,2m+1\). By a conformal mapping one can arrange that \(\sigma\) coincide with the semicircle \(\sigma_0\) with endpoints at the points \(A\) and \(B\) \((^6)\). We shall assume that \(\sigma\) coincides with \(\sigma_0\).

Denote by \(\Phi(z)\) the function \(u(x,y)+iv(x,y)\), holomorphic in the domain \(D_1\) and satisfying the condition \(\Phi(-1)=0\).

The conditions (5) may be written in the form

\[ \operatorname{Re}(1-i)\Phi'(x)=f(x)\ \text{on }L_2,\qquad \operatorname{Im}(1-i)\Phi'(x)=-f(x)\ \text{on }L_1. \tag{6} \]

By virtue of the condition \(u=0\) on \(\sigma_0\), we conclude that the function \(\Phi(z)\) is analytically continued through \(\sigma_0\) to the entire upper half-plane, and

\[ \Phi(z)= \begin{cases} u(x,y)+iv(x,y), & \text{inside }D_1,\\ -u\!\left[\dfrac{x}{x^2+y^2},\dfrac{y}{x^2+y^2}\right] +i\,v\!\left[\dfrac{x}{x^2+y^2},\dfrac{y}{x^2+y^2}\right], & \text{outside }D_1. \end{cases} \tag{7} \]

Hence it follows that the function \(\Phi(z)\) must satisfy the condition

\[ \overline{\Phi(1/\bar z)}=-\Phi(z). \tag{8} \]

At infinity \(\Phi'(z)\) has a zero of second order owing to the boundedness of \(u(x,y)\) (7).

Let \(a_{2j-1}<0<a_{2j}\). From (7) and (6) we obtain

\[ \begin{aligned} \operatorname{Re}(1-i)\Phi'(x)&=(1/x^2)\,f(1/x)\quad &&\text{on }\overline L_1,\\ \operatorname{Im}(1-i)\Phi'(x)&=-(1/x^2)\,f(1/x)\quad &&\text{on }\overline L_2, \end{aligned} \tag{9} \]

where \(\overline L_1\) and \(\overline L_2\) denote, respectively, the union of the intervals \((b_{2k-1},b_{2k-2})\), \(k=1,\ldots,m+1\), and \((b_{2k},b_{2k-1})\), \(k=1,\ldots,j-1,j+1,\ldots,m\), \((-\infty,b_{2j-1})\), \((b_{2j},\infty)\), and \(b_k=1/a_k\).

Thus, the determination of the function \(\Phi'(z)\) is reduced to determining, in the upper half-plane, a piecewise holomorphic function \(\Phi'(z)\) having a zero of second order at infinity and satisfying the boundary conditions (6) and (9).

The solution of this problem of class \(h_0\) is given by the well-known Keldysh–Sedov formula \((^7,^8)\)

\[ (1-i)\Phi'(z)=\frac{1}{\pi i}\frac{R_1(z)}{R_2(z)} \int_{-\infty}^{\infty}\frac{R_2(t)}{R_1(t)}\,\frac{g(t)}{t-z}\,dt +\frac{C_0+C_1z+\cdots+C_{2m-1}z^{2m-1}}{R(z)}, \tag{10} \]

where \(g(x)=f(x)\) on \(L_2\); \(g(x)=-if(x)\) on \(L_1\); \(x^2g(x)=f(1/x)\) on \(\overline{L}_1\);

\[ x^2g(x)=-if(1/x)\quad \text{on } \overline{L}_2 \quad \text{and} \quad R_1(z)=\left[(z-1)\prod_{1}^{m}(z-a_{2k-1})(z-b_{2k-1})\right]^{1/2}; \]

\[ R_2(z)=\left[(z+1)\prod_{1}^{m}(z-a_{2k})(z-b_{2k})\right]^{1/2},\quad R(z)=\left[(z^2-1)\prod_{1}^{2m}(z-a_k)(z-b_k)\right]^{1/2}, \]

where by \(R_1(z)/R_2(z)\) we mean the branch holomorphic in the plane cut along \(L_2,\overline{L}_1\), taking the value \(1\) at infinity, and by \(R(z)\) the branch holomorphic in the plane cut in the same way, taking positive values on \(Ox\) for \(x>b_{2j}\); \(C_0,C_1,\ldots,C_{2m-1}\) are arbitrary real constants.

After determining \(\Phi'(z)\), the function \(\Phi(z)\) is found from the formula

\[ \Phi(z)=\int_{-1}^{z}\Phi'(\zeta)\,d\zeta . \tag{11} \]

It is easy to see that, for condition (8) to hold, it is necessary and sufficient that

\[ C_k=-C_{2m-k-1},\quad k=0,1,\ldots,m-1. \]

To determine \(C_k,\ k=0,1,\ldots,m-1\), we have the following conditions:

\[ \operatorname{Re}\Phi(a_{2k-1})=\psi_{2k-1}(a_{2k-1}),\quad k=1,\ldots,m. \tag{12} \]

These conditions constitute a system of \(m\) linear equations with respect to \(C_k,\ k=0,1,\ldots,m-1\):

\[ \sum_{j=0}^{m-1}\gamma_{kj}C_j=\gamma_k,\quad k=1,\ldots,m, \tag{13} \]

where \(\gamma_{kj}\) do not depend on \(\psi_k(x)\), while \(\gamma_k=0\) when \(\psi_k(x)\equiv0\).

From the uniqueness of the solution of problem \(T_1^1\) it follows directly that system (13) is uniquely solvable.

The real part of the function \(\Phi'(z)\) gives the required function \(u(x,y)\) in the domain \(D_1\). In the domain \(D_2\), the solution \(u(x,y)\) is given by formula (4), where \(\tau(x)\) and \(\nu(x)\) are determined respectively from (11) and (10).

Remark. To prove the existence of a solution one may use the method of integral equations. Just as in problem \(T_1\) \((^2)\), to determine the function \(\nu(x)\) (it is assumed that \(\sigma\) coincides with \(\sigma_0\) and \(\varphi=0\)) one obtains the singular integral equation

\[ \lambda(x)\nu(x)+\frac{1}{\pi}\int_{-1}^{1}\left(\frac{1}{t-x}-\frac{t}{1-tx}\right)\nu(t)\,dt=-f(x). \tag{14} \]

Analogously to paper \((^9)\), we conclude that the solution* of this equation, belonging to the class \(h_0\) and satisfying the Hölder condition, is given by the formula

\[ \nu(x)=-\operatorname{Im}\Phi'^{+}(x), \]

where \(\Phi'^{+}(x)\) is determined from (10).

Problem \(T_0\). 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 \(D\) everywhere except at the points of the segment \(AB\) and the characteristics \(E_kA_k,\ E_kB_k\); 2) \(u(x,y)\) is continuous in the closed domain \(\overline{D}\); 3) the partial derivatives \(u_x\) and \(u_y\) are continuously matched at all points of the segment \(AB\), except, possibly, the points \(E_k,\ k=1,\ldots,n\), at which \(u_x\) and \(u_y\) may tend to infinity of logarithmic type, and the points \(A,B\), at which \(u_x\) and \(u_y\) may tend to infin—

* In paper \((^2)\) only one particular solution of integral equation (11), belonging to the class of sought solutions, was obtained. With its help it is impossible to construct a solution of problem \(T_1\) for \(n>1\). This is obvious if \(\psi_k=\alpha_k,\ k=0,1,\ldots,2m\), where \(\alpha_k\) are constants, and \(\varphi=0\).

of finiteness of order less than unity; 4) \(u(x,y)\) assumes the prescribed values

\[ u=\varphi \ \text{on } \sigma,\qquad u=\psi_k(x) \ \text{on } A_kA_{k+1},\quad k=0,1,\ldots,n, \tag{15} \]

where \(\varphi\) is continuous, and the \(\psi_k(x)\) are twice differentiable functions whose second derivatives satisfy Hölder’s condition, with
\[ \varphi(-1)=\psi_0(-1),\quad \psi_k\!\left[\frac12(a_{k+1}-1)\right] = \psi_{k+1}\!\left[\frac12(a_{k+1}-1)\right], \quad k=0,1,\ldots,n-1. \]

This boundary-value problem is a generalization of the Tricomi problem \((^{1,3})\) in the case when the first derivative of the boundary function in the domain \(D_2\) has a finite number of discontinuities of the first kind.

In essence, \(T_0\) is a problem of the type of problem \(T_1^{1}\) and is investigated analogously to it. For the function \(\Phi'(z)\) we obtain

\[ \Phi'(z)=\frac{1-i}{2\pi}\left(\frac{z+1}{z-1}\right)^{1/2} \int_{-1}^{1} \left(\frac{t-1}{t+1}\right)^{1/2} \left(\frac{1}{t-z}-\frac{t}{1-tz}\right) f(t)\,dt, \]

where
\[ f(x)=\psi'_k\!\left[\frac12(x-1)\right],\qquad a_k<x<a_{k+1},\quad k=0,1,\ldots,n, \]
and by \(\left[(z+1)/(z-1)\right]^{1/2}\) is meant the branch holomorphic in the plane cut along \((-1,1)\) and taking the value \(1\) at infinity.

The function \(\Phi'(z)\) is bounded at \(z=-1\), becomes infinite of order \(1/2\) at \(z=1\), and has a logarithmic singularity at the points \(a_k,\ k=1,\ldots,n\).

In an analogous way one can generalize problem \(T_1^{1}\).

Mathematical Institute with Computing Center
of the Bulgarian Academy of Sciences

Received
6 X 1962

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, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 41 (1953).
\(^{4}\) T. D. Dzhuraev, Izv. AN UzSSR, ser. fiz.-matem. nauk, 6, 3 (1961).
\(^{5}\) S. Gellerstedt, Ark. Math., Astr. och Phys., 26 A, 3, 1 (1937).
\(^{6}\) A. V. Bitsadze, Equations of Mixed Type, Moscow, 1959.
\(^{7}\) N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
\(^{8}\) M. V. Keldysh, L. I. Sedov, DAN, 16, No. 1, 7 (1937).
\(^{9}\) A. V. Bitsadze, UMN, 12, issue 5, 185 (1957).