Abstract Generated abstract
The paper studies a modified Tricomi-type boundary value problem for the Lavrent’ev-Bitsadze equation \(u_{xx}+\operatorname{sign} y\,u_{yy}=0\) in a mixed elliptic-hyperbolic domain with prescribed data on selected boundary arcs and characteristics, allowing certain constants to be determined as part of the solution. It proves uniqueness by deriving an extremum principle in the elliptic part and relating boundary conditions on the line of type change to characteristic data in the hyperbolic part. For a semicircular elliptic boundary with zero data, the construction is reduced to a Keldysh-Sedov boundary value problem for a piecewise holomorphic function, yielding an explicit integral representation with constants fixed by a uniquely solvable linear system. The remaining hyperbolic-domain solution is then obtained by the standard characteristic formula.
Full Text
MATHEMATICS
G. KARATOPRAKLIEV
ON A MODIFIED PROBLEM \(T_1\) FOR THE EQUATION
\[ u_{xx}+\operatorname{sign} y\, u_{yy}=0 \]
(Presented by Academician M. A. Lavrent’ev on 29 XII 1962)
In the present note we consider* a modified problem \(T_1\) for the Lavrent’ev–Bitsadze equation
\[
u_{xx}+\operatorname{sign} y\, u_{yy}=0.
\tag{1}
\]
Let \(D\) be a simply connected domain of the \(xy\)-plane, 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 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
\[
A_k\left[\frac12(a_k-1),-\frac12(a_k+1)\right]
\quad\text{and}\quad
B_k\left[\frac12(a_k+1),\frac12(a_k-1)\right],
\]
\(k=0,1,\ldots,n+1\) \((a_0=-1,\ a_{n+1}=1)\), lie respectively on the characteristics \(AC\) and \(BC\). Denote by \(D_1\) and \(D_2\) respectively the elliptic and hyperbolic parts of the mixed domain \(D\).
Modified problem \(T_1\). 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 the points \(E_k\), \(k=0,1,\ldots,n+1\), at which \(u_x\) and \(u_y\) may become infinite of order less than one; 4) \(u(x,y)\) assumes the prescribed values
\[
u=\varphi \quad \text{on } \sigma;
\tag{2}
\]
\[
u=\psi_k \text{ on } A_kA_{k+1}\ \text{for even } k;\qquad
u=\psi_k+\alpha_k \text{ on } B_kB_{k+1}\ \text{for odd } k,
\tag{3}
\]
where \(\varphi\) is continuous, while \(\psi_k(x)\), \(k=0,1,\ldots,n\), are twice differentiable functions whose second derivatives satisfy the Hölder condition, with \(\psi_0(-1)=\varphi(-1)\); \(\alpha_k\) are real constants not prescribed in advance.
Let \(n=2m\). The case \(n=2m-1\) is investigated analogously. The modified problem \(T_1\) cannot have more than one solution. This assertion reduces to the fact that if a solution \(u(x,y)\) of the modified problem \(T_1\) assumes the values \(u=0\) on \(\sigma\), \(u=0\) on \(A_{2k}A_{2k+1}\), \(k=0,1,\ldots,m\), \(u=\alpha_{2k-1}\) on \(B_{2k-1}B_{2k}\), \(k=1,\ldots,m\), then necessarily \(u=0\) in the domain \(D\) and \(\alpha_{2k-1}=0\), \(k=1,\ldots,m\). In the domain \(D_2\) the function \(u(x,y)\) has the form
\[
u(x,y)=\frac{\tau(x+y)+\tau(x-y)}{2}
+\frac12\int_{x-y}^{x+y} \nu(t)\,dt,
\tag{4}
\]
where \(\tau(x)=u(x,0)\), \(-1\le x\le 1\), \(\nu(x)=u_y(x,0)\), \(-1<x<1\).
By virtue of (3), from (4) we obtain
\[
u_x-\lambda(x)u_y=f(x),\qquad
y=0,\qquad
a_k<x<a_{k+1},\quad k=0,1,\ldots,2m,
\tag{5}
\]
where \(\lambda(x)=-1\) on \(L_1\); \(\lambda(x)=1\) on \(L_2\); \(f(x)=\psi'_{2k-1}\!\left[\frac12(x+1)\right]\) on \(L_1\);
* If no additional restrictions are imposed on the functions \(\psi_k\), then problem \(T_1\) may fail to have a solution of the required form, a fact not indicated in works \((^{1-8})\).
\(f(x)=\psi_{2k}\bigl[{}^{1}/_{2}(x-1)\bigr]\) on \(L_2\); \(L_1\) and \(L_2\) denote respectively the union of the intervals \((a_{2k-1},a_{2k})\), \(k=1,\ldots,m\), and \((a_{2k},a_{2k+1})\), \(k=0,1,\ldots,m\).
Hence, just as in problem \(\mathrm{T}_1\), we conclude that if \(\psi_k(x)\equiv 0\), \(k=0,1,\ldots,2m\), then the solution \(u(x,y)\) of the modified problem \(\mathrm{T}_1\) in the closed domain \(\overline D_1\) cannot attain a nonzero extremum in the intervals \(a_k<x<a_{k+1}\), \(k=0,1,\ldots,2m\), of the segment \(AB\). It is easy to see that the solution \(u(x,y)\) cannot have a nonzero extremum also at the points \(E_k\), \(k=1,\ldots,2m\). A known property of the characteristic quadrilateral for the string equation allows one to assert that \(u(a_{2k-1},0)=u(a_{2k},0)=\alpha_{2k-1}\), \(k=1,\ldots,m\). Suppose that the function \(u(x,y)\) attains a nonzero extremum at some point \(E_{2k}\), for example at the point \(E_{2k_0}\). Then the value of the function \(u(x,y)\) at the point \(E_{2k_0-1}\) will also be extremal. Separate* the point \(E_{2k_0-1}\) from the remaining points \(E_k\) by a level line \(\Gamma: u(x,y)=\mathrm{const}\), with endpoints on the segment \(AB\) and lying entirely in \(D_1\). To the domain bounded by the line \(\Gamma\) and a segment of the real axis, we apply Green’s formula:
\[ \iint (u_x^2+u_y^2)\,dx\,dy = -\int (u-\mathrm{const})\frac{\partial u}{\partial n}\,ds, \]
where \(n\) is the inward normal. From this formula, by virtue of the equalities \(u_x+u_y=0\) on \(L_1\) and \(u_x-u_y=0\) on \(L_2\), we conclude that \(u(x,y)=\mathrm{const}\) throughout the domain \(D_1\), and this is impossible when \(\varphi\not\equiv 0\).
Consequently, if \(\psi_k(x)\equiv 0\), \(k=0,1,\ldots,2m\), the solution \(u(x,y)\) of the modified problem \(\mathrm{T}_1\) 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 the modified problem \(\mathrm{T}_1\) follows immediately.
For simplicity we shall assume that \(\sigma\) coincides with the semicircle \(\sigma_0\) with endpoints at the points \(A,B\) and \(u=0\) on \(\sigma_0\). In addition, we shall assume that \(u_x\) and \(u_y\) are continuous in the closed domain \(\overline D_1\) everywhere except, possibly, at the points \(E_k\), \(k=0,1,\ldots,2m+1\).
Denote by \(\Phi(z)\) the function \(u(x,y)+iv(x,y)\), holomorphic in the domain \(D_1\) and satisfying the condition \(\Phi(-1)=0\). Analogously to problem \(\mathrm{T}_1\) \((^4)\), the determination of the function \(\Phi'(z)\) reduces to the determination of a function \(\Phi'(z)\) that is piecewise holomorphic in the upper half-plane, has a zero of the second order at infinity, and satisfies the boundary conditions
\[ \operatorname{Re}(1-i)\Phi'(x)=f(x)\quad \text{on } L_2, \qquad \operatorname{Im}(1-i)\Phi'(x)=-f(x)\quad \text{on } L_1, \tag{6} \]
\[ \operatorname{Re}(1-i)\Phi'(x)=\frac{1}{x^2}f(1/x)\quad \text{on } \overline L_1, \qquad \operatorname{Im}(1-i)\Phi'(x)=-\frac{1}{x^2}f(1/x)\quad \text{on } \overline L_2, \]
where \(\overline L_1\) and \(\overline L_2\) denote respectively the union of the intervals \((b_{2k},b_{2k-1})\), \(k=1,\ldots,m\), and \((b_{2k+1},b_{2k})\), \(k=0,1,\ldots,j-1,j+1,\ldots,m\), \((-\infty,b_{2j})\), \((b_{2j+1},\infty)\), it being assumed that \(a_{2j}<0<a_{2j+1}\); \(b_k=1/a_k\).
The solution of this problem of class \(h_0\) is given by the Keldysh–Sedov formula \((^{5,6})\)
\[ (1-i)\Phi'(z) = \frac{1}{\pi i}\frac{R_1(z)}{R_2(z)} \int_{-\infty}^{\infty} \frac{R_2(t)}{R_1(t)}\frac{g(t)}{t-z}\,dt + \frac{C_0+C_1z+\cdots+C_{2m-1}z^{2m-1}}{R(z)}, \tag{7} \]
where \(g(x)=f(x)\) on \(L_2\); \(g(x)=-if(x)\) on \(L_1\); \(x^2g(x)=f(1/x)\) on \(\overline L_1\);
\[ x^2g(x)=-if(1/x)\quad \text{on } \overline L_2 \]
and
\[ R_1(z)= \left[(z+1)\prod_{1}^{m}(z-a_{2k})(z-b_{2k})\right]^{1/2}, \]
\[ R_2(z)= \left[(z-1)\prod_{1}^{m}(z-a_{2k-1})(z-b_{2k-1})\right]^{1/2}, \qquad R(z)= \left[(z^2-1)\prod_{1}^{2m}(z-a_k)\times \right. \]
\[ \left. \times (z-b_k)\right]^{1/2}, \]
where by \(R_1(z)/R_2(z)\) is meant the branch holomorphic
* If one separates the point \(E_{2k_0}\) by a level line, then one cannot conclude from Green’s formula that \(u(x,y)=\mathrm{const}\) in \(D_1\).
on the plane cut along \(L_2,\ \overline{L}_1\), taking the value 1 at infinity, and by \(R(z)\) the branch, holomorphic on the plane cut in the same way, taking positive values on \(Ox\) for \(x>b_{2j+1}\); \(C_0, C_1,\ldots,C_{2m-1}\) are arbitrary real constants.
We find the function \(\Phi(z)\) by the formula \(\Phi(z)=\int_{-1}^{z}\Phi'(\xi)\,d\xi\), and it is necessary that \(\Phi(z)\) satisfy the condition \(\Phi(1/\overline{z})=-\Phi(z)\) \((^4)\). It is easy to see that, in order for this condition to be fulfilled, it is necessary and sufficient that \(C_k=C_{2m-k-1}\), \(k=0,1,\ldots,m-1\). To determine \(C_k\) and \(\alpha_{2k+1}\), \(k=0,1,\ldots,m-1\), we have the following conditions:
\[ \operatorname{Re}\Phi(a_{2k+1}) = \psi_{2k}\!\left[\frac12(a_{2k+1}-1)\right] + \psi_{2k+1}\!\left[\frac12(a_{2k+1}+1)\right] -\psi_{2m}(0)+\alpha_{2k+1}, \qquad k=0,1,\ldots,m-1, \]
\[ \tag{8} \operatorname{Re}\Phi(a_{2k+2}) = \psi_{2k+2}\!\left[\frac12(a_{2k+2}-1)\right] + \psi_{2k+1}\!\left[\frac12(a_{2k+2}+1)\right] -\psi_{2m}(0)+\alpha_{2k+1}, \qquad k=0,1,\ldots,m-1. \]
These conditions constitute a system of \(2m\) linear equations with respect to \(C_k\) and \(\alpha_{2k+1}\), \(k=0,1,\ldots,m-1\):
\[ \sum_{j=0}^{m-1}\beta_{kj}C_j-\alpha_{2k+1}=\beta_k, \qquad k=0,1,\ldots,m-1, \]
\[ \tag{9} \sum_{j=0}^{m-1}\gamma_{kj}C_j-\alpha_{2k+1}=\gamma_k, \qquad k=0,1,\ldots,m-1, \]
where \(\beta_{kj}\) and \(\gamma_{kj}\) do not depend on \(\psi_k(x)\), while \(\beta_k=0\) and \(\gamma_k=0\) when \(\psi_k(x)\equiv 0\).
From the uniqueness of the solution of the modified problem \(T_1\) it follows directly that system (9) 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 constructed by a known method.
Remark. In the case \(n=2m-1,\ m>1\), the values of the function \(u(x,y)\) at the points \(E_k,\ k=1,\ldots,2m-1\), are known up to a constant equal to the value of \(u(x,y)\) at the point \(C(0,-1)\). Denote \(u(0,-1)=\delta\). To determine \(C_k,\ k=0,1,\ldots,m-2\) (in this case \(C_k=-C_{2m-k-2}\), \(k=0,1,\ldots,m-2\), and \(C_{m-1}=0\)), \(\alpha_{2k-1}\), \(k=1,\ldots,m-1\), and \(\delta\), one obtains a uniquely solvable system of \(2m-1\) linear equations with respect to \(C_k,\alpha_{2k-1}\), and \(\delta\).
For \(n=1\) the modified problem \(T_1\) coincides with problem \(T_1\). In this case problem \(T_1\) is well posed (from the condition \(\Phi(1/\overline{z})=-\Phi(z)\) it follows that the only arbitrary constant \(C_0\) appearing in the expression for \(\Phi(z)\) is equal to zero).
The author expresses his gratitude to V. V. Aleksandrov for useful discussions.
Mathematical Institute with Computing Center
of the Bulgarian Academy of Sciences
Received
6 X 1962
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