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BRIEF COMMUNICATIONS
ON THE REPRESENTABILITY OF SPATIAL BOUNDARY VALUE PROBLEMS IN THE FORM OF A SYSTEM OF PLANE BOUNDARY VALUE PROBLEMS
Yu. A. Dement'ev
In a domain \(D\) of the variables \(x, y, z\), the classical boundary value problems of Dirichlet, Goursat, and Tricomi are considered. Their equivalence to systems of corresponding plane boundary value problems for characteristic sections of the domain \(D\) is proved\(^*\).
- In the domain \(D\) with surface \(S\), let us consider the following boundary value problems.
\(D_1\). Find a solution of the Laplace equation
\[ u_{xx}+u_{yy}+u_{zz}=0, \tag{1} \]
satisfying the boundary condition
\[ u=\varphi \quad \text{for } (x,y,z)\in S. \tag{2} \]
\(D_2\). Find a solution of the system of plane Dirichlet problems
\[ u_{1xx}+u_{1zz}=-u_{2xx}, \tag{3} \]
\[ u_{2yy}+u_{2zz}=-u_{1yy}, \tag{4} \]
satisfying the condition
\[ \left.u_1\right|_S=\varphi,\qquad \left.u_2\right|_S=0. \tag{5} \]
Here the functions \(u_1, u_2\) are functions of three independent variables \(x,y,z\). In equation (3) the parameter is \(y\), while in equation (4) the parameter is \(x\).
For a fixed value of the parameter, for example \(y=y_0\), one obtains a plane Dirichlet problem for the Poisson equation in the domain \(D_{y_0}\), which is the section of the body \(D\) by the plane \(y=y_0\), with boundary conditions \(u_1(x,y_0,z)=\varphi\) for \((x,y_0,z)\in S\).
Theorem 1. From the existence of a solution of problem \(D_1\) there follows the existence of a solution of problem \(D_2\), and, conversely, knowing the solution of problem \(D_2\), it is easy to find the solution of problem \(D_1\).
Indeed, it is obvious: the sum \(u(x,y,z)=u_1(x,y,z)+u_2(x,y,z)\) is a solution of problem \(D_1\).
Suppose now that the solution \(u(x,y,z)\) of problem \(D_1\) is known. Construct functions \(u_1(x,y,z)\), \(u_2(x,y,z)\) such that they satisfy the relation
\[ u_1(x,y,z)+u_2(x,y,z)=u(x,y,z) \tag{6} \]
and are a solution of problem \(D_2\).
From equations (3) and (4) we find
\[ u_{1zz}=-(u_{1xx}+u_{2xx})=-u_{xx}, \tag{7} \]
\[ u_{2zz}=-(u_{1yy}+u_{2yy})=-u_{yy}. \tag{8} \]
Equations (7) and (8) are integrated; the unknown constants arising here are uniquely determined by condition (5):
\[ u_1(x,y,z)=-\iint u_{xx}(x,y,z)\,dz\,dz+X_1(x,y)z+Y_1(x,y), \tag{9} \]
\(^*\) It is assumed that these characteristic sections have a “normal” contour in their derivative, i.e., a contour for which the Dirichlet problem for the Laplace equation is uniquely solvable.
\[ u_2(x,y,z)=-\iint u_{yy}(x,y,z)\,dz\,dz+X_2(x,y)z+Y_2(x,y). \tag{10} \]
For fixed \((x,y)=(x_0,y_0)\), the determinant of the system obtained for the unknowns, for example for \(X_1(x_0,y_0)\), \(Y_1(x_0,y_0)\), is not equal to zero if the straight line \(x=x_0\), \(y=y_0\) intersects the surface \(S\) at two distinct points \(z_1\) and \(z_2\), and is equal to
\[
\left|\begin{array}{cc}
z_1 & 1\\
z_2 & 1
\end{array}\right|=z_1-z_2\ne 0.
\]
At points where such a straight line is tangent to the surface \(S\), the functions \(X_1,Y_1,X_2,Y_2\) are determined by continuity.
Differentiating (9), (10), we obtain
\[
(u_1+u_2)_{zz}=u_{1zz}+u_{2zz}=-u_{xx}-u_{yy}=u_{zz},
\]
which, together with condition (5), is equivalent to condition (6).
Theorem 2. From the uniqueness of the solution of problem \(D_1\) follows the uniqueness of the solution of problem \(D_2\) and, conversely, from the uniqueness of the solution of problem \(D_2\) follows the uniqueness of the solution of problem \(D_1\).
Suppose the contrary to the direct assertion: there are two solutions \(u_{11},u_{21}\) and \(u_{12},u_{22}\). Then their difference \(u_1=u_{11}-u_{12}\), \(u_2=u_{21}-u_{22}\) satisfies equations (3) and (4) and the homogeneous boundary condition. Consequently, the sum \(u=u_1+u_2\) satisfies equation (1) with the homogeneous boundary condition (2). By assumption \(u\equiv0\), consequently, \(u_1=-u_2\), and from (3) and (4) we obtain \(u_{1zz}=u_{2zz}=0\), i.e.
\[
u_1=f_1(x,y)z+f_2(x,y),\qquad u_2=g_1(x,y)z+g_2(x,y).
\]
The only linear function equal to zero for two distinct values of the argument is identically zero.
Suppose the contrary to the converse assertion: there are two solutions \(v\) and \(w\) of problem \(D_1\). Then their difference \(u=v-w\) satisfies equation (1) with the homogeneous boundary condition (2). To this problem there corresponds problem \(D_2\) with the homogeneous boundary condition (5). By assumption it has only the zero solution; however, its nonzero solution can be obtained from the function \(u=v-w\) by the method described in the proof of Theorem 1. This is what was required to prove.
- Inside the characteristic cone \(K: z^2+y^2-(z+1)^2=0\) for \(-1\le z\le0\), consider the problems.
\(G_1\). Find a solution of the equation
\[
u_{rr}+\frac{1}{r^2}u_{\theta\theta}+\frac{1}{r}u_r-u_{zz}=0,
\tag{1'}
\]
satisfying the boundary condition
\[
u|_K=\varphi.
\tag{2'}
\]
\(G_2\). Find a solution of the system
\[
u_{1r'r'}-u_{1zz}=u_{2zz},
\tag{3'}
\]
\[
u_{2rr}+\frac{1}{r^2}u_{2\theta\theta}+\frac{1}{r}u_{2r}=-\frac{1}{r^2}u_{1\theta\theta}-\frac{1}{r}u_{1r},
\tag{4'}
\]
satisfying the boundary condition
\[
u_1|_K=\varphi,\qquad u_2|_K=0.
\tag{5'}
\]
Here \(r,\theta,z\) is the cylindrical coordinate system
\[
x=r\cos\theta,\qquad y=r\sin\theta,\qquad z=z;
\tag{11}
\]
in equation (3′) the role of the parameter is played by \(\theta\), and in equation (4′) the role of the parameter is played by \(z\).
For fixed \(\theta=\theta_0\) we have the Goursat problem in the triangle which is the section of the conoid by the plane \(\theta=\theta_0,\ \theta=\theta_0+\pi;\ r'=r\) for \(\theta=\theta_0,\ r'=-r\) for \(\theta=\theta_0+\pi\).
For fixed \(z=z_0\) we have the planar Dirichlet problem in the disk of the section of the conoid by the plane \(z=z_0\). The equivalence theorems for problems \(G_1\) and \(G_2\), analogous to Theorems 1 and 2 of the preceding point, are proved by the same scheme.
- The circle \(K\) of unit radius, lying in the plane \(z=0\), divides the domain \(D\) into two subdomains \(D_1\) and \(D_2\). \(D_1\) is a simply connected domain \((z>0)\) with surface \(S_1\) and \(K\). \(D_2\) is bounded for \(z<0\) by the characteristic cone \(K_1\) and by the circle of the plane \(z=-1/2\).
In the domain \(D\) we consider boundary-value problems.
\(T_1.\) Find a solution of the equation
\[ u_{rr}+\frac{1}{r^2}u_{\theta\theta}+\frac{1}{r}u_r \pm u_{zz}=0 \quad(\text{“+” for } z>0,\ \text{“−” for } z<0), \tag{1''} \]
continuous in \(\overline D\), with derivative \(u_z\) continuously joined along \(K\), satisfying the boundary conditions
\[ u\big|_{S_1}=\varphi,\qquad u\big|_{K_1}=\psi . \tag{2''} \]
\(T_2.\) Find a solution of the system of plane problems
\[ u_{1r'r'} \pm u_{1zz}=\mp u_{2zz}, \tag{3''} \]
\[ u_{2rr}+\frac{1}{r^2}u_{2\theta\theta}+\frac{1}{r}u_{2r} =-\frac{1}{r^2}u_{1\theta\theta}-\frac{1}{r}u_{1r}\,{}^*) \tag{4''} \]
with derivative \(u_{1z}\) continuously joined along \(K\), and satisfying the boundary conditions
\[ u_1\big|_{S_1}=\varphi,\quad u_2\big|_{S_1}=0,\quad u_1\big|_{K_1}=\psi,\quad u_2\big|_{K_1}=0 . \tag{5''} \]
For \(\theta=\theta_0,\ \theta=\theta_0+\pi\), equation \((3'')\) and conditions \((5'')\) give the Lavrent’ev–Bitsadze problem.
For \(z=z_0\), equation \((4'')\) and conditions \((5'')\) give the plane Dirichlet problem.
The equivalence theorems are proved analogously to the theorems of item 1.
In the present note we do not address the question of existence and uniqueness of the problems posed.
The representations obtained are convenient for constructing iterative difference schemes that have faster convergence than the usual Seidel iterations [1]. In the case of the Dirichlet problem this is confirmed by numerical computations. For constant boundary conditions the solution is obtained in one iteration step, independently of the number of points.
\[ \text{*) Notation of the preceding item.} \]
References
- Berezin I. S., Zhidkov N. P. Methods of Computation, 2. Moscow, Fizmatgiz, 1960.
Received by the editors
31 July 1966.
Moscow