UDC 517.947.42

ON CERTAIN IMPROPERLY POSED PROBLEMS IN POTENTIAL THEORY

Yu. T. Antokhin

The paper studies a number of improperly posed problems arising in the theory of equations of elliptic type (the Cauchy problem and the Dirichlet problem), and also shows that some important improperly posed problems “with large perturbations” can be reduced, by a simple and uniform device, to analogous problems, but already with “small perturbations.” The problems presented below arose in an attempt to extend the range of application of methods developed in the study of inverse problems in the theory of the Newtonian potential.

§ 1. On nonuniqueness of the solution of the Dirichlet problem for elliptic systems

In this section we shall give examples of nonuniqueness of the solution of the Dirichlet problem for equations of elliptic type. In particular, we shall consider multidimensional analogues of the well-known system of A. V. Bitsadze

\[ \frac{\partial^2 w}{\partial z^2}=0, \tag{1} \]

where

\[ w=u+iv,\quad u=u(x,y),\quad v=v(x,y),\quad z=x+iy, \]

\[ z \in D,\quad D\text{ is a certain domain in the plane},\quad \frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right), \]

(see [5], p. 60).

Let us give examples of nonuniqueness of the solution of the Dirichlet problem for elliptic systems of the 2nd order in \(n\)-dimensional space, in which the number of equations and of unknown functions is equal to \(n\). We write system (1) in matrix form

\[ A(p)u=0, \tag{2} \]

where \(u=u(x)\) is a column with components \(u_1(x),\ldots,u_n(x)\), \(x=(x_1,\ldots,x_n)\),

\[ p=\left(\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_n}\right),\quad A(\lambda)=\|a_{ik}(\lambda)\| \]

is a matrix of size \(n\times n\), the \(a_{ik}(\lambda)\) are polynomials of the 2nd order in \(\lambda=(\lambda_1,\ldots,\lambda_n)\), \(\det A(\lambda)\ne0\) for \(\lambda\ne0\).

As the matrix \(A(\lambda)\), let us first take a matrix \(A_1(\lambda)\) such that
\[ a_{ii}=(-1)^n\left(|\lambda|^2-2\lambda_i^2\right),\quad 1\le i\le n,\quad a_{ik}=2(-1)^{n+1}\lambda_i\lambda_k,\quad i\ne k,\quad n\ge2. \]
The matrix \(A_1(\lambda)\) will be orthogonal and \(\det A_1(\lambda)=|\lambda|^{2n}\). System (2)

for \(A(p)=A_1(p)\) has a nontrivial solution \((n-2)x_1^2+|x|^2-1, 0, \ldots, 0\), equal to zero on the ellipsoid \(|x|^2+(n-2)x_1^2=1\) for any \(n\geqslant 2\).

Let us consider multidimensional analogues of A. V. Bitsadze’s system. Such systems can be constructed for \(n=4\) and \(n=8\). For this purpose we use quaternions \((n=4)\) and Cayley numbers \((n=8)\), by means of which multiplication of vectors in \(n\)-dimensional space is defined for \(n=4\), \(n=8\). The product of columns \(\lambda\) and \(\mu\) with components \(\lambda_1,\ldots,\lambda_n\) and \(\mu_1,\ldots,\mu_n\), respectively, is called the column \(\lambda\mu\) of the form

\[ \lambda\mu=A_n(\lambda)\mu . \tag{3} \]

Here the matrix \(A_n(\lambda)\) for \(n=4\) has the form

\[ A_4(\lambda)= \left\| \begin{array}{rrrr} \lambda_1 & -\lambda_2 & -\lambda_3 & -\lambda_4\\ \lambda_2 & \lambda_1 & -\lambda_4 & \lambda_3\\ \lambda_3 & \lambda_4 & \lambda_1 & -\lambda_2\\ \lambda_4 & -\lambda_3 & \lambda_2 & \lambda_1 \end{array} \right\|, \tag{4} \]

and for \(n=8\) \(A_n(\lambda)\) in block notation is expressed through \(A_4(\lambda)\),

\[ A_8(\lambda)= \left\| \begin{array}{rr} A_4 & -A_4'\\ A_4' & A_4 \end{array} \right\|, \tag{5} \]

where \(A_4\) is given by formula (4), and \(A_4'\) is obtained from \(A_4\) by replacing \(\lambda_i\) by \(\lambda_{i+4}\) and changing the sign in the last three columns.

Consider the matrix \(B_n(\lambda)=A_n^2(\lambda)\). If in system (2) one takes \(B_n(p)\) as \(A(p)\), then the system thus constructed will have the nontrivial solution \(|x|^2+(n-2)x_1^2-1, 0,\ldots,0\), equal to zero on the ellipsoid \(|x|^2+(n-2)x_1^2=1\).

Let us construct the matrix \(C_n(\lambda)\), replacing in the matrix \(A_n(\lambda)\) the numbers \(\lambda_{2k}\) by \(2\lambda_{2k}\lambda_{2k-1}\), and \(\lambda_{2k-1}\) by \(\lambda_{2k-1}^2-\lambda_{2k}^2\) \((k=1,\ldots,n/2;\ n=4,\ n=8)\). If in system (2) one takes \(C_n(p)\) instead of \(A(p)\), then the obtained system will have infinitely many linearly independent solutions, equal to zero on the boundary of some domain. Indeed, it can be proved that a system of functions of the form \(f(x), g(x), 0,\ldots,0\) satisfies the system \(C_n(p)u=0\), if the function \(w(x)=f(x)+ig(x)\) satisfies the conditions

\[ \frac{\partial^2 w}{\partial z_1^2} = \frac{\partial^2 w}{\partial z_3^2} = \frac{\partial^2 w}{\partial z_5^2} = \frac{\partial^2 w}{\partial z_7^2} =0, \tag{6} \]

where \(z_k=x_k+ix_{k+1}\),

\[ \frac{\partial}{\partial z_k} = \frac{\bar{\partial}}{\partial z_k} = \frac{1}{2}\left( \frac{\partial}{\partial x_k} +i\frac{\partial}{\partial x_{k+1}} \right) \qquad (k=1,3,5,7). \]

As the function \(w\), for \(n=4\) take \(w=w_1\overline{w}_3\), and for \(n=8\)

\[ w=w_1\overline{w}_3 w_5\overline{w}_7, \]

where \(w_k=(1-|z_k|^2)z_k^m\) \((k=1,3,5,7,\ m=1,2,3,\ldots)\). Then, for all \(z_k\), conditions (6) are satisfied and \(w=0\) when \(|z_1|=|z_3|=1\) and \(|z_1|=\cdots=|z_7|=1\), respectively.

It would be interesting to apply to the study of the systems considered above the methods of the works [2, 4, 7] ([6], p. 77 and appendix).

§ 2. THE CAUCHY PROBLEM FOR ELLIPTIC AND PARABOLIC EQUATIONS WITH ANALYTIC COEFFICIENTS

In this section we shall consider the formulation of the Cauchy problem for equations of elliptic and parabolic type with analytic coefficients. The treatment will be based on formulas solving the Cauchy problem when the Cauchy data are known exactly on part of the boundary of the domain. The formulas obtained below, (14), (8), (10), (19), which solve the Cauchy problem, reduce this problem to that of the analytic continuation of real analytic functions. The latter problem has by now been studied sufficiently well; literature on this question is given in the book [15]. The problem in the formulation to which the fundamental formula (19) leads for the equation $\Delta^2 u=f$ was studied in [1]. It should be noted that, in addition to the known applications of the Cauchy problem for the Laplace equation in geophysics and hydrodynamics (see the literature in [15], [16, a]), there are works of A. I. Lur'e (see [12]), where the Cauchy problem serves as a tool for studying boundary-value problems of the theory of elasticity in cylindrical domains. We also mention the work of G. V. Efimov [10], where applications of the Cauchy problem for the equation $u_{xx}=u_{yy}$ to problems of field theory were studied.

1. Let us first consider the problem of constructing, in a finite domain $D$, an analytic function $f(z)$, if its values are known on a part $\Gamma_1$ of the boundary $\Gamma$. The following holds.

Theorem 2.1. If the function $f(\zeta)$, $\zeta\in\Gamma_1$, is the boundary value of a function $f(z)$ analytic in the domain $D$ and continuous in $\overline D$, then the Cauchy-type integral

\[ g(z)=\frac{1}{2\pi i}\int_{\Gamma_1}\frac{f(\zeta)\,d\zeta}{\zeta-z} \tag{7} \]

is analytically continued through the arc $\Gamma_1$ into the interior of the domain $D$, and for all $z\in D$ the formula

\[ f(z)=g(z)-g^*(z), \tag{8} \]

holds, where $g^*(z)$ is the analytic continuation of $g(z)$ through the arc $\Gamma_1$ into $D$.

Proof. Let the symbol $\displaystyle \int_{\Gamma'}$ denote integration with such an orientation along $\Gamma'\subset\Gamma$ that the domain $D$ remains to the left of $\Gamma'$. By the Cauchy formula,

\[ f(z)=g(z)+\frac{1}{2\pi i}\int_{\Gamma-\Gamma_1}\frac{f(\zeta)\,d\zeta}{\zeta-z} \]

for $z\in D$. From the last formula we see that the function $f-g$ is analytically continued from the domain $D$ through the arc $\Gamma_1$ to the exterior of $D$. But for $z\notin\overline D$, by the Cauchy formula,

\[ g(z)+\frac{1}{2\pi i}\int_{\Gamma-\Gamma_1}\frac{f(\zeta)\,d\zeta}{\zeta-z}=0, \]

whence the analytic continuation of $f-g$ through the arc $\Gamma_1$ from the interior to the exterior of $D$ is $-g(z)$. Formula (8) is proved.

In connection with formula (8) we mention the work of Grinshtein [8]. In that work the question of analytic continuation of Cauchy–Stieltjes integrals is studied.

\[ J(z)=\frac{1}{2\pi i}\int_{-\infty}^{+\infty}\frac{d\sigma(t)}{t-z} \]

from the upper half-plane into the lower one \((\sigma(t)\) is a real function), and for piecewise-analytic \(\sigma(t)\) formulas of type (8) have been obtained (see also [18], p. 610), which, however, have a different meaning.

Formula (8) admits broad generalizations. For example, if \(D\) is a bounded domain in three-dimensional space and \(u(x,y,z)\) is a harmonic function in this domain, then from the known formulas

\[ \frac{1}{4\pi}\int_{\Gamma}\left\{u\frac{\partial}{\partial n}\frac{1}{r}-\frac{1}{r}\frac{\partial}{\partial n}u\right\}\,d\Gamma = \begin{cases} u, & (x,y,z)\in D,\\ 0, & (x,y,z)\notin D, \end{cases} \tag{9} \]

where \(r^{2}=(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}\), \((x_{0},y_{0},z_{0})\in\Gamma\), and \(\dfrac{\partial}{\partial n}\) is the derivative along the exterior normal to \(\Gamma\), by analogous reasoning one can obtain that for all \((x,y,z)\in D\)

\[ u(x,y,z)=v-v^{*}, \tag{10} \]

where \(v^{*}\) is the analytic continuation of the potential \(v\),

\[ v=\frac{1}{4\pi}\int_{\Gamma_{1}}\left\{u\frac{\partial}{\partial n}\frac{1}{r}-\frac{1}{r}\frac{\partial}{\partial n}u\right\}\,d\Gamma, \tag{11} \]

from outside the domain \(D\) through \(\Gamma_{1}\) into \(D\). Let us obtain a formula analogous to (8) and (10) for parabolic equations. Suppose that in the domain \(D=\{(x,t):0\le x\le 1,\ 0\le t\le T\}\) a function \(u(x,t)\) is given such that

\[ u_{t}=u_{x^{2}}. \tag{12} \]

Consider the points \(A=(0,0)\), \(B=(1,0)\), \(P=(0,t)\), \(Q=(1,t)\). Let \(u(0,t)=\mu(t)\), \(u_{x}(0,t)=\nu(t)\), \(0\le t\le T\), and let the function \(u(x,t)\) be continuous in \(\overline{D}\), and for \(\tau<t\)

\[ G=G(x,t;\xi,\tau)=\frac{1}{2\sqrt{\pi(t-\tau)}}\exp\left(-\frac{(x-\xi)^{2}}{4(t-\tau)}\right). \]

We shall prove that the heat potential \(v(x,t)\),

\[ v(x,t)=-\int_{0}^{t}\{G\nu(\tau)-G_{\xi}\mu(\tau)\}\bigg|_{\xi=0}\,d\tau, \tag{13} \]

is analytically continued from the domain \(x<0\) into the domain \(0<x<1,\ 0<t<T\). If this continuation of the potential (13) is denoted by \(v^{*}(x,t)\), then the formula

\[ u(x,t)=v(x,t)-v^{*}(x,t),\qquad (x,t)\in D \tag{14} \]

holds. Formula (14) is obtained by the reasoning applied in deriving formula (8) from the known equalities

\[ \int_{AB}u(\xi,0)G(x,t;\xi,0)\,d\xi + \int_{PA+BQ}\{u_{\xi}G-uG_{\xi}\}\,d\tau= \]

\[ =\begin{cases} u(x,t), & (x,t)\in D,\\ 0, & (x,t)\in \overline D . \end{cases} \]

Indeed, on the basis of these formulas, for \(0<x<1,\ 0<t<T\),

\[ u-v=\int_0^1 u(\xi,0)G(x,t;\xi,0)\,d\xi+ \]

\[ +\int_0^t\{u_\xi(1,\tau)G(x,t;1,\tau)-u(1,\tau)G_\xi(x,t;1,\tau)\}\,d\tau, \]

i.e. \(u-v\) is analytically continued into the domain \(x<0\). On the basis of the same formulas, the right-hand side of the last equality for \(x<0\) coincides with \(-v(x,t)\). Formula (14) is proved.

The significance of formulas (14), (10), (8) and their generalizations consists in the fact that they reduce the construction of a solution of the Cauchy problem to the problem of analytic continuation of real analytic functions, such as the potentials (13), (11), (7). It should be noted that these potentials are known to us in domains of the same dimension as the original space itself.

2. In deriving (10), (14), (8) we used only formulas of the form (9), i.e. we used Green’s formulas applied to the fundamental solution and to the sought solution of the Cauchy problem. Consequently, our results also carry over to elliptic systems of arbitrary order in which the coefficients at the highest derivatives are constant, while the coefficients at the lower derivatives are entire functions, since formulas of the form (9) are valid for them (see [9], p. 70).

Let us explain how the Cauchy problem for such systems should be constructed using the example of a single equation of higher order. Let \(u(x)\) be a real-valued function of the variable \(x=(x_1,\ldots,x_n)\), and suppose that in a bounded domain \(D\) with sufficiently smooth boundary \(\Gamma\)

\[ L(u)=\sum_{k=0}^{m}\sum_{i_1,\ldots,i_k=1}^{n} A_{i_1\ldots i_k}(x)\, \frac{\partial^k u}{\partial x_{i_1}\cdots \partial x_{i_k}}=0, \tag{15} \]

where \(A_{i_1\ldots i_k}\) are entire functions, \(A_{i_1\ldots i_m}=\mathrm{const}\), and, for \((\lambda_1,\ldots,\lambda_n)\ne0\), always

\[ \sum_{i_1,\ldots,i_m=1}^{n} A_{i_1\ldots i_m}\lambda_{i_1}\cdots\lambda_{i_m}>0. \]

Then

\[ \int_{\Gamma} M[u(y),K(y,x)]\,d\Gamma_y= \begin{cases} u(x), & x\in D,\\ 0, & x\in \overline D, \end{cases} \tag{16} \]

where \(K(y,x)\) is a fundamental solution of the equation adjoint to equation (15); \(M[u,K]\) is a certain bilinear form with respect to \(u(y)\), \(K(y,x)\) and their derivatives up to order \(m-1\) (see [9], p. 55).

By the Cauchy problem we shall mean the problem of finding a solution of equation (15) if, on a part \(\Gamma_1\) of the boundary \(\Gamma\), those derivatives of the sought solution \(u(x)\) and their linear combinations are known which determine the form \(M[u(y),K(y,x)]\), \(y\in\Gamma_1\). For example, if the equation under consideration is

\[ \Delta^2 u=0, \tag{17} \]

then, for \(x \in D\), always

\[ u(x)=\int_{\Gamma}\{u(\Delta K)_{\nu}-u_{\nu}\Delta K+\Delta u\cdot K_{\nu}-(\Delta u)_{\nu}K\}\,d\Gamma_y, \]

where \((\cdot)_{\nu}\) denotes differentiation along the exterior normal; \(K=K(x-y)\) is the fundamental solution of the equation \(\Delta^2 u=0\) in all space. The Cauchy data for equation (17) will be

\[ u|_{\Gamma_1},\quad u_{\nu}|_{\Gamma_1},\quad \Delta u|_{\Gamma_1},\quad (\Delta u)_{\nu}|_{\Gamma_1}. \tag{18} \]

Problem (17)—(18) was studied in [1].

Analogously to formula (8), proceeding from the equalities (16), we establish that the required solution is representable in the form

\[ u(x)=v(x)-v^*(x), \tag{19} \]

where \(x\in D\), \(v^*(x)\) is the analytic continuation of \(v(x)\) from outside the domain \(D\) through \(\Gamma_1\) into \(D\), and \(v(x)\) is a function known for all \(x\in \Gamma_1\),

\[ v(x)=\int_{\Gamma_1} M[u(y),K(y,x)]\,d\Gamma. \tag{20} \]

In the class of functions for which all derivatives up to order \(m-1\) on the boundary \(\Gamma\) are uniformly bounded in absolute value, the Cauchy problem is posed correctly in any subdomain \(D'\subset D\) in the sense that the modulus of the sought solution is estimated by a quantity tending to zero when the Cauchy data tend to zero.

Indeed, from (19) it follows that it suffices to verify the correctness of the construction of \(v^*(x)\). Let us take some domain \(D_1\) outside \(D\) and join it with \(D'\) by a domain \(D_2\) so that \(D_2\) intersects \(\Gamma_1\), and the domain \(D'+D_1+D_2\) is simply connected and lies at a positive distance from \(\Gamma-\Gamma_1\). Since

\[ v^*(x)=\int_{\Gamma-\Gamma_1} M[u,K]\,d\Gamma_y \]

for \(x\in D'+D_1+D_2\), and, consequently, is uniformly bounded in \(D'+D_1+D_2\), while in \(D_1\) \(v^*(x)\) is known and coincides with \(v(x)\), the problem of finding the analytic function \(v^*\) is posed correctly (see [15]).

§ 3. On Certain Problems with “Large Perturbations”

In this section we shall show that certain problems in the theory of potential, of conformal mappings, and of dynamical systems on the torus, which originally were problems “with a large perturbation,” can by a simple transformation be reduced to analogous problems, but already with a “small perturbation.”

1. Let us consider the well-known incorrect problem of transforming an analytic homeomorphism of the circle into a rotation (see [3], p. 180). The problem is as follows. Let a real analytic function \(f(z)\) be given, with \(f(z+2\pi)=f(z)\) for \(-\infty<z<\infty\), \(f'(z)>-1\). Let \(Tz=z+f(z)\), and let \(2\pi\mu\) be the rotation number of the mapping \(z\to Tz\), i.e.

\[ 2\pi\mu=\lim_{n\to\infty}\frac{f(z)+f(Tz)+\cdots+f(T^{n-1}z)}{n}. \]

It is required to construct a real analytic function \(\varphi(z)\), \(\varphi'(z)>0\), such that for all \(z\) the conditions

\[ \varphi(z+f(z))=\varphi(z)+2\pi\mu, \tag{21} \]

are fulfilled.

\[ \varphi(z+2\pi)=\varphi(z)+2\pi . \tag{22} \]

In the work of V. I. Arnol'd [3, 6], the formulated problem was studied under the assumption that the function \(f(z)\) is small in modulus for real \(z\). We shall show that this does not reduce the generality. Indeed, it is not difficult to verify that the equalities (21) and (22) preserve their form when \(\varphi(z)\) is replaced by \(\omega(z)=\varphi(Nz)/N\), \(f(z)\) by \(F(z)=f(Nz)/N\), and \(\mu\) by \(\nu=\mu/N\), where \(N\) is any positive integer. It may be shown that \(2\pi \nu\) is the rotation number of the mapping \(z\to z+F(z)\). For the new desired function \(\omega(z)\), the perturbation \(F(z)\) can be made arbitrarily small for large \(N\).

1. Consider the problem of constructing a conformal mapping of the disk \(|z|<1\) onto a star-shaped domain \(D\) that has an axis of symmetry. We shall show that here one may restrict oneself to the case when the domain \(D\) is, in a certain sense, arbitrarily close to a disk. Indeed, let the indicated mapping be realized by the function \(w=f(z)\), let the domain \(D\) be symmetric with respect to the straight line \(\operatorname{Im} w=0\), star-shaped with respect to the point \(w=0\), and let the equation of its boundary be \(\rho=\varphi(\theta)\) in polar coordinates of the plane \(\{w\}\).

Consider the function \(w_1=\sqrt{f(z^2)}\), \(0\le \arg z\le \pi/2\), \(|z|\le 1\), \(0\le \arg w_1\le \pi/2\). By the principle of symmetry, we continue the mapping \(z\to w_1(z)\) to the domain \(-\pi/2\le \arg z\le 0\), \(|z|\le 1\), and then to the domain \(\operatorname{Re} z\le 0\), \(|z|\le 1\). We obtain a mapping of the disk \(|z|\le 1\) onto a domain \(D_1\), which will have two axes of symmetry \(\operatorname{Re} w_1=\operatorname{Im} w_1=0\), and whose boundary is given by the equation \(\rho=\sqrt{\varphi(2\theta)}\), \(|\theta|\le \pi/2\). Repeating this construction, one can ensure that, instead of the original mapping \(w=f(z)\), it is sufficient to be able to construct a mapping of the unit disk onto a domain with boundary \(\rho=\psi(\theta)\) such that
\[ \max_{[0,\,2\pi]} |1-\psi(\theta)| \]
is arbitrarily small.

1. Consider the inverse problem of the logarithmic potential and show that a similar circumstance also occurs here.

Let a star-shaped domain \(D\) be given with boundary \(r=\varphi(\theta)\) in polar coordinates \((r,\theta)\), \(|\theta|\le \pi\). The inverse problem of the logarithmic potential for the domain \(D\), filled with matter of density one, consists in determining the boundary \(r=\varphi(\theta)\), knowing for large
\[ R=\sqrt{x^2+y^2} \]
the potential

\[ u=\iint\limits_D \ln \Delta\, d\xi d\eta, \tag{23} \]

where \(\Delta^2=(x-\xi)^2+(y-\eta)^2\). Expanding \(\ln\Delta\) in (23), for large \(R\), into a series by the formula

\[ \ln\Delta=\ln R-\sum_{m=1}^{\infty}\frac{r^m}{mR^m}\cos m(\theta-\varphi), \]

where \(\xi+i\eta=r\exp(i\theta)\), \(x+iy=R\exp(i\varphi)\), and computing the coefficients of \(1/R^n\), we obtain the equivalent problem of finding \(\varphi(\theta)\) from the known numbers \(c_n\),

\[ c_n=\int_{|\theta|\le \pi} \varphi^{2+n}(\theta)\exp(in\theta)\,d\theta \quad (n=0,1,2,\ldots). \tag{24} \]

Consider the domain \(D_N\), the equation of whose boundary is \(r=\omega_N(\theta)\), where \(\omega_N(\theta)=[\varphi(N\theta)]^{1/N}\), \(N\) is a positive integer. Fill

the domain \(D_N\) with material of variable density \(\rho=r^m\), where \(m=2(N-1)\). Simple computations show that knowledge of the numbers (24) makes it possible to compute, for large \(R\), the potential
\[ u_N=\iint_{D_N} r^m \ln \Delta\, d\xi\, d\eta . \]
This follows from the equalities
\[ \int_{|\theta|<\pi} \omega_N^{m+n+2}\exp(in\theta)\,d\theta = \begin{cases} c_k, & n=Nk,\\ 0, & n\ne Nk, \end{cases} \]
where \(m=2(N-1)\), \((n=0,1,2,\ldots)\).

Thus, we see that the inverse problem for the logarithmic potential of a domain filled with material of unit density reduces to the analogous problem for a domain whose boundary \(r=\omega_N\) is arbitrarily close to the unit circle, but which is filled with material of variable density. This circumstance is not noted in [13, 14] and [16, b)].

The number of problems in which a large perturbation is replaced, in some sense, by a small perturbation can be increased. For example, this can be done in the problem of finding the function \(\varphi(x,y)\) from the nonlinear equation of the first kind
\[ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \frac{\sigma(\xi,\eta)\,d\xi\,d\eta} {\sqrt{(x-\xi)^2+(y-\eta)^2+(z-\varphi(\xi,\eta))^2}} =f_z(x,y), \tag{25} \]
where \(f_z(x,y)\) is a known harmonic function for \(z>0\), \(-\infty<x<\infty\), \(-\infty<y<\infty\), and \(\sigma(\xi,\eta)\) ensures the existence of the integral (25). Replacing the unknowns by the formulas \(x=Nx_1,\ldots,\eta=N\eta_1\), \(\varphi_1=\varphi_1(\xi_1,\eta_1)=\varphi(N\xi_1,N\eta_1)/N\) makes it possible to regard the function \(\varphi\) in equation (25) as arbitrarily small in comparison with any fixed \(z\). Equations of type (25) are important in applications (see [17, 11]).

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Received by the editors
4 November 1965.