UDC 517.948.35 : 947.42

ON CERTAIN INCORRECT PROBLEMS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS

Yu. T. Antokhin

The present work serves as a continuation and illustration of the author’s work [1]. Some new incorrect problems are considered which arise in the theory of analytic continuation, in the theory of harmonic mappings in space, and also in the theory of equations of composite type.

§ 1. ON THE ANALYTIC CONTINUATION OF HARMONIC FUNCTIONS

1. In [1] an equation of the form

\[ Ax=f, \tag{1} \]

was investigated, where \(A\) is a positive self-adjoint operator in a Hilbert space \(H\), \(x\in H\), \(f\in H\). It was assumed that equation (1) is solvable, the solution is unique, and zero serves as a point of the spectrum of the operator \(A\). In the case when the desired solution is “sourcewise representable,” i.e.

\[ x=Ay \tag{2} \]

for some \(y\in H\), approximate formulas with an indication of the accuracy of approximation were proposed for finding \(x\). In what follows it will be shown that condition (2) is natural and is fulfilled in many problems of applied character.

We shall show that condition (2) is useful in studying the question of estimating \(\|x\|\) in terms of \(\|f\|\) in equation (1).

Theorem 1.1. Let \(A=A^*\), \(x\in D(A)\), \(y\in D(A)\), \(\|f\|\leq \varepsilon\), \(\|y\|\leq C\), and let \(x\) be a solution of equation (1). Then the following assertions are valid:

\(1^\circ.\) If \(x=Ay\), then

\[ \|x\|\leq (\varepsilon C)^{1/2}. \tag{3} \]

\(2^\circ.\) If \(x=A^2y\), then

\[ \|x\|\leq (\varepsilon \sqrt{C})^{2/3}. \tag{4} \]

Proof. We prove inequality (4). Since \(\|x\|^2=(Ax,Ay)\leq \|f\|\,\|Ay\|\), and also \(\|Ay\|^2=(x,y)\leq \|x\|\,\|y\|\), (4) is established. Inequality (3) is verified analogously.

As an example of the use of Theorem 1.1, consider the following problem. Let the function \(u(x,y)\) be harmonic in the half-plane \(y>0\), with \(u\to 0\) as \(y\to +\infty\), and \(u(x,0)\in L_2(-\infty,+\infty)\). It is required to estimate \(\|u(x,0)\|\), if \(\|u(x,1)\|\leq \varepsilon\). For this purpose consider the operator \(A\), defined as follows.

If \(u(x)\in L_2(-\infty,+\infty)\), then

\[ Au(x)=u(x,1), \tag{5} \]

where \(u(x,1)\) is the value at \(y=1\) of the function \(u(x,y)\), harmonic for \(y \geqslant 0\), equal to \(u(x)\) for \(y=0\). It is not hard to verify that assertions \(1^\circ\) and \(2^\circ\) of Theorem 1.1 will hold if the function \(u(x,y)\) is harmonically continued into the strip \(-1 \leqslant y \leqslant 0\) and \(-2 \leqslant y \leqslant 0\), respectively. To estimate \(u(x)\) one may use inequalities (3) and (4), where one should put \(\varepsilon=\|u(x,1)\|\), \(C=\|u(x,-1)\|\) and \(\varepsilon=\|u(x,1)\|\), \(C=\|u(x,-2)\|\), respectively.

Theorem 1.1 should be regarded as a generalization of some results of M. M. Lavrent’ev [3] (p. 36, Theorem 2 for \(r=1/4\), \(\rho=1/2\)) and M. Keith [9], obtained for functions harmonic in a disk, a ball, and a cylinder with a rectangular base. Let us explain this with the following example. Let the function \(u(r,\vartheta)\) be harmonic in the disk \(0 \leqslant r \leqslant 1\), where \(r,\vartheta\) are polar coordinates, \(0 \leqslant \vartheta \leqslant 2\pi\). In the space \(L_2(0,2\pi)\) consider the operator \(A\) defined as follows. If \(f(\vartheta)\in L_2(0,2\pi)\), then

\[ Af=u\left(\frac12,\vartheta\right), \]

where

\[ u\left(\frac12,\vartheta\right)=u(r,\vartheta)\bigg|_{r=\frac12}, \qquad \Delta u=0 \quad \text{for } 0 \leqslant r < 1,\quad u(1,\vartheta)=f(\vartheta). \]

It is not hard to verify that \(A=A^*\) and that

\[ A^2 f=u\left(\frac14,\vartheta\right). \]

Applying inequality (3), we obtain

\[ \left\|u\left(\frac12,\vartheta\right)\right\|\leqslant (\varepsilon C)^{1/2}, \]

where

\[ \left\|u\left(\frac14,\vartheta\right)\right\|\leqslant \varepsilon,\qquad \|u(1,\vartheta)\|\leqslant C. \]

Analogous applications of Theorem 1.1 can be made to biharmonic functions in a ball and in a half-space, as well as to functions harmonic inside a cone and taking zero values on the cone itself. We note that Theorem 1.1 was proved in the work of L. V. Mayorov [8] for a completely continuous operator \(A\), which is immaterial in the present question; moreover, the connection with the results of M. M. Lavrent’ev and M. Keith is not revealed, but conclusions are drawn that are of interest in the theory of measuring instruments.

§ 2. Consider the problem of determining the function \(u(x)\equiv u(x,0)\) from equation (5), if the function \(u(x,1)=f(x)\in L_2\) is known. Using this problem as an example, we shall show how condition (2) is used to estimate the rate of convergence of the series by which the solution of problem (5) is expressed. Problem (5) was studied in [5] for functions of interest in geophysics. In that work a process of successive approximations, considered below, was proposed and justified, but the causes influencing the speed of convergence were not clarified. A simple argument shows that the operator \(A\) in equation (5) is not completely continuous in \(L_2=L_2(-\infty,+\infty)\), and therefore the methods of [2–4, 6] are not applicable here. The following is true.

Theorem 1.2. If equation (5) is solvable, then

\[ u(x)=\lim_{N\to\infty} u_N(x), \tag{6} \]

where

\[ u_N(x)=\sum_{n=0}^{N}(E-A)^n f, \tag{7} \]

\(E\) is the identity operator, \(f=f(x)=u(x,1)\). If it is additionally known that the function \(u(x,y)\) from equation (5) is harmonically continued into the strip \(-m\leq y\leq 0\), where \(m\) is an integer, \(m>0\), then

\[ \|u-u_N\|\leq \lambda_0^{m}(1-\lambda_0)^{N+1}C_1 \tag{8} \]

for \(\lambda_0=m/(m+N+1)\), \(C_1=\|u(x,-m)\|\).

Proof. We prove the equalities (6), (7). Consider in \(L_2(-\infty,+\infty)\) the semigroup of operators \(A_y\), where \(y>0\),

\[ A_y u(x)=\frac{y}{\pi}\int_{-\infty}^{+\infty}\frac{u(\xi)\,d\xi}{y^2+(x-\xi)^2}. \tag{9} \]

It is obvious that \(A_yu(x)=u(x,y)\), where \(u(x,y)\) is harmonic for \(y>0\), \(u(x,0)=u(x)\). By the definition of the operator \(A\) we have

\[ Au(x)=A_1u(x). \tag{10} \]

It is easy to verify that

\[ A=A_{1/2}^{2}. \tag{11} \]

It follows from (9) that \(A_y=A_y^*\), \(y>0\). Therefore \(A>0\), \(A=A^*\) by virtue of (10), (11). Moreover, \(\|A\|\leq 1\), since \(\|u(x,y)\|\) decreases monotonically as \(y\) increases. Therefore the methods of [1] apply to equation (5), giving the equalities (6), (7). Let us prove the inequality (8). As indicated in Sec. 1, from the condition that \(u(x,y)\) is extendable into the strip \(-m\leq y\leq 0\) it follows that

\[ u(x)=A^m v(x), \tag{12} \]

where \(v(x)=u(x,-m)\in L_2\). If \(\{E_\lambda\}\) is the family of projection operators for \(A\), then from (7) it follows that

\[ \|u-u_N\|^2=\int_0^1 \lambda^{2m}(1-\lambda)^{2N+2}\,d(E_\lambda v,v)\leq \]

\[ \leq \lambda_0^{2m}(1-\lambda_0)^{2N+2}\|v\|^2, \]

where \(\lambda_0=m/(m+N+1)\). The theorem is completely proved.

We note that in the inequality (8), for fixed \(N\) and \(C_1\), the factor \(\lambda_0^m(1-\lambda_0)^{N+1}\to 0\) as \(m\to\infty\). All assertions of the theorem are valid for any \(A=A^*\), \(0\leq A\leq E\).

Sec. 3. Consider a generalization of problem (5). Let \(u(x)\in L_2(-\infty,+\infty)\). Consider the function \(u(x,y)\), harmonic for \(y>0\), with \(u(x,0)=u(x)\). Let \(\sigma(x)\) be a continuous function for \(-\infty<x<\infty\), \(0<\sigma_0\leq\sigma(x)\leq\sigma_1\), where \(\sigma_0,\sigma_1\) are constants. Consider in \(L_2\) the operator \(B\), defined by the formula

\[ Bu(x)=u(x,y)\big|_{y=\sigma(x)}=u(x,\sigma(x)). \tag{13} \]

We pose the problem of finding the function \(u(x)\) from the equation

\[ Bu(x)=f(x), \tag{14} \]

where \(f(x)\) is a given function, \(f(x)\in L_2\). Equation (14) was considered in

work of V. N. Strakhov and V. M. Devitsyn [7], where it was solved without justification by the method of B. A. Andreev (6), (7).

Let us solve equation (14) by the methods of [1]. It is easy to verify that the adjoint operator \(B^*\) is given by the formula

\[ B^*u(x)=\frac{1}{\pi}\int_{-\infty}^{+\infty} \frac{\sigma(t)u(t)\,dt}{\sigma^2(t)+(x-t)^2}. \tag{15} \]

Instead of equation (14), consider the equation

\[ \widetilde{B}u(x)=g(x), \tag{16} \]

where

\[ g(x)=\frac{\sigma_0}{\sigma_1}B^*f,\qquad \widetilde{B}=\frac{\sigma_0}{\sigma_1}B^*B, \]

\[ \widetilde{B}u(x)= \]

\[ =\frac{\sigma_0}{\pi\sigma_1}\int_{-\infty}^{+\infty} \frac{\sigma(x)+\sigma(t)} {[\sigma(x)+\sigma(t)]^2+(x-t)^2}\,u(t)\,dt. \tag{17} \]

Using the estimates \(\sigma_0\leq\sigma(x)\leq\sigma_1\) and the known property of the operator (9), \(\|A_y\|\leq 1\), it is easy to establish that \(\|g\|\leq\|f\|\), \(\|\widetilde{B}\|\leq 1\), and since \(\widetilde{B}=\widetilde{B}^*\), equation (16) can be solved by formulas (6) and (7), where \(\widetilde{B}\) should be taken instead of \(A\). These latter formulas also solve equation (14).

p. 4. Let us note that the problem of solving equation (5) is equivalent to the ill-posed problem of finding an element \(f\in D(B)\), if \(B\) is a positive unbounded operator. For equation (5), \(B=A^{-1}\). Generalizing the “method of the extinguishing multiplier,” known in the theory of divergent integrals, it is not difficult to establish that if \(B=B^*\), \(B>0\), \(f\in D(B)\), then

\[ Bf=\lim_{\varepsilon\to +0} B\exp(-\varepsilon B)f, \tag{18} \]

where convergence is understood in the strong sense and the operator \(B\exp(-\varepsilon B)\) is now already bounded,

\[ \|B\exp(-\varepsilon B)\|\leq \frac{1}{\varepsilon e}. \]

If, in addition, one assumes that \(\|B^2f\|\leq C\), then

\[ \|Bf-B\exp(-\varepsilon B)f\|\leq \varepsilon C. \tag{19} \]

We shall not dwell on applications of formulas (18), (19).

§ 2. ON HARMONIC MAPPINGS IN SPACE

As indicated in [1], a number of problems for equations of type (1) arise in the study of boundary value problems for the Laplace equation in an unbounded domain. Let us consider one such problem.

Let a domain \(D\) of three-dimensional space of points \((x,y,z)\) be mapped homeomorphically onto a domain \(\widetilde{D}\), where

\[ D=\{(x,y,z):z\geq f(x,y)\},\qquad \widetilde{D}=\{(q_1,q_2,q_3):q_3\geq g(q_1,q_2)\}, \]

where \(f\) and \(g\) are twice continuously differentiable functions, defi-

given for all values of their arguments, \(q_i=q_i(x,y,z)\), \((x,y,z)\in \overline{D}\), \(i=1,2,3\). Let the mapping satisfy the Cauchy–Riemann condition in space, i.e.

\[ \operatorname{rot}\,\overline{q}=\operatorname{div}\,\overline{q}=0 \tag{20} \]

for \(\overline{q}=q_1e_1+q_2e_2+q_3e_3\), where \(e_1,e_2,e_3\) are the unit vectors of the coordinate axes.

Condition (20) means geometrically that under the mapping \((x,y,z)\leftrightarrow(q_1,q_2,q_3)\) an infinitesimal sphere passes into an ellipsoid for which one semiaxis is equal to the sum of the other two.

Let \(q_i\) be twice continuously differentiable in \(\overline{D}\), and let the function \(q_3\) have the form

\[ q_3=G+h, \tag{21} \]

where \(G=z-H(x,y,z)\), \(\Delta H=0\) in \(D\), \(H|_{z=f}=f(x,y)\), \(H\to 0\) as \(z\to+\infty\), and \(h\) is a function harmonic in \(D\), \(h\to 0\) as \(z\to\infty\). Put further \(\zeta=x+iy\),

\[ u(\zeta)=u(x,y)\equiv q_1(x,y,f(x,y)), \tag{22} \]

\[ v(\zeta)=v(x,y)\equiv q_2(x,y,f(x,y)), \tag{23} \]

\[ w(\zeta)=u(\zeta)-iv(\zeta),\qquad g(w)=g(u(\zeta),v(\zeta)), \]

\[ \frac{\partial}{\partial \zeta} =\frac12\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right), \qquad \frac{\partial}{\partial \overline{\zeta}} =\frac12\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right), \]

\[ \omega\equiv 1+\left(\frac{\partial f}{\partial x}\right)^2 +\left(\frac{\partial f}{\partial y}\right)^2, \]

where the symbol \(\equiv\) means, for example in (22), that \(u(x,y)\), by definition, is \(q_1(x,y,f(x,y))\), etc., and \(\dfrac{\partial}{\partial x}\), \(\dfrac{\partial}{\partial y}\), \(\dfrac{\partial}{\partial z}\) are partial derivatives with respect to \(x,y,z\).

The following is true.

Theorem 2.1. The homeomorphism \(D\leftrightarrow \overline{D}\) considered above generates a homeomorphism of the following equation:

\[ \frac{\partial w}{\partial \overline{\zeta}} = 2\frac{\partial f}{\partial \overline{\zeta}} \frac{\partial g(w)}{\partial \zeta} -\sqrt{\omega}\,P(w) -\frac12\,\omega\,\frac{\partial G}{\partial z}, \tag{24} \]

where

\[ \frac{\partial G}{\partial z} = 1-\left.\frac{\partial H}{\partial z}\right|_{z=f(x,y)}, \qquad P(w)=\frac12\sqrt{\omega}\left.\frac{\partial h}{\partial z}\right|_{z=f}, \]

the function \(h=h(x,y,z)\) is harmonic in \(D\), \(h|_{z=f}=g(w)\), \(h\to0\) as \(z\to+\infty\). If \(h\) decreases sufficiently rapidly as \(|x|+|y|+|z|\to\infty\), then the operator \(P(w)\) has the property

\[ \int_{-\infty}^{+\infty}\!\!\int_{-\infty}^{+\infty} |P(w)|^2\,dx\,dy \le M \int_{-\infty}^{+\infty}\!\!\int_{-\infty}^{+\infty} \left|\frac{\partial w}{\partial \overline{\zeta}}\right|^2\,dx\,dy, \tag{25} \]

where

\[ M=\sup\left[ \left(\frac{\partial g}{\partial u}\right)^2+ \left(\frac{\partial g}{\partial v}\right)^2 \right], \qquad -\infty<u<\infty,\quad -\infty<v<\infty. \]

Proof. The derivation of equation (24) presents no difficulties. Differentiating (22) and (23) with respect to \(x\) and \(y\), taking into account that under the mapping \(D\leftrightarrow \overline{D}\), for all \(x,y\) one always has

\[ q_3(x,y,f(x,y))=g(u(x,y),v(x,y)), \tag{26} \]

from equations (20) we obtain

\[ \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x} =\frac{\partial f}{\partial y}\frac{\partial g}{\partial x} -\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}, \tag{27} \]

\[ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} =\frac{\partial f}{\partial x}\frac{\partial g}{\partial x} +\frac{\partial f}{\partial y}\frac{\partial g}{\partial y} -\omega\frac{\partial q_3}{\partial z}. \tag{28} \]

From (27) and (28), with the aid of (21), passing to the derivatives \(\dfrac{\partial}{\partial \xi}\), \(\dfrac{\partial}{\partial \zeta}\), we obtain (24).

Let us prove inequality (25). Consider the function \(h\) from the condition of the theorem and the function \(\bar h=h(x,y,t+f(x,y))\). From the harmonicity of \(h\) it follows that, for \(|x|<\infty\), \(|y|<\infty\), we have

\[ \bar h_{x^2}+\bar h_{y^2}+\bar\omega \bar h_{t^2} -2\bigl(\bar f_x \bar h_{xt}+\bar f_y \bar h_{yt}\bigr) -\bar h_t\Delta f=0. \tag{29} \]

Consider the integral \(J(t)\):

\[ J(t)=\int\limits_{-\infty}^{+\infty}\!\!\int \{\bar\omega \bar h_t^2-(\bar h_x^2+\bar h_y^2)\}\,dxdy. \]

From equation (29) and the sufficiently good decrease of \(\bar h\) as \(|x|+|y|+t\to\infty\), it is easy to establish that \(J'(t)=J(t)=0\) for \(t>0\). For \(t=0\), for the function \(h\) we have the identity

\[ \int\limits_{-\infty}^{+\infty}\!\!\int h_z^2\,dxdy = \int\limits_{-\infty}^{+\infty}\!\!\int \{h_x^2+h_y^2+2h_z(f_xh_x+f_yh_y)\}\,dxdy, \tag{30} \]

where all derivatives are evaluated at \(z=f(x,y)\). From (30) and the equalities \(h_x=g_x-h_zf_x\), \(h_y=g_y-h_zf_y\) for the operator \(P(\omega)\), we obtain

\[ \int\limits_{-\infty}^{+\infty}\!\!\int |P(\omega)|^2\,dxdy = \frac{1}{4}\int\limits_{-\infty}^{+\infty}\!\!\int (g_x^2+g_y^2)\,dxdy, \]

whence inequality (25) follows. The theorem is proved.

Note that equation (24), for small \(f_x, f_y, g_u, g_v\), belongs to the class of quasilinear elliptic integro-differential equations. The methods developed for integrating such equations (see [11], p. 350) are not applicable here, since we do not have the inequality

\[ \int\limits_{-\infty}^{+\infty}\!\!\int |w|^2\,dxdy \le K\int\limits_{-\infty}^{+\infty}\!\!\int |w_\xi|^2\,dxdy, \]

which is the cause of the incorrectness of the problem in the space \(W_2^1(E_2)\), where \(E_2\) is the two-dimensional plane.

§ 3. AN INCORRECT PROBLEM FOR AN EQUATION OF COMPOSITE TYPE

Boundary-value problems for composite-type equations are, generally speaking, incorrect in the class of those functions for which the existence and uniqueness theorem holds.

Let us consider an example. Let \(D\) be a bounded domain of three-dimensional space that is star-shaped with respect to the origin; \((x,y,z)\) are Cartesian coordinates, \((r,\lambda,\theta)\) are spherical coordinates, \(0\le \lambda\le 2\pi\), \(0\le \theta\le \pi\), \(\Gamma\) is the bou—

of the domain \(D\), \(n\) is the exterior normal to \(\Gamma\). Consider in \(D\) the problem of solving the equation

\[ \frac{d\Delta u}{dr}=0 \tag{31} \]

under the boundary condition

\[ u|_{\Gamma}=\varphi,\qquad u_n|_{\Gamma}=\psi. \tag{32} \]

In what follows it will be shown that, if \(\Gamma,\varphi\) and \(\psi\) are sufficiently smooth, then problem (31), (32) is essentially reduced, in an equivalent way, to the problem of determining the density of a certain simple-layer potential. A consequence of this is that problem (31), (32) is ill-posed in the following sense. There exist solutions of equation (31), continuously differentiable in \(\overline D\) and three times differentiable for \(r\ne0\), such that \(\varphi,\psi,|u|_D\) are arbitrarily small, while the number \(\max_D|\Delta u|\) is arbitrarily large (see the remark at the end of the paper).

A problem similar to (31), (32) was investigated in the work of T. D. Dzhuraev [10] for the equation \(\partial(\Delta u)/\partial x=0\), but the ill-posedness of such problems was not noted.

We shall prove the existence and uniqueness theorem for problem (31), (32), and then indicate the condition for its well-posedness. Let \(r=\Phi(\lambda,\theta)\) be the equation of the boundary \(\Gamma\). We shall seek the solution in the class of functions continuously differentiable in \(\overline D\) and three times continuously differentiable for \(r\ne0\). In the class of such functions the following is valid.

Theorem 3.1. If the functions \(\varphi,\psi,\Phi\) are differentiable a sufficient number of times, then the solution of problem (31), (32) exists and is unique.

Proof. Condition (31) is equivalent to the fact that

\[ u=v+\int_D \frac{\rho\,dm}{\Delta}, \tag{33} \]

where \(\Delta v=0\) in \(D\), \(\rho=\rho(\lambda',\theta')\), \(dm=r'^2\,dr'\,d\lambda'\,d\theta'\), \((r',\lambda',\theta')\) is the variable point of \(D\), \(\Delta\) is the distance between the point \((r,\lambda,\theta)\) and \((r',\lambda',\theta')\), \((r,\lambda,\theta)\in D\). Let

\[ v|_{\Gamma}=f,\qquad \left.\int_D \frac{\rho\,dm}{\Delta}\right|_{\Gamma}=g. \tag{34} \]

Consider the operators \(A\) and \(B\), acting on sufficiently smooth functions \(h\) defined on \(\Gamma\).

\(1^\circ.\)

\[ Ah=h_n^+|_{\Gamma}, \]

where \(\Delta h^+=0\) in \(D\), \(h^+|_{\Gamma}=h\).

\(2^\circ.\)

\[ Bh=h_n^-|_{\Gamma}, \]

\(\Delta h^-=0\) outside \(D\), \(h^-|_{\Gamma}=h\), \(h^-=O\!\left(\frac1r\right)\). If \(g\) is defined by equality (34), then, obviously,

\[ Bg = B\left(\left.\int_D \frac{\rho\,dm}{\Delta}\right|_{\Gamma}\right) = \left.\left(\frac{\partial}{\partial n}\int_D \frac{\rho\,dm}{\Delta}\right)\right|_{\Gamma}. \tag{35} \]

Problem (31), (32) is, in view of (33)—(35), equivalent to the problem of finding the functions \(f\) and \(g\) from the conditions

\[ f+g=\varphi; \tag{36} \]

\[ Af+Bg=\psi. \tag{37} \]

The system (36), (37) has a unique solution, since otherwise, for some sufficiently smooth functions \(\tilde f,\tilde g\) on \(\Gamma\), \(|\tilde f|+|\tilde g|\not\equiv 0\), one could construct functions \(F\) and \(G=O(1/r)\), harmonic inside and outside \(D\), respectively, such that \(F|_{\Gamma}=-G|_{\Gamma}\), \(F_n|_{\Gamma}=-G_n|_{\Gamma}\). From the latter conditions it follows, by a known theorem, that the function \(-G\) is the harmonic continuation of \(F\) to the exterior of \(D\), but since \(G=O(1/r)\), it follows that \(F\equiv G\equiv 0\), \(\tilde f\equiv \tilde g\equiv 0\) on \(\Gamma\). We have obtained a contradiction. At the same time the uniqueness of the solution of problem (31), (32) has also been proved.

We shall prove that a solution of the system (35), (37) exists and is given by the formulas

\[ f=(A-B)^{-1}(\psi-B\varphi); \tag{38} \]

\[ g=(A-B)^{-1}(A\varphi-\psi). \tag{39} \]

For this it is enough to show that the formulas

\[ (A-B)^{-1}\varphi= \left\{ \frac{1}{4\pi}\int_{\Gamma}\varphi\,\frac{d\Gamma}{\Delta} \right\}_{\Gamma}; \tag{40} \]

\[ B(A-B)^{-1}A\varphi=A(A-B)^{-1}B\varphi. \tag{41} \]

are valid.

We prove (40). If we denote by \(h\) the potential generated by \(\varphi\), computed on \(\Gamma\),

\[ h= \left\{ \frac{1}{4\pi}\int_{\Gamma}\varphi\,\frac{d\Gamma}{\Delta} \right\}_{\Gamma}, \]

then, by the theorem on the jump of the normal derivative of a simple-layer potential, \(h_n^+-h_n^-=(A-B)h=\varphi\), which gives formula (40).

We prove formula (41). First note that if \(\Delta\varphi^+=0\) in \(D\), \(\varphi^+|_{\Gamma}=f\), then by Green’s formula

\[ \frac{1}{4\pi}\int_{\Gamma}\varphi_n^+\,\frac{d\Gamma}{\Delta} = \frac{1}{4\pi}\int_{\Gamma}\varphi\,\frac{\partial}{\partial n}\left(\frac{1}{\Delta}\right)d\Gamma, \tag{42} \]

where \(\Delta\) is the distance between the point \((r,\lambda,\theta)\in D\) and the point \((r',\lambda',\theta')\in\Gamma\). From the equalities (40) and (42) we obtain

\[ (A-B)^{-1}A\varphi=(A-B)^{-1}\varphi_n^+= \]

\[ = \left\{ \frac{1}{4\pi}\int_{\Gamma}\varphi_n^+\,\frac{d\Gamma}{\Delta} \right\}_{\Gamma} = \lim_{r\to\Phi(\lambda,\theta)+0} \frac{1}{4\pi}\int_{\Gamma}\varphi_n^+\,\frac{d\Gamma}{\Delta} = \]

\[ = \lim_{r\to\Phi+0}\frac{1}{4\pi} \int_{\Gamma}\varphi\,\frac{\partial}{\partial n}\left(\frac{1}{\Delta}\right)d\Gamma = \frac{1}{4\pi}F_e(\Phi(\lambda,\theta),\lambda,\theta), \]

where \(F_e\) is the limiting value from outside \(\Gamma\) of the double-layer potential \(F\) of density \(\varphi\),

\[ F=\int_\Gamma \varphi\,\frac{\partial}{\partial n}\left(\frac{1}{\Delta}\right)d\Gamma . \]

Therefore

\[ B(A-B)^{-1}A\varphi=\frac{1}{4\pi}\frac{\partial F}{\partial n_e}, \tag{43} \]

where \(\partial F/\partial n_e\) is the exterior normal derivative of the double layer \(F\). Similarly we show that

\[ A(A-B)^{-1}B\varphi=\frac{1}{4\pi}\frac{\partial F}{\partial n_i}, \tag{44} \]

where \(\partial F/\partial n_i\) is the interior normal derivative of the layer \(F\). Since, by Lyapunov’s theorem, for sufficiently smooth \(\varphi\) and \(\Gamma\) one always has

\[ \frac{\partial F}{\partial n_i}=\frac{\partial F}{\partial n_e}, \]

the identity (41) and formulas (38), (39) are proved.

To complete the proof of existence of a solution, it remains, according to (34), to find a harmonic function \(v\) in \(D\) with \(v|_\Gamma=f\), and also to determine the density \(\rho\) of the potential \(V\) from (33):

\[ V=\int_D \rho\,\frac{dm}{\Delta}, \tag{45} \]

under condition (34)

\[ V|_\Gamma=g. \tag{46} \]

Let us solve problem (45), (46). We find \(\overline V\) from the conditions

\[ \Delta \overline V=0\quad \text{outside }D,\qquad \overline V|_\Gamma=g. \tag{47} \]

Consider outside \(D\) the harmonic function \(\overline w\):

\[ \overline w=-r^3\bigl(r^{-2}\overline V\bigr)'_r . \tag{48} \]

Let \(\alpha\) be the angle between the normal \(n\) and the vector \((r',\lambda',\theta')\), whose endpoint lies on \(\Gamma\), \(0\le \alpha<\pi/2\). Consider the function \(\overline\rho\):

\[ \overline\rho=\frac{\overline w_n^{+}-\overline w_n}{4\pi\Phi\cos\alpha}, \tag{49} \]

where \(\overline w_n,\overline w_n^{+}\) are the derivatives along the exterior normal \(n\) to the surface \(\Gamma\), respectively, of the function (48) and of the function \(\overline w^{+}\):

\[ \Delta \overline w^{+}=0\ \text{ in }D,\qquad \overline w^{+}|_\Gamma=\overline w|_\Gamma . \tag{50} \]

We shall prove that \(\overline\rho\) from (49) solves problem (45), (46). For this we establish that

\[ \tilde V|_\Gamma=g, \tag{51} \]

if

\[ \widetilde V=\int_D \rho\,\frac{-dm}{\Delta}. \tag{52} \]

Consider, for \(r>\Phi(\lambda,\theta)\), the function \(w\):

\[ w=-r^3(r^{-2}\widetilde V)'_r . \tag{53} \]

Using (52), (53), and the well-known expansion

\[ \frac1\Delta=\sum_{n=0}^{\infty}\frac{r'^n}{r^{n+1}}P_n(\cos\gamma), \tag{54} \]

where \(\gamma\) is the angle between the vectors \((r',\lambda',\theta')\) and \((r,\lambda,\theta)\), and \(P_n\) is the Legendre polynomial, for sufficiently large \(r\) we obtain

\[ w=\int_{\Gamma}\mu\,\frac{d\Gamma}{\Delta},\qquad \mu=\overline{\rho}\Phi\cos\alpha. \tag{55} \]

From (55) and (49) we have

\[ w=\frac1{4\pi}\int_{\Gamma}(\overline w_n^{+}-\overline w_n)\frac{d\Gamma}{\Delta}. \tag{56} \]

Consider the previously constructed function \(\overline w\) (formulas (47), (48)). By Green’s formula,

\[ \overline w=\frac1{4\pi}\int_{\Gamma}\overline w\,\frac{\partial}{\partial n}\left(\frac1\Delta\right)d\Gamma -\frac1{4\pi}\int_{\Gamma}\overline w_n\,\frac1\Delta\,d\Gamma . \tag{57} \]

From (57), (50), formula (42) applied to \(\overline w^{+}\), and formula (56), we obtain that for large \(r\), \(w=\overline w\). Hence it follows that everywhere outside \(D\), \(w=\overline w\), \(V=\widetilde V\), and (51) follows from (47). The solution of problem (45), (46) has been constructed. The theorem is completely proved.

Remark. From formula (33) it is clear that a particular solution of equation (31) is the function \(U\):

\[ U=\int_D \rho\,\frac{dm}{\Delta}, \tag{58} \]

where the notation is the same as in (33). Put in (58) \(\rho=\sqrt[3]{n}\,P_n(\cos\theta')\), and take as \(D\) the ball \(r\leqslant1\). Then, using expansion (54) to compute \(U|_{\Gamma}\), \(U_n|_{\Gamma}\), the estimate \(\int_D\rho^2\,dm=O(n^{-1/3})\), and Schwarz’s inequality in estimating \(|U|\), \(|\operatorname{grad}U|\), we obtain, for large \(n\), a solution of equation (31) such that \(|U|\), \(|\operatorname{grad}U|\) are arbitrarily small, whereas \(\max_D|\Delta U|=4\pi\sqrt[3]{n}\) is arbitrarily large. This example shows that, for equations with partial derivatives, a boundary-value problem can be considered well posed only under the condition that a small change in the boundary conditions corresponds to a small change in the solution and in all its derivatives of those orders that enter the equation.

References

1. Yu. T. Antokhin. On certain problems of the analytic theory of first-order equations. Differential Equations, vol. II, No. 2, 1966.

2. V. M. Fridman. UMN, vol. XI, issue 1 (67), 233—234, 1956.

3. M. M. Lavrent'ev. On certain ill-posed problems of mathematical physics. Novosibirsk, 1962.

4. H. Z. Bialy. Angew. Math. und Mech. 38, Nos. 7, 8, 261—263, 1958.

5. V. A. Andreev. Izvestiya AN SSSR, Ser. Geogr. i Geofiz., 13, No. 3, 256—267, 1949.

6. E. T. Gordadze. Proceedings of the Georgian Polytechnic Institute, 8 (93), 23—31, 1963.

7. V. N. Strakhov, V. M. Devitsyn. Izv. AN SSSR, Physics of the Earth, No. 4, 60—72, 1965.

8. L. F. Maiorov. Journal of Computational Mathematics and Mathematical Physics, 5, No. 2, 363—365, 1965.

9. M. Keith. Arch. Rational Mech. and Analysis, 16, No. 2, 126—154, 1964.

10. T. D. Dzhuraev. Izv. AN UzSSR, Ser. Phys.-Math., 3, 5—10, 1964.

11. I. N. Vekua. Generalized Analytic Functions. Fizmatgiz, 1958.

Received by the editors
November 4, 1965.