517.946 : 519.3

ON A VARIATIONAL PROBLEM WITH A SMALL PARAMETER. II*)

V. D. KOPCHENOV

The present work is a continuation of the article [1], in which a certain variational problem with a small parameter was considered. There the uniqueness and existence of the solution of the variational problem were proved, and some of its differential properties were also studied.

In the present paper a method is presented for obtaining the estimate (0.6) (see [1]); at the same time it is shown that this estimate is sharp with respect to order in \(\varepsilon\). An essential point of this method is the derivation of estimates for the higher generalized derivatives of the Neumann problem for a bounded domain near its boundary (see § 4).

In the appendix examples are given of problems of mathematical physics which lead to the differential problem (0.1), (0.3), (0.4) (see [1]).

§ 3. ESTIMATES OF THE SECOND GENERALIZED DERIVATIVES OF \(u\) AND OF THE SOLUTION OF THE VARIATIONAL PROBLEM (1.5) [1] NEAR THE BOUNDARY \(\Lambda\) OF THE DOMAIN \(G\)

3.1. In this section we shall assume that \(A_{ik}\in C^1(\overline G)\), \(f\in C^1(\overline G)\), and that the boundary \(\Lambda\) of the domain \(G\) belongs to \(C^2\).

Consider the cylinder \(H_{\rho,-a,a}\), introduced by us in item 1.10 [1]. Let \(\delta\) be a number satisfying the inequalities \(0<\delta<a,\rho\). In the variables \((y_1,\ldots,y_n)\), write equation (1.21) [1] for a variation \(v\), equal to zero outside the finite cylinder \(H_{\rho-\delta,-a+\delta,a-\delta}\):

\[ \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \sum_{i,k=1}^{n} B_{ik}\frac{\partial u}{\partial y_i}\frac{\partial v}{\partial y_k}\,dy + \frac{1}{\varepsilon^2} \int_{H_{\rho-\delta,0,a-\delta}} uv\,dy = \int_{H_{\rho-\delta,0,a-\delta}} fv\,dy, \tag{3.1} \]

where \(B_{ik}\) are defined in (1.41) [1].

We perform with equation (3.1) all the operations that we performed with equation (1.40) [1], taking into account that \(v\) must be taken in the form

\[ v=\eta u_{h_j}\qquad (j=1,\ldots,n-1), \]

where \(\eta\) is defined in (1.43), (1.44) [1].

As a result we obtain in the cylinder \(H_{\rho-2\delta,-a+2\delta,a-2\delta}\) estimates of the second generalized derivatives of \(u\), completely analogous to estimates (1.34) [1]:

\[ \left\| \frac{\partial^2 u}{\partial y_i\partial y_j} \right\|_{H_{\rho-2\delta,-a+2\delta,a-2\delta}} \le \overline B \sum_{i=1}^{n} \left\| \frac{\partial u}{\partial y_i} \right\|_{H_{\rho,-a,a}} + \]

*) Communication I, see the journal Differential Equations, No. 6, 1966.

\[ +\overline C\sqrt{\sum_{i=1}^n \left\|\frac{\partial u}{\partial y_i}\right\|_{H_{\rho,-a,a}}} \qquad (j=1,\ldots,n-1), \tag{3.2} \]

where \(\overline B\) and \(\overline C\) are constants that do not depend on the adjacent multipliers.

It follows from inequality (3.2) that the second generalized derivatives of the solution \(u\) of the variational problem (1.5) [1], with the exception of \(\dfrac{\partial^2 u}{\partial y_n^2}\), are summable with square over \(H_{\rho-2\delta,-a+2\delta,a-2\delta}\).

3.2. Let us prove an analogous inequality for \(\dfrac{\partial^2 u}{\partial y_n^2}\). If \(\Delta\) is a rectangular parallelepiped lying strictly inside the half-cylinder \(H_{\rho-2\delta,-a+2\delta,0}\), and \(v\) is a variation equal to zero outside \(\Delta\), then equality (3.1) takes the form

\[ \int_\Delta \sum_{i,k=1}^n B_{ik}\frac{\partial u}{\partial y_i}\frac{\partial v}{\partial y_k}\,dy=0 \]

and since, by Lemma 2 (p. 1.8 [1]), the function \(u\) has on \(\Delta\) second derivatives belonging to \(L_2(\Delta)\), integrating the last equality by parts we obtain

\[ \int_\Delta v\sum_{i,k=1}^n \frac{\partial}{\partial y_k}\left(B_{ik}\frac{\partial u}{\partial y_i}\right)\,dy=0. \]

Consequently, the function \(u\) satisfies the differential equation

\[ \sum_{i,k=1}^n B_{ik}\frac{\partial^2 u}{\partial y_i\partial y_k} + \sum_{i,k=1}^n \frac{\partial B_{ik}}{\partial y_k}\frac{\partial u}{\partial y_i} =0, \]

whence

\[ B_{nn}\frac{\partial^2 u}{\partial y_n^2} = -\sum_{\substack{i,k=1\\ i=k+n}}^n B_{ik}\frac{\partial^2 u}{\partial y_i\partial y_k} - \sum_{i,k=1}^n \frac{\partial B_{ik}}{\partial y_k}\frac{\partial u}{\partial y_i} =F. \]

We note that \(F\in L_2(\Delta)\), and by the ellipticity condition \(B_{nn}\geq \beta_1>0\), so that

\[ \left\|\frac{\partial^2 u}{\partial y_n^2}\right\|_\Delta < \frac{1}{\beta_1}\|F\|_\Delta . \]

Since the open set \(H_{\rho-2\delta,-a+2\delta,0}\) is the sum of a countable number of mutually nonintersecting \(\Delta\)’s and \(\|F\|_{H_{\rho-2\delta,-a+2\delta,0}}<\infty\), the inequality holds (see (3.2))

\[ \left\|\frac{\partial^2 u}{\partial y_n^2}\right\|_{H_{\rho-2\delta,-a+2\delta,0}} \leq \frac{1}{\beta_1}\|F\|_{H_{\rho-2\delta,-a+2\delta,0}} \leq \]

\[ \leq \overline B_1\sum_{i=1}^n \left\|\frac{\partial u}{\partial y_i}\right\|_{H_{\rho,-a,a}} + \overline C_1 \sqrt{\sum_{i=1}^n \left\|\frac{\partial u}{\partial y_i}\right\|_{H_{\rho,-a,a}}}, \tag{3.3} \]

where \(\overline B_1\) and \(\overline C_1\) are constants that do not depend on the adjacent multipliers.

In an analogous way it is proved that the second derivatives of the solution \(u\) of the variational problem (1.5) [1] belong to \(L_2(H_{\rho-2\delta,\,0,\,a-2\delta})\). Let us pass to the old coordinates and take into account that, by Borel’s lemma, the surface \(\Lambda\) is covered by a finite number of regions of the type \(H_{\rho-2\delta,\,-a+2\delta,\,a-2\delta}\).

As a result we obtain that there exists a bounded domain \(\Omega\), covering \(\Lambda\) (a layer containing \(\Lambda\)), and a constant \(A>0\) such that, for any second-order partial derivative of \(u\), the inequality
\[ \left\| \frac{\partial^2 u}{\partial x_i \partial x_j} \right\|_{\Omega} \le A \left( \sum_{i=1}^{n} \left\| \frac{\partial u}{\partial x_i} \right\|_{R_n} + \sqrt{ \sum_{i=1}^{n} \left\| \frac{\partial u}{\partial x_i} \right\|_{R_n} } \right), \tag{3.4} \]
holds, where the constant \(A\) does not depend on \(u\).

We may always assume that the boundary of the domain \(\Omega\) is a smooth surface, which we shall denote by \(\Gamma\). In view of what has been said, the limiting values
\[ \left( \frac{\partial u}{\partial \nu} \right)_{\Lambda+}, \qquad \left( \frac{\partial u}{\partial \nu} \right)_{\Lambda-}, \qquad \left( \frac{\partial u}{\partial \nu} \right)_{\Gamma} \]
make sense, where \(\nu\) is the conormal to the corresponding surface.

In view of what has been said, the function \(u\) satisfies in \(G\) the differential equation
\[ \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left( A_{ik}\frac{\partial u}{\partial x_k} \right) - \frac{1}{\varepsilon^2}u = -f \tag{3.5} \]
and in \(R_n\setminus \overline{G}\) the equation
\[ \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left( A_{ik}\frac{\partial u}{\partial x_k} \right) =0. \tag{3.6} \]

Let us write the variational equation for our problem, where we shall assume that \(v=0\) outside \(\Omega\):
\[ \int_{\Omega} \sum_{i,k=1}^{n} A_{ik}\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_k}\,dx + \frac{1}{\varepsilon^2} \int_{\Omega\cap G} uv\,dx = \int_{\Omega\cap G} fv\,dx. \]

Integrating this equation by parts, we obtain:
\[ \int_{\Lambda} v\mu \left( \left( \frac{\partial u}{\partial \nu} \right)_{\Lambda+} - \left( \frac{\partial u}{\partial \nu} \right)_{\Lambda-} \right) \,d\sigma + \int_{\Gamma} v\mu \left( \frac{\partial u}{\partial \nu} \right)_{\Gamma} \,d\Gamma - \]
\[ - \int_{\Omega\cap G} v \left[ \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left( A_{ik}\frac{\partial u}{\partial x_k} \right) - \frac{u}{\varepsilon^2} + f \right] \,dx - \]
\[ - \int_{\Omega\setminus \overline{G}} v \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left( A_{ik}\frac{\partial u}{\partial x_k} \right) \,dx = J_1+J_2+J_3+J_4=0. \]

Here
\[ \frac{\partial}{\partial \nu} = \frac{1}{\mu} \sum_{i,k=1}^{n} A_{ik}\cos(\widehat{x_k,n})\frac{\partial}{\partial x_i}, \qquad \mu = \left[ \sum_{i=1}^{n} \left( \sum_{k=1}^{n} A_{ik}\cos(\widehat{x_k,n}) \right)^2 \right]^{\frac12}. \tag{3.7} \]

\(J_2\) is equal to zero, because \(\left. v \right|_{\Gamma}=0\), \(J_3=J_4=0\), because \(u\) satisfies equations (3.5) and (3.6). And since \(v\) is an arbitrary variation, it follows that

\[ \left(\frac{\partial u}{\partial \nu}\right)_{\Lambda+} = \left(\frac{\partial u}{\partial \nu}\right)_{\Lambda-} \]

for almost all points \(Q \in \Lambda\). Thus we have proved the theorem.

Theorem 2. Among functions \(g\) of class \(\mathbf M\) there exists, and moreover uniquely, a function \(u\) for which the minimum of the variational problem (1.5) [1] is attained. The function \(u\) has on \(R \setminus \Lambda\) second generalized derivatives, integrable together with their square, satisfies the differential equation (3.5), (3.6), and the condition on the boundary \(\Lambda\):

\[ u_+ = u_-, \qquad \left(\frac{\partial u}{\partial \nu}\right)_{\Lambda+} = \left(\frac{\partial u}{\partial \nu}\right)_{\Lambda-}. \]

4. ESTIMATES OF THE SOLUTION OF THE NEUMANN PROBLEM NEAR THE BOUNDARY \(\Lambda\) OF THE DOMAIN \(G\)

4.1. Consider the functional

\[ H(v)=A(v)+2\varepsilon^2(\varphi,v)_{\Lambda}-2\varepsilon^2(f,v)_G, \tag{4.1} \]

where \(\varphi\) is a function prescribed on the boundary \(\Lambda\) of the domain \(G\), \(\varphi \in C^{\left[\frac n2\right]+2}(\Lambda)\). Here \(A(v)=A(v,v)\), \(A(v,w)\) is the bilinear functional

\[ A(v,w)=\int_G \left(\sum_{i,k=1}^{n} A_{ik}\frac{\partial v}{\partial x_i}\frac{\partial w}{\partial x_k}+\frac{1}{\varepsilon^2}vw\right)\,dG, \]

\(f \in C^{\left[\frac n2\right]+2}(\overline G)\), \(\varepsilon^2\) is a small parameter \(\left(\varepsilon \ll \frac12\right)\).

Lemma 4. If \(v \in W_2^{(1)}(G)\), then \(H(v)\) is bounded below

\[ \begin{aligned} H(v) &=A(v)+2\varepsilon^2(\varphi,v)_{\Lambda}-2\varepsilon^2(f,v)_G \\ &> \beta D_G(v)+\frac{1}{\varepsilon^2}\|v\|_G -2\varepsilon^2|(\varphi,v)_{\Lambda}|-2\varepsilon^2|(f,v)_G| \\ &> \beta\left( D_G(v)+\frac{1}{\beta\varepsilon^2}\|v\|_G^2 -\frac{2\varepsilon^2}{\beta}\|\varphi\|_{\Lambda}\|v\|_{\Lambda} -\frac{2\varepsilon^2}{\beta}\|f\|_G\|v\|_G \right) \\ &> \beta\left( D_G(v)+\frac{1}{\beta\varepsilon^2}\|v\|_G^2 -\frac{2c\varepsilon^2}{\beta}\|\varphi\|_{\Lambda}\bigl(\|v\|_G+\sqrt{D_G(v)}\bigr) -\frac{2\varepsilon^2}{\beta}\|f\|_G\|v\|_G \right). \end{aligned} \]

Here \(c\) is a constant depending on the domain \(G\) (see Lemma 2.9 [2]). Put

\[ K=\max\left(\frac{2c\|\varphi\|_{\Lambda}\varepsilon^2}{\beta},\, \frac{2\varepsilon^2\|f\|_G}{\beta}\right). \]

Then we have

\[ H(v)>\beta\left( D_G(v)+\frac{1}{\beta\varepsilon^2}\|v\|_G^2 -K\bigl(2\|v\|_G+\sqrt{D_G(v)}\bigr) \right). \]

Hence, under the condition \(\varepsilon \ll \frac12\), \(\beta<1\), we obtain

\[ H(v)>\beta\left( D_G(v)+\frac{1}{\beta\varepsilon^2}\|v\|_G^2 -K\left(\frac{1}{\sqrt{\beta}\,\varepsilon}\|v\|_G+\sqrt{D_G(v)}\right) \right)> \]

\[ > \beta \left( D_G(v) + \frac{1}{\beta \varepsilon^2}\|v\|_G^2 - \sqrt{2}\,K \sqrt{\frac{1}{\beta \varepsilon^2}\|v\|_G + D_G(v) + \frac{K^2}{2} - \frac{K^2}{2}} \right) = \]

\[ = \beta \left( \sqrt{D_G(v) + \frac{1}{\beta \varepsilon^2}\|v\|_G^2} - \frac{K}{\sqrt{2}} \right)^2 - \frac{\beta K^2}{2} > -\frac{\beta K^2}{2}. \]

The lemma is proved.

Consequently, there exists an exact lower bound of \(H(v)\), which we denote by \(-d\):

\[ \inf_{v \in W_2^{(1)}(G)} H(v) = -d . \tag{4.2} \]

Lemma 5. There exists a function \(u \in W_2^{(1)}(G)\) such that \(H(u) = -d\). Moreover, for any \(\xi \in W_2^{(1)}(G)\) the equality

\[ A(u,\xi) + \varepsilon^2(\varphi,\xi)_\Lambda = \varepsilon^2(f,\xi)_G . \]

holds.

Proof. Let \(\{v_n\}\) be a minimizing sequence, i.e. \(v_n \in W_2^{(1)}(G)\) and \(\lim\limits_{n\to\infty} H(v_n) = -d\).

In what follows we use the obvious equality

\[ A\left(\frac{v_n - v_m}{2}\right) = \frac{1}{2}\left[A(v_n) + 2\varepsilon^2(\varphi,v_n)_\Lambda - 2\varepsilon^2(f,v_n)_G\right] + \]

\[ + \frac{1}{2}\left[A(v_m) + 2\varepsilon^2(\varphi,v_m)_\Lambda - 2\varepsilon^2(f,v_m)_G\right] - \]

\[ - \left[ A\left(\frac{v_m + v_n}{2}\right) + 2\varepsilon^2\left(\varphi,\frac{v_n + v_m}{2}\right)_\Lambda - 2\varepsilon^2\left(f,\frac{v_n + v_m}{2}\right)_G \right] = \]

\[ = \frac{1}{2}H(v_n) + \frac{1}{2}H(v_m) - H\left(\frac{v_n + v_m}{2}\right). \tag{4.3} \]

Choosing \(n\) and \(m\) so large that \(H(v_n) < -d+\delta\) and \(H(v_m) < -d+\delta\) \((\delta \to 0)\), and taking into account that
\(-H\left(\dfrac{v_n+v_m}{2}\right) \le d\), we have

\[ A\left(\frac{v_n-v_m}{2}\right) < \frac{-d+\delta}{2} + \frac{-d+\delta}{2} + d = \delta, \]

i.e. \(A(v_n - v_m) < 4\delta\).

By condition (0.2) [1] we obtain

\[ \beta D_G(v_n-v_m) + \frac{1}{\varepsilon^2}\|v_n-v_m\|_G^2 < A(v_n-v_m) < 4\delta . \tag{4.4} \]

Inequality (4.4) shows that the minimizing sequence \(\{v_n\}\) converges in \(W_2^{(1)}(G)\).

Let \(u \in W_2^{(1)}(G)\) be the limiting function; then we obtain

\[ |H(u)-H(v_n)| < |A(u)-A(v_n)| + 2\varepsilon^2|(\varphi,u-v_n)_\Lambda| + 2\varepsilon^2|(f,u-v_n)_G| < \]

\[ < \left|A^{\frac12}(u)-A^{\frac12}(v_n)\right| \cdot \left|A^{\frac12}(u)+A^{\frac12}(v_n)\right| + \]

\[ + 2\varepsilon^2\|\varphi\|_\Lambda\|u-v_n\|_\Lambda + 2\varepsilon^2\|f\|_G\|u-v_n\|_G . \]

For \(u, v_n \in W_2^{(1)}(G)\), by virtue of inequality (4.4) we shall have (see Lemma 2.9 in [2])

\[ |H(u)-H(v_n)|< A^{\frac12}(u-v_n)\cdot |A^{\frac12}(u)+A^{\frac12}(v_n)|+ \]

\[ +2\varepsilon^2\|\varphi\|_{\Lambda}\left(\sqrt{D_G(u-v_n)}+\|u-v_n\|_G\right)+2\varepsilon^2\|f\|_G\|u-v_n\|_G \to 0 \quad \text{as } n\to\infty, \]

whence it follows that

\[ H(u)=\lim_{n\to\infty} H(v_n)=-d. \tag{4.5} \]

Let \(\xi\in W_2^{(1)}(G)\). Then \(H(u+\lambda \xi)=\lambda^2 A(\xi)+2\lambda[A(u,\xi)+\varepsilon^2(\varphi,\xi)_{\Lambda}-\varepsilon^2(f,\xi)_G]+H(u)\) has a minimum at \(\lambda=0\), and by Fermat’s theorem we have

\[ A(u,\xi)+\varepsilon^2(\varphi,\xi)_{\Lambda}=\varepsilon^2(f,\xi)_G. \tag{4.6} \]

The lemma is proved.

We show that \(u\) is the unique function in \(W_2^{(1)}(G)\) realizing equality (4.5).

Suppose that, in addition to \(u\), there exists a function \(\bar u\in W_2^{(1)}(G)\) such that

\[ H(\bar u)=-d. \tag{4.7} \]

By virtue of equality (4.3) we have

\[ A\left(\frac{u-\bar u}{2}\right)=\frac12 H(u)+\frac12 H(\bar u)-H\left(\frac{u+\bar u}{2}\right), \]

taking into account \(-H\left(\dfrac{u+\bar u}{2}\right)\le d\), as well as (4.5) and (4.7), we obtain

\[ A\left(\frac{u-\bar u}{2}\right)\le -\frac d2-\frac d2+d\le 0. \]

Hence it follows that

\[ D_G(u-\bar u)=0,\qquad \|u-\bar u\|_G=0. \]

The assertion is proved.

Theorem 3. If \(A_{ik}, f, \varphi\in C^1\), \(\Lambda\in C^2\), then the function \(u\) that realizes the minimum of the functional (4.1) belongs to \(W_2^{(2)}(G)\) and satisfies the equation

\[ L(u)=-\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i}\left(A_{ik}\frac{\partial u}{\partial x_k}\right)+\frac{1}{\varepsilon^2}u=\varepsilon^2 f, \tag{4.8} \]

and on \(\Lambda\) satisfies the boundary condition

\[ \left(\frac{\partial u}{\partial \nu}\right)_+=-\varepsilon^2\varphi. \tag{4.9} \]

Here the equality is understood “for almost all points \(Q\in\Lambda\).”

Proof. The fact that \(u\in W_2^{(2)}(G)\) is proved analogously to how this was done in § 3.2. We shall use equation (4.6), valid for any function \(\xi\in W_2^{(1)}(G)\). After integrating it by parts, we obtain

\[ \int_\Lambda \xi\mu \left(\frac{\partial u}{\partial \nu}\right)_+ d\Lambda +\varepsilon^2 \int_\Lambda \varphi \xi\, d\Lambda +\int_G \left( -\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial u}{\partial x_k}\right) +\frac{u}{\varepsilon^2}-f \right)\xi\, dG=0 . \tag{4.10} \]

If \(\xi|_\Lambda=0\), then the first and second integrals on the left-hand side of this equality are equal to zero. Consequently, the third integral is equal to zero for all \(\xi\in W_2^{(1)}(G)\), \(\xi|_\Lambda=0\). This implies (4.8). If now \(\xi\) is regarded as an arbitrary function from \(W_2^{(1)}(G)\), then, since the third integral on the left-hand side of (4.10) is equal to zero, we obtain

\[ \int_\Lambda \xi\mu \left[ \left(\frac{\partial u}{\partial \nu}\right)_+ + \varepsilon^2\varphi \right] d\Lambda=0, \]

whence (4.9) follows. The theorem is proved.

4.2. Substituting \(\xi=u\) in (4.6), we obtain

\[ A(u)+\varepsilon^2(\varphi,u)_\Lambda=\varepsilon^2(f,u)_G . \]

Using (0.2), from [1] we shall have

\[ \beta\left[D_G(u)+\frac{1}{\beta\varepsilon^2}\|u\|_G^2\right] \leq \varepsilon^2|(\varphi,u)_\Lambda|+\varepsilon^2|(f,u)_G| . \]

On the right-hand side we use the inequality for norms

\[ D_G(u)+\frac{1}{\beta\varepsilon^2}\|u\|_G^2 < \frac{\varepsilon^2}{\beta}\left[ c(G)\|\varphi\|_\Lambda\left(\|u\|_G+\sqrt{D_G(u)}\right) +\|f\|_G\|u\|_G \right]. \tag{4.11} \]

Introduce the notation

\[ T=\max\left\{ \frac{c(G)\|\varphi\|_\Lambda}{\beta}, \frac{\|f\|_G}{\beta} \right\}. \]

Under the condition \(\varepsilon\ll \dfrac12\), \(\beta<1\), we obtain from (4.11)

\[ D_G(u)+\frac{1}{\beta\varepsilon^2}\|u\|_G^2 < T\varepsilon^2\left( \frac{1}{\sqrt{\beta}\,\varepsilon}\|u\|_G+\sqrt{D_G(u)} \right) < \]

\[ < \sqrt{2}\,T\varepsilon^2 \sqrt{ D_G(u)+\frac{1}{\beta\varepsilon^2}\|u\|_G^2 } \]

or

\[ \sqrt{ D_G(u)+\frac{1}{\beta\varepsilon^2}\|u\|_G^2 } < \sqrt{2}\,T\varepsilon^2, \]

whence

\[ \sqrt{ \int_G \sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_i}\right)^2 dx } < \sqrt{2}\,T\varepsilon^2, \qquad \sqrt{\int_G u^2 dx} < \sqrt{2\beta}\,T\varepsilon^3 . \tag{4.12} \]

4.3. We study the behavior of the solution of the variational problem (4.2) near the boundary \(\Lambda\) of the domain \(G\). Just as was done in Sec. 1.10 [1], introduce, in a ball of sufficiently small radius \(r\) with center at a point \(Q\in\Lambda\), local coordinates \((y_1,\ldots,y_n)\) and use the notation introduced at the beginning

p. 1.10 [1]. Let us write, in the coordinates \((y_1,\ldots,y_n)\), equation (4.6) for a function \(\xi\) equal to zero outside \(H_{\rho-\delta,0,a-\delta}\):

\[ \int_{H_{\rho-\delta,0,a-\delta}} \sum_{i,k=1}^{n} B_{ik}\frac{\partial u}{\partial y_i}\frac{\partial \xi}{\partial y_k}\,dy +\frac{1}{\varepsilon^2} \int_{H_{\rho-\delta,0,a-\delta}}u\xi\,dy +\varepsilon^2 \int_{\sigma_{\rho-\delta}}\varphi\xi\,d\sigma = \varepsilon^2 \int_{H_{\rho-\delta,0,a-\delta}}f\xi\,dy, \tag{4.13} \]

where \(\sigma_{\rho-\delta}\) is the part of the surface \(y_n=0\) belonging to \(H_{\rho-\delta,0,a-\delta}\).

We perform with equation (4.13) all the operations that we performed with equation (1.40) in [1]; then we obtain an equation analogous to (1.42) [1],

\[ \int_{H_{\rho-\delta,0,a-\delta}} \sum_{i,k=1}^{n} \frac{\partial \xi}{\partial y_k} \left( B_{ik}\frac{\partial u_{h_j}}{\partial y_i} + \frac{\partial u(y+\bar h_j)}{\partial y_i}B_{ikh_j} \right)\,dy + \]

\[ +\frac{1}{\varepsilon^2} \int_{H_{\rho-\delta,0,a-\delta}}u_{h_j}\xi\,dy +\varepsilon^2 \int_{\sigma_{\rho-\delta}}\varphi_{h_j}\xi\,d\sigma = \varepsilon^2 \int_{H_{\rho-\delta,0,a-\delta}}f_{h_j}\xi\,dy. \tag{4.14} \]

Here

\[ \varphi_{h_j} = \frac{ \varphi(y_1,\ldots,y_{j-1},y_j+h_j,y_{j+1},\ldots,y_n) -\varphi(y_1,\ldots,y_n) }{h_j}; \]

\(j=1,\ldots,n-1;\ |\bar h_j|<\delta\); the remaining notation is analogous to the notation introduced in p. 1.11 [1].

Take as \(\xi\) the function \(\xi=\eta u_{h_j}\), where \(\eta\) is a continuous, nonnegative function on \(H_{\rho,0,a}\) such that

\[ \eta= \begin{cases} 1, & \text{if } y\in H_{\rho-2\delta,0,a-2\delta},\\ 0, & \text{if } y\notin \overline{H}_{\rho-\delta,0,a-\delta}, \end{cases} \]

and the condition

\[ \left(\frac{\partial \eta}{\partial y_i}\right)^2<c(\delta)\eta \qquad (i=1,\ldots,n) \]

is satisfied.

Substituting \(\xi=\eta u_{h_j}\) into (4.14), we obtain

\[ \int_{H_{\rho-\delta,0,a-\delta}} \eta \sum_{i,k=1}^{n} B_{ik}\frac{\partial u_{h_j}}{\partial y_i} \frac{\partial u_{h_j}}{\partial y_k} + \frac{1}{\varepsilon^2} \int_{H_{\rho-\delta,0,a-\delta}} \eta u_{h_j}^{\,2}\,dy + \]

\[ + \int_{H_{\rho-\delta,0,a-\delta}} u_{h_j}\sum_{i,k=1}^{n} B_{ik}\frac{\partial u_{h_j}}{\partial y_i} \frac{\partial \eta}{\partial y_k}\,dy + \]

\[ + \int_{H_{\rho-\delta,0,a-\delta}} \eta\sum_{i,k=1}^{n} B_{ikh_j} \frac{\partial u(y+\bar h_j)}{\partial y_i} \frac{\partial u_{h_j}}{\partial y_k}\,dy + \int_{H_{\rho-\delta,0,a-\delta}} u_{h_j}\sum_{i,k=1}^{n}B_{ikh_j}\times \]

\[ {}\times \frac{\partial \eta}{\partial y_k}\, \frac{\partial u(y+\bar h_j)}{\partial y_i}\,dy +\varepsilon^2\int_{\rho-\delta}^{\sigma}\eta\varphi_{h_j}u_{h_j}\,d\sigma -\varepsilon^2\int_{H_{\rho-\delta,0,a-\delta}}\eta f_{h_j}u_{h_j}\,dy \]
\[ =J_1+\cdots+J_7=0. \tag{4.15} \]

Let us estimate the integrals \(J_k\) \((k=1,\ldots,5)\):

\[ J_1>\beta_1\int_{H_{\rho-\delta,0,a-\delta}} \eta\sum_{i=1}^n\left(\frac{\partial u_{h_j}}{\partial y_i}\right)^2dy =\beta_1\Lambda_{h_j}^{2}, \]

\[ J_2=\frac{1}{\varepsilon^2}\bar\Lambda_{h_j}^{\,2}>0, \]

\[ |J_3|\leq c(\delta)\bar M n^{3/2} \sqrt{\int_{H_{\rho-\delta,0,a-\delta}}u_{h_j}^{2}\,dy}\,\Lambda_{h_j} \leq A_1\varepsilon^2\Lambda_{h_j}, \]

where \(\bar M\) is defined in Sec. 1.12 [1]; \(A_1\) and, in what follows, \(A_2,\ldots,A_n,\ldots\) do not depend on \(h_j\); moreover, in deriving the estimate the inequality (4.12) was used,

\[ |J_4|\leq \bar M n^{3/2} \sqrt{\int_{H_{\rho,0,a}}\sum_{i=1}^n \left(\frac{\partial u}{\partial y_i}\right)^2dy}\,\Lambda_{h_j} \leq A_2\varepsilon^2\Lambda_{h_j}, \]

\[ |J_5|\leq c(\delta)\bar M n^{3/2} \sqrt{\int_{H_{\rho,0,a}}\sum_{i=1}^n \left(\frac{\partial u}{\partial y_i}\right)^2dy}\, \sqrt{\int_{H_{\rho-\delta,0,a-\delta}}\eta u_{h_j}^{2}\,dy} \leq A_3\varepsilon^2\bar\Lambda_{h_j}. \]

Introduce the notation

\[ L=\max_l\left\{\|\varphi\|_{\sigma_\rho},\ldots, \left\| \frac{\partial^l\varphi} {\partial y_1^{\alpha_1},\ldots,\partial y_{n-1}^{\alpha_{n-1}}} \right\|_{\sigma_\rho}\right\} \qquad \left(l\leq\left[\frac n2\right]+2\right). \]

Let us estimate \(J_6\):

\[ |J_6|<L\varepsilon^2\left[ \sqrt{\int_{H_{\rho-\delta,0,a-\delta}}\eta u_{h_j}^{2}\,dy} + \sqrt{\int_{H_{\rho-\delta,0,a-\delta}} \sum_{i=1}^n \left(\frac{\partial(\eta u_{h_j})}{\partial y_i}\right)^2dy} \right]< \]

\[ < A_4\varepsilon^2\left( \sqrt{\int_{H_{\rho-\delta,0,a-\delta}}\eta u_{h_j}^{2}\,dy} + \sqrt{\int_{H_{\rho-\delta,0,a-\delta}} \eta\sum_{i=1}^n \left(\frac{\partial u_{h_j}}{\partial y_i}\right)^2dy} \right) \leq \]

\[ \leq A_4\varepsilon^2(\bar\Lambda_{h_j}+\Lambda_{h_j}). \]

Introduce the notation

\[ \bar N=\max_l\left\{ \|f\|_{H_{\rho,0,a,\ldots}}, \left\| \frac{\partial^l f} {\partial y_1^{\alpha_1},\ldots,\partial y_{n-1}^{\alpha_{n-1}}} \right\|_{H_{\rho,0,a}} \right\} \qquad \left(l\leq\left[\frac n2\right]+2\right). \]

Let us estimate \(J_7\):

\[ |J_7|\leq \bar N\varepsilon^2 \sqrt{\int_{H_{\rho-\delta,0,a-\delta}}\eta u_{h_j}^{2}\,dy} = A_5\varepsilon^2\bar\Lambda_{h_j}. \]

From (4.15) it then follows that

\[ \Lambda_{h_j}^{2}+\frac{1}{\beta_1\varepsilon^2}\overline{\Lambda}_{h_j}^{2} \leq A_6\varepsilon^2(\Lambda_{h_j}+\overline{\Lambda}_{h_j}) \leq A_7\varepsilon^2\sqrt{\Lambda_{h_j}^{2}+\overline{\Lambda}_{h_j}^{2}} \leq \]

\[ \leq A_7\varepsilon^2 \sqrt{\Lambda_{h_j}^{2}+\frac{1}{\beta_1\varepsilon^2}\overline{\Lambda}_{h_j}^{2}} . \tag{4.16} \]

The last inequality is obtained for \(\varepsilon \ll \dfrac12\), \(\beta_1<1\). Dividing the left- and right-hand sides of inequality (4.16) by the same factor, we obtain

\[ \sqrt{ \int_{H_{\rho-\delta,\,0,\,a-\delta}} \eta\sum_{i=1}^{n} \left(\frac{\partial u_{h_j}}{\partial y_i}\right)^2\,dy + \frac{1}{\beta_1\varepsilon^2} \int_{H_{\rho-\delta,\,0,\,a-\delta}} \eta u_{h_j}^{2}\,dy } \leq A_7\varepsilon^2, \tag{4.17} \]

where \(A_7\) is a constant depending on \(\overline N,\ \overline M,\ L,\ n,\ \delta,\ \beta_1\), but not depending on \(u,\ \varepsilon,\ h_j\).

Taking the integrals in the left-hand side of (4.17) over the smaller cylinder
\(H_{\rho-2\delta,\,0,\,a-2\delta}\subset H_{\rho-\delta,\,0,\,a-\delta}\), and also taking into account Lemma 1 from [5], we obtain

\[ \left\| \frac{\partial^2 u}{\partial y_i\partial y_k} \right\|_{H_{\rho-2\delta,\,0,\,a-2\delta}} \leq A_7\varepsilon^2, \]

\[ \left\| \frac{\partial u}{\partial y_k} \right\|_{H_{\rho-2\delta,\,0,\,a-2\delta}} \leq A_8\varepsilon^3 \qquad (i=1,\ldots,n;\quad k=1,\ldots,n-1). \]

Let \(\lambda\) be a number satisfying the inequalities \(0<\lambda<a,\rho\). Using the induction method, we shall have, under the assumption that
\(\varphi\in C^l(\Lambda)\), \(f\in C^l(\overline G)\), and \(\Lambda\in C^{l+1}\),

\[ \left\| \frac{\partial^{l+1}u} {\partial y_1^{\alpha_1},\ldots,\partial y_n^{\alpha_n}} \right\|_{H_{\rho-\lambda,\,0,\,a-\lambda}} < A\varepsilon^2 \left( \alpha_1+\cdots+\alpha_n=l+1; \right. \]

\[ \left. \alpha_n=0,1;\quad l\leq \left[\frac n2\right]+2 \right), \]

\[ \left\| \frac{\partial^{l}u} {\partial y_1^{\alpha_1},\ldots,\partial y_{n-1}^{\alpha_{n-1}}} \right\|_{H_{\rho-\lambda,\,0,\,a-\lambda}} < A\varepsilon^3 \left(\alpha_1+\cdots+\alpha_{n-1}=l\right), \tag{4.18} \]

where \(A\) depends on \(\lambda,\ \varphi,\ B_{ik},\ f\) and their derivatives up to order \(l\) inclusive, but not on \(\varepsilon\).

In the arguments given above, we could have introduced, in a sphere with center at \(Q\) and of sufficiently small radius, new coordinates \((y_1,\ldots,y_n)\) such that \(y_n\) denotes the distance from the point to the surface \(\Lambda\) in the direction corresponding to the conormal to \(\Lambda\); then the derivative with respect to \(y_n\) would be the derivative in the conormal direction to \(\Lambda\), while the derivatives with respect to \(y_1,\ldots,y_{n-1}\) would be derivatives in directions parallel (locally) to the tangential directions on \(\Lambda\). For such coordinates, estimates (4.18) can be obtained in a completely analogous way, requiring, however, that the surface \(\Lambda\in C^{l+2}\).

§ 5. Estimates of the solution \(u\) of the variational problem (1.5) [1] in the space \(C(R_n)\)

5.1. We shall prove the following lemma.

Lemma 6. The solution \(u_0\) of the differential problem

\[ L(u_0)=-\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial u_0}{\partial x_k}\right) +\frac{1}{\varepsilon^2}u_0=f,\quad \text{if } x\in G, \]

\[ L(u_0)=\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial u_0}{\partial x_k}\right)=0,\quad \text{if } x\in \overline{G}, \]

\[ u_{0+}-u_{0-}=0,\quad \text{if } x\in \Lambda, \]

\[ \left(\frac{\partial u_0}{\partial \nu}\right)_{+} - \left(\frac{\partial u_0}{\partial \nu}\right)_{-} =0,\quad \text{if } x\in \Lambda, \]

\[ \lim_{\rho\to\infty}u_0(Q,\rho)=0 \tag{5.1} \]

in the class \(W_2^{(2)}\) (for any bounded domain), having a finite Dirichlet integral, is unique. It coincides with the solution \(u\in \mathbf{M}\) of the variational problem (1.5) [1]. Here \(\nu\) is the conormal to the surface \(\Lambda\) (see (3.7)).

Proof. Let \(u\) be another solution of the differential problem satisfying the conditions of the lemma. Taking the difference of the corresponding equations (5.1) for \(u_0\) and \(u\), we obtain

\[ L(u_0-u)= -\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial (u_0-u)}{\partial x_k}\right) +\frac{u_0-u}{\varepsilon^2}=0,\quad \text{if } x\in G, \]

\[ L(u_0-u)= \sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial (u_0-u)}{\partial x_k}\right) =0,\quad \text{if } x\in \overline{G}. \tag{5.2} \]

Multiply both equalities in (5.2) by \((u_0-u)\) and integrate over the ball \(|x|<\rho\), containing \(G\),

\[ \int_{|x|<\rho} (u_0-u)L(u_0-u)\,dx = -\int_{|x|<\rho}(u_0-u) \sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial (u_0-u)}{\partial x_k}\right)\,dx + \]

\[ +\frac{1}{\varepsilon^2}\int_{G}(u_0-u)^2\,dx =J_1+J_2=0. \tag{5.3} \]

The derivatives of the form

\[ \frac{\partial^2(u_0-u)}{\partial x_i\,\partial x_k} \]

are summable; therefore the factors

\[ \frac{\partial (u_0-u)}{\partial x_k} \]

and \(u_0-u\) are absolutely continuous along the axis \(x_i\) for almost all

\[ (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n), \]

and we may integrate \(J_1\) by parts:

\[ J_1= -\int_{|x|=\rho}(u_0-u) \sum_{i,k=1}^{n}A_{ik}\frac{\partial}{\partial x_k}(u_0-u)\cos(\widehat{n,x_i})\,d\sigma + \]

\[ +\int_{|x|<\rho}\sum_{i,k=1}^{n} A_{ik}\, \frac{\partial (u_0-u)}{\partial x_i}\, \frac{\partial (u_0-u)}{\partial x_k}\,dx =J_3+J_4 . \]

We transform the integral \(J_3\), introducing the direction of the conormal \(\nu\), whose direction cosines are given in (3.7):

\[ J_3=-\int_{|x|=\rho} \mu (u_0-u)\, \frac{\partial (u_0-u)}{\partial \nu}\,d\sigma . \]

Applying Cauchy’s inequality to \(J_3\), we obtain

\[ \left| \int_{|x|=\rho}\mu (u_0-u)\, \frac{\partial (u_0-u)}{\partial \nu}\,d\sigma \right|^2 \leq N\int_{|x|=\rho}(u_0-u)^2\,d\sigma \times \]

\[ \times \int_{|x|=\rho} \left( \frac{\partial (u_0-u)}{\partial \nu} \right)^2 d\sigma, \tag{5.4} \]

where

\[ N=\max_{x\in \Lambda}|\mu|. \]

For the first integral in (5.4), an estimate is given in the work of S. V. Uspenskii [3]:

\[ \int_{|x|=\rho}(u_0-u)^2\,d\sigma=o(\rho) \qquad (\rho\to\infty). \]

Let us estimate the second integral in (5.4). We shall prove that there exists a sequence \(\rho_j\) such that

\[ \int_{|x|=\rho_j} \left( \frac{\partial (u_0-u)}{\partial \nu} \right)^2 d\sigma < \frac{1}{\rho_j} \quad \text{as } \rho_j\to\infty,\quad \text{when } j\to\infty . \tag{5.5} \]

Suppose this is not true. Then, beginning with some \(\rho\), we would have

\[ \int_{|x|=\rho}\sum_{i=1}^{n} \left( \frac{\partial (u_0-u)}{\partial x_i} \right)^2 d\sigma > \int_{|x|=\rho} \left( \frac{\partial (u_0-u)}{\partial \nu} \right)^2 d\sigma \geq \frac{1}{\rho}, \]

and after integrating the left- and right-hand sides with respect to \(\rho\) on the interval \([R,\infty]\),

\[ \int_{\Omega_R}\sum_{i=1}^{n} \left( \frac{\partial (u_0-u)}{\partial x_i} \right)^2 dx > \int_R^\infty \frac{d\rho}{\rho}, \]

where \(\Omega_R\) is the exterior of the sphere of radius \(R\). But this is impossible, since the integral on the right diverges, while that on the left converges. Inequality (5.5) is proved. Hence it follows that the left-hand side of inequality (5.4) tends to zero along this sequence of spheres, and this means that

\[ \beta \int_{|x|<\rho_j}\sum_{i=1}^{n} \left( \frac{\partial (u_0-u)}{\partial x_i} \right)^2 dx + \frac{1}{\varepsilon^2}\int_G (u_0-u)^2 dx \leq \]

\[ = \int_{|x|<\rho_j}\sum_{i,k=1}^{n} A_{ik}\, \frac{\partial (u_0-u)}{\partial x_i}\, \frac{\partial (u_0-u)}{\partial x_k}\,dx + \]

\[ +\frac{1}{\varepsilon^{2}}\int_G (u_0-u)^2\,dx \to 0 \quad \text{as } \rho_j \to \infty . \]

Therefore \(u_0=u\).

The second part of the lemma follows from the fact that the solution of the variational problem also satisfies the conditions of the lemma by Theorem 2, item 3.2.

5.2. Consider the differential problem (5.1):

\[ L(u)=A(u)+\frac{1}{\varepsilon^{2}}u=f, \quad \text{if } x\in G, \]

\[ L(u)=A(u)=0, \quad \text{if } x\in \overline{\overline{G}}, \]

\[ u_+-u_-=0, \quad \text{if } x\in \Lambda, \]

\[ \left(\frac{\partial u}{\partial \nu}\right)_+ - \left(\frac{\partial u}{\partial \nu}\right)_- =0, \quad \text{if } x\in \Lambda, \]

\[ \lim_{\rho\to\infty} u(Q,\rho)=0, \]

where

\[ A(u)=-\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial u}{\partial x_k}\right). \]

Represent the solution \(u\) of problem (5.1) in the form

\[ u=\varepsilon^2 f+u_1, \quad \text{if } x\in G, \tag{5.6} \]

\[ u=u_1, \quad \text{if } x\in \overline{\overline{G}}. \]

We obtain for \(u_1\) the following differential problem:

\[ L(u_1)=A(u_1)+\frac{1}{\varepsilon^{2}}u_1=\varepsilon^2 f_1, \quad \text{if } x\in G, \]

\[ L(u_1)=A(u_1)=0, \quad \text{if } x\in \overline{\overline{G}}, \]

\[ u_{1+}-u_{1-}=-\varepsilon^2 f, \quad \text{if } x\in \Lambda, \]

\[ \left(\frac{\partial u_1}{\partial \nu}\right)_+ - \left(\frac{\partial u_1}{\partial \nu}\right)_- = -\varepsilon^2 \frac{\partial f}{\partial \nu}, \quad \text{if } x\in \Lambda, \]

\[ \lim_{\rho\to\infty} u_1(Q,\rho)=0. \]

Here \(f_1=-A(f)\).

Everywhere below in this paragraph we shall assume that

\[ A_{ik}\in C^{\left[\frac n2\right]+2}(R_n) \]

and

\[ f\in C^{\left[\frac n2\right]+4}(\overline{G}). \]

Represent the solution \(u_1\) in the form:

\[ u_1=u_2+u_3, \quad \text{if } x\in G, \tag{5.7} \]

\[ u_1=u_2, \quad \text{if } x\in \overline{\overline{G}}, \]

where \(u_3\in C^2(\overline{G})\) is the solution of the Neumann problem:

\[ L(u_3)=A(u_3)+\frac{1}{\varepsilon^{2}}u_3=\varepsilon^2 f_1, \quad \text{if } x\in G, \tag{5.8} \]

\[ \left(\frac{\partial u_3}{\partial \nu}\right)_+ = -\varepsilon^2 \frac{\partial f}{\partial \nu}, \quad \text{if } x\in \Lambda. \]

We obtain for \(u_2\), respectively, the differential problem

\[ L(u_2)=A(u_2)+\frac{1}{\varepsilon^2}u_2=0,\quad \text{if } x\in G, \]

\[ L(u_2)=A(u_2)=0,\quad \text{if } x\in \overline{G}, \tag{5.9} \]

\[ u_{2+}-u_{2-}=-\varepsilon^2 f-u_3=\varepsilon^2\omega,\quad \text{if } x\in \Lambda, \]

\[ \left(\frac{\partial u_2}{\partial \nu}\right)_{+} - \left(\frac{\partial u_2}{\partial \nu}\right)_{-} =0,\quad \text{if } x\in \Lambda, \tag{5.10} \]

\[ \lim_{\rho\to\infty}u_2(Q,\rho)=0. \]

5.3. Let us estimate the solution \(u_3\) of the Neumann problem (5.8). To this end we construct a nonnegative function \(\psi\in C^2(\overline{G})\) such that the following conditions are satisfied:

\[ -L(\psi)=-A(\psi)-\frac{1}{\varepsilon^2}\psi<0,\quad \text{if } x\in G, \tag{5.11} \]

\[ \frac{1}{\psi_+}\left(\frac{\partial \psi}{\partial \nu}\right)_{+}\leq -q<0,\quad \text{if } x\in \Lambda. \tag{5.12} \]

Then the estimate of \(u_3\) follows from Theorem V of [6], namely

\[ \|u_3\|_{C(\overline{G})} \leq \left( \frac{\varepsilon^2}{q} \left\| \frac{\partial f}{\partial \nu}\frac{1}{\psi} \right\|_{C(\Lambda)} + \varepsilon^2\|\psi\|_{C(\overline{G})} \left\| \frac{f_1}{\psi} \right\|_{C(\overline{G})} \right) \|\psi\|_{C(\overline{G})}. \tag{5.13} \]

In a sufficiently small \((n\)-dimensional) neighborhood of the surface \(\Lambda\) one can introduce coordinates

\[ (x_1,\ldots,x_n,\nu)=(Q,\nu), \tag{5.14} \]

which denote the following.

If from each point of \(\Lambda\) one draws the conormal inside \(G\), then at some sufficiently small distance \(h\) from \(\Lambda\) all of them do not intersect. Therefore each point \(P\in G\) from a sufficiently small neighborhood of \(\Lambda\) is uniquely determined by the coordinates (5.14), where \(Q(x_1,\ldots,x_n)\) is the point of \(\Lambda\) from which the conormal to \(\Lambda\) emanates, such that the point \(P\) lies on it, and \(\nu\) is the distance from \(P\) to \(Q\). Denote by \(G_\delta\) the \(n\)-dimensional layer adjacent from the inside to the surface \(\Lambda\), whose thickness along the conormal \(\nu\) is equal to \(\delta>0\).

Construct a nonnegative twice continuously differentiable function \(\psi(P)\) according to the following rule:

\[ \psi(P)= \begin{cases} 1-q\nu, & \text{if } P=(Q,\nu)\in G_{\frac{h}{2}},\\[4pt] \Phi(P), & \text{if } P=(Q,\nu)\in G_h\setminus G_{\frac{h}{2}},\\[4pt] \dfrac{1}{2}, & \text{if } P\in G\setminus G_h, \end{cases} \]

where \(\Phi(P)\in C^2\!\left(G_h\setminus G_{\frac{h}{2}}\right)\), and the conditions are satisfied:

\[ \Phi'(P)= \begin{cases} -q, & \text{if } P=\left(Q,\dfrac{h}{2}\right),\\[6pt] 0, & \text{if } P=(Q,h). \end{cases} \]

for any point \(Q\in\Lambda\).

The function \(\psi\) satisfies condition (5.12) by virtue of its construction. Condition (5.11) can be satisfied for sufficiently small \(\varepsilon>0\). Consequently, by (5.13), for \(u_3\) we obtain the estimate

\[ \|u_3\|_{C(\bar G)} \leq \varepsilon^2 \left( \frac{1}{q}\left\|\frac{\partial f}{\partial \nu}\right\|_{C(\Lambda)} + \|f_1\|_{C(\bar G)} \right). \tag{5.15} \]

5.4. Consider the cylinder \(H_{\rho_0,-a_0,a_0}\) introduced by us in Sec. 1.13 [1]. In the variables \((y_1,\ldots,y_n)\) introduce the auxiliary function \(\omega\) according to the following rule (cf. 5.10):

\[ \omega(y_1,\ldots,y_n)= \begin{cases} 0, & \text{if } y\in H_{\rho_0,0,a_0},\\[6pt] -f(y_1,\ldots,y_{n-1},-y_n) -\dfrac{1}{\varepsilon^2}\, u_3(y_1,\ldots,y_{n-1},-y_n), & \text{if } y\in H_{\rho_0,-a,0}. \end{cases} \tag{5.16} \]

The estimate \(\omega\in C(H_{\rho_0,-a_0,0})\) does not depend on \(\varepsilon\) in view of (5.15). Norms of the form

\[ \left\| \frac{\partial^{\,l+1}\omega} {\partial y_1^{\alpha_1},\ldots,\partial y_n^{\alpha_n}} \right\|_{H_{\rho_0,-a_0,a_0}} \left( \alpha_1+\cdots+\alpha_n=l+1;\ \alpha_n=0,1;\ l\leq \left[\frac{n}{2}\right]+2;\ H_{\rho_0,-a_0,a_0}\subset H_{\rho,-a,a} \right) \]

also do not depend on \(\varepsilon\), according to the first of inequalities (4.18). The differential problem (5.9) with boundary conditions (5.10) corresponds to the variational problem (1.38) with conditions (1.36) and (1.37) (see [1]); moreover, as the function \(\omega\) introduced by us at the beginning of Sec. 1.11 [1], one may take the function defined in (5.16).

In the space of variables \((x_1,\ldots,x_n)\) this function may turn out to be multivalued, since one and the same point \(P\in R_h\setminus G\) may belong to two cylinders; however, for obtaining estimate (1.47) from [1], which we shall need below, this is immaterial.

5.5. Multiply the second equation in (5.9) by \(u_2\) and integrate over the domain \(\omega_{\rho_j}\setminus G\) \((G\subset \omega_{\rho_j},\ \omega_{\rho_j}\) is a ball of radius \(\rho_j)\):

\[ \int_{\substack{|x|<\rho_j,\, x\in \bar G}} u_2 A(u_2)\,dx = - \int_{|x|=\rho_j} \mu u_{2+} \left(\frac{\partial u_2}{\partial \nu}\right)_{+} \,d\sigma + \int_{\Lambda} \mu u_{2-} \left(\frac{\partial u_2}{\partial \nu}\right)_{-} \,d\Lambda + \int_{\substack{|x|<\rho_j,\, x\in \bar G}} \sum_{i,k=1}^{n} A_{ik}\, \frac{\partial u_2}{\partial x_i} \frac{\partial u_2}{\partial x_k} \,dx =0. \tag{5.17} \]

Here \(\mu\) is defined in (3.7).

Let \(\rho_j\) run through the sequence (5.5); then the first integral on the right-hand side of (5.17) vanishes:

\[ \lim_{j\to\infty}\left|\int_{\sigma_{\rho_j}}\mu u_{2+}\left(\frac{\partial u_2}{\partial \nu}\right)_+\,d\sigma\right|=0. \]

Therefore, from (5.17) it follows that

\[ \int_{R_n\setminus G} u_2 R(u_2)\,dx = \int_{\Lambda}\mu u_{2-}\left(\frac{\partial u_2}{\partial \nu}\right)_-\,d\Lambda + \int_{R_n\setminus G}\sum_{i,k=1}^{n} A_{ik}\frac{\partial u_2}{\partial x_i}\frac{\partial u_2}{\partial x_k}\,dx =0. \tag{5.18} \]

Multiply the first equation in (5.9) by \(u_2\) and integrate over the domain \(G\):

\[ \int_G u_2 A(u_2)\,dx+\frac{1}{\varepsilon^2}\int_G u_2^2\,dx = -\int_{\Lambda}\mu u_{2+}\left(\frac{\partial u_2}{\partial \nu}\right)_+\,d\Lambda + \]

\[ + \int_G \sum_{i,k=1}^{n} A_{ik}\frac{\partial u_2}{\partial x_i}\frac{\partial u_2}{\partial x_k}\,dx + \frac{1}{\varepsilon^2}\int_G u_2^2\,dx =0. \tag{5.19} \]

Adding (5.18) and (5.19), and moving the integral over the surface \(\Lambda\) to the right, taking (5.10) into account, we obtain

\[ \int_{R_n\setminus \Lambda}\sum_{i,k=1}^{n} A_{ik}\frac{\partial u_2}{\partial x_i}\frac{\partial u_2}{\partial x_k}\,dx + \frac{1}{\varepsilon^2}\int_G u_2^2\,dx = \varepsilon^2\int_{\Lambda}\mu\frac{\partial u_2}{\partial \nu}\omega\,d\Lambda. \tag{5.20} \]

Here

\[ \frac{\partial u_2}{\partial \nu} = \left(\frac{\partial u_2}{\partial \nu}\right)_+ = \left(\frac{\partial u_2}{\partial \nu}\right)_- \]

for almost all \(Q\in\Lambda\).

On the right-hand side of (5.20) we use Cauchy’s inequality; as a result we obtain

\[ \int_{R_n}\sum_{i,k=1}^{n} A_{ik}\frac{\partial u_2}{\partial x_i}\frac{\partial u_2}{\partial x_k}\,dx + \frac{1}{\varepsilon^2}\int_G u_2^2\,dx \le \varepsilon^2 N \left\|\frac{\partial u_2}{\partial \nu}\right\|_{\Lambda} \|\omega\|_{\Lambda}, \tag{5.21} \]

where \(N\) is defined in (5.4).

5.6. Introduce in the ball \(\omega_r\) of radius \(r\) with center at the point \(Q\in\Lambda\) (see Sec. 1.10 in [1]) a local coordinate system \((x_1,\ldots,x_{n-1},\nu)\), where \(\nu\) is the distance from the point to \(\Lambda\) along the conormal to \(\Lambda\), and, by analogy with Sec. 1.10 [1], consider in the space \((x_1,\ldots,x_{n-1},\nu)\) the truncated cylinder \(H_{\rho,0,a}^{(\nu)}\), where \(\rho=\dfrac r2\), and \(a>0\) is sufficiently small. We assume that values \(\nu>0\) correspond to points of \(R_n\setminus \overline G\). Thus, the domain in the space \((x_1,\ldots,x_n)\) corresponding to \(H_{\rho,0,a}^{(\nu)}\) belongs to \(R_n\setminus G\). The piece of the surface \(\Lambda\) bounding this domain and corresponding to \(\nu=0\) will be denoted by \(\sigma\).

We have the inequality (see [2])

\[ \left\|\frac{\partial u_2}{\partial \nu}\right\|_{\sigma} < c_1\left( \left\|\frac{\partial u_2}{\partial \nu}\right\|_{H_{\rho,0,a}^{(\nu)}} + \sqrt{ D_{H_{\rho,0,a}^{(\nu)}}\left(\frac{\partial u_2}{\partial \nu}\right) } \right) < \]

\[ < c_2\left( \sum_{i=1}^{n}\left\|\frac{\partial u_2}{\partial x_i}\right\|_{H^{(\nu)}} + \sum_{i,k=1}^{n}\left\|\frac{\partial^2 u_2}{\partial x_i\partial x_k}\right\|_{H^{(\nu)}} \right), \tag{5.22} \]

where the constants \(c_1,c_2\), and in what follows \(c_3,\ldots,c_n,\ldots\), do not depend on \(\varepsilon\).

In the last inequality we have denoted by \(H^{(\nu)}\) the domain in the space \((x_1,\ldots,x_n)\) into which the cylinder \(H_{\rho,0,a}^{(\nu)}\) has passed. Denote by \(H\) the domain in the space of the variables \((x_1,\ldots,x_n)\) into which the cylinder \(H_{\rho_0,-a_0,0}\) passes (see item 1.13 in [1]). Let \(H^{(\nu)} \subset H\). We write the estimates in \(L_2(H^{(\nu)})\) for the second generalized derivatives of \(u_2\). By virtue of inequality (1.47) from [1], for \(l=2\),

\[ \left\| \frac{\partial^2 u_2}{\partial y_1^{\alpha_1},\ldots,\partial y_n^{\alpha_n}} \right\|_{H_{\rho_0,-a_0,0}} \leq c_3 \left( \sum_{i=1}^n \left\| \frac{\partial u_2}{\partial y_i} \right\|_{H_{\rho,-a,a}} +\varepsilon^2 \right). \]

Here \(\alpha_n=0,1\). But by formula (3.3), which in our case will have the form

\[ \left\| \frac{\partial^2 u_2}{\partial y_n^2} \right\|_{H_{\rho_0,-a_0,0}} \leq c_4 \left( \sum_{i=1}^n \left\| \frac{\partial u_2}{\partial y_i} \right\|_{H_{\rho,-a,a}} +\varepsilon^2 \right), \]

the last estimate is also valid for the case \(\alpha_n=2\).

Passing to the variables \((x_1,\ldots,x_n)\), we have

\[ \left\| \frac{\partial^2 u_2}{\partial x_i\,\partial x_k} \right\|_{H^{(\nu)}} \leq c_5 \left( \sum_{i=1}^n \left\| \frac{\partial u_2}{\partial x_i} \right\|_{R_n} +\varepsilon^2 \right). \tag{5.23} \]

By Borel’s lemma there is a finite number of domains \(H^{(\nu)}\) covering the surface \(\Lambda\).

Using (5.22) and (5.23), from (5.21) we obtain

\[ \int_{R_n} \sum_{i,k=1}^n A_{ik} \frac{\partial u_2}{\partial x_i} \frac{\partial u_2}{\partial x_k}\,dx + \frac{1}{\varepsilon^2} \int_G u_2^2\,dx < c_6\varepsilon^2 \left( \sum_{i=1}^n \left\| \frac{\partial u_2}{\partial x_i} \right\|_{R_n} +\varepsilon^2 \right). \tag{5.24} \]

Hence, in view of the ellipticity condition and the positivity of the second integral on the left-hand side of (5.24), the inequality follows immediately:

\[ \int_{R_n} \sum_{i=1}^n \left( \frac{\partial u_2}{\partial x_i} \right)^2 dx \leq c_7\varepsilon^2 \sqrt{ \int_{R_n} \sum_{i=1}^n \left( \frac{\partial u_2}{\partial x_i} \right)^2 dx } + c_7\varepsilon^4 . \]

Thus, in any case,

\[ \frac{1}{\varepsilon} \sqrt[4]{ \int_{R_n} \sum_{i=1}^n \left( \frac{\partial u_2}{\partial x_i} \right)^2 dx } \leq c_8, \]

where \(c_8\) coincides with the largest positive root of the equation

\[ \xi^4=c_8(\xi^2+1). \]

And finally

\[ \left\| \frac{\partial u_2}{\partial x_i} \right\|_{R_n} \leq c_9\varepsilon^2 \quad (i=1,\ldots,n), \tag{5.25} \]

where \(c_9=c_8^2\) is a constant independent of \(u_2\) and \(\varepsilon\).

Omitting the first integral on the left-hand side of (5.24) and estimating the right-hand side of (5.24) by means of (5.25), we obtain

\[ \|u_2\|_G<c_{10}\varepsilon^3 . \tag{5.26} \]

Hence, from (5.25), with the aid of Lemma 2.9 of [2], there follows the inequality

\[ \|u_{2+}\|_{\Lambda}<c_{11}\varepsilon^{2}, \]

and also, invoking the first condition in (5.10), the inequality

\[ \|u_{2-}\|_{\Lambda}<c_{12}\varepsilon^{2}. \tag{5.27} \]

5.7. Let us consider once again the cylinder \(H_{\rho_0,-a_0,a}\) (see Sec. 1.13 in [1]). We shall show that estimate (1.47) from [1] can be extended to arbitrary derivatives of \(u_2\) of the form

\[ \frac{\partial^l u_2}{\partial y_1^{\alpha_1}\cdots \partial y_n^{\alpha_n}}, \tag{5.28} \]

where \(\alpha_1+\cdots+\alpha_n=l,\quad 2\le l\le \left[\frac{n}{2}\right]+3,\quad y\in H_{\rho_0,-a_0,0}\).

For the derivative \(\dfrac{\partial^2 u_2}{\partial y_n^2}\) we proved this in Sec. 3.2.

Let \(\Delta\) be an \(n\)-dimensional cube lying strictly inside the cylinder \(H_{\rho_0,-a_0,0}\). Then, by Lemma 2 (Sec. 1.8, [1]), the function \(u_2\) has third generalized derivatives of the form

\[ \frac{\partial^3 u_2}{\partial y_n^2\,\partial y_j} \qquad (j=1,\ldots,n-1), \]

belonging to \(L_2(\Delta)\). Consequently, the function \(u_2\) satisfies on \(\Delta\) the differential equation

\[ \sum_{i,k=1}^{n} B_{ik}\frac{\partial^3 u_2}{\partial y_i\,\partial y_k\,\partial y_j} + \sum_{i,k=1}^{n}\frac{\partial B_{ik}}{\partial y_j} \frac{\partial^2 u}{\partial y_i\,\partial y_k} + \]

\[ + \sum_{i,k=1}^{n}\frac{\partial B_{ik}}{\partial y_k} \frac{\partial^2 u}{\partial y_i\,\partial y_j} + \sum_{i,k=1}^{n}\frac{\partial^2 B_{ik}}{\partial y_k\,\partial y_j} \frac{\partial u_2}{\partial y_i} =0, \]

where \(j=1,\ldots,n-1\). Let us isolate \(\dfrac{\partial^3 u_2}{\partial y_n^2\,\partial y_j}\):

\[ B_{nn}\frac{\partial^3 u_2}{\partial y_n^2\partial y_j} = -\sum_{\substack{i,k=1\\ i=k\ne n}}^{n} B_{ik}\frac{\partial^3 u_2}{\partial y_i\,\partial y_k\,\partial y_j} - \sum_{i,k=1}^{n}\frac{\partial B_{ik}}{\partial y_j} \frac{\partial^2 u_2}{\partial y_i\,\partial y_k} - \]

\[ - \sum_{i,k=1}^{n}\frac{\partial B_{ik}}{\partial y_k} \frac{\partial^2 u_2}{\partial y_i\,\partial y_j} - \sum_{i,k=1}^{n}\frac{\partial^2 B_{ik}}{\partial y_k\,\partial y_j} \frac{\partial u_2}{\partial y_i} =F_1. \tag{5.29} \]

We note that \(F_1\in L_2(\Delta)\).

Using the ellipticity condition \(B_{nn}\ge \beta_1>0\), from (5.29) we obtain

\[ \left\| \frac{\partial^3 u_2}{\partial y_n^2\partial y_j} \right\|_{\Delta} \le \frac{1}{\beta_1}\|F_1\|_{\Delta}. \]

Repeating literally the reasoning of Sec. 3.2, taking into account (1.47) from [1], we obtain the inequality

\[ \left\| \frac{\partial^3 u_2}{\partial y_n^2\partial y_j} \right\|_{H_{\rho_0,-a_0,0}} \le c_{13}\left( \sum_{i=1}^{n} \left\| \frac{\partial u_2}{\partial y_i} \right\|_{H_{\rho,-a,a}} + \varepsilon^2 \right). \]

After this, in the same way we estimate the derivative \(\dfrac{\partial^3 u_2}{\partial y_n^3}\), etc.

As a result, for derivatives of the form (5.28) in the cylinder \(H_{\rho_0,-a_0,0}\) we shall have the estimate

\[ \left\| \frac{\partial^l u_2}{\partial y_1^{\alpha_1},\ldots,\partial y_n^{\alpha_n}} \right\|_{H_{\rho_0,-a_0,0}} \leq c_{14} \left( \sum_{i=1}^{n} \left\| \frac{\partial u_2}{\partial y_i} \right\|_{H_{\rho,-a,a}} +\varepsilon^2 \right). \]

We pass to the old coordinates and take into account that the surface \(\Lambda\), on the basis of Borel’s lemma, is covered by a finite number of domains of the type \(H_{\rho_0,-a_0,0}\).

As a result we obtain that there exists a bounded domain \(\Omega \subset R_n \times G\), covering \(\Lambda\) (a layer adjacent to \(\Lambda\)), and a constant \(c>0\) such that, for any partial derivative of order \(l\) \(\left(l \leq \left[\dfrac{n}{2}\right]+3\right)\) of \(u_2\), taking (5.25) into account, the inequality

\[ \left\| \frac{\partial^l u_2}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right\|_{\Omega} \leq c\varepsilon^2, \tag{5.30} \]

holds, where the constant \(c\) does not depend on \(u_2\) and \(\varepsilon\).

5.8. From (5.6) and (5.7) it follows that \(u=u_2\) if \(x \in R_n \times G\). Let \(\Omega_R\) be the exterior of the sphere \(\omega_R\) of radius \(R\), in which estimate (2.5) from [1] holds, and let \(g=\omega_R \times G\).

Covering the domain \(g \times \Omega\) by a finite number of \(n\)-dimensional cubes with the aid of Lemma 2 (Sec. 1.8, [1]) and estimate (5.25), we obtain the inequality

\[ \left\| \frac{\partial^l u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right\|_{g \times \Omega} \leq A\varepsilon^2 \left( \alpha_1+\cdots+\alpha_n=l,\, l\leq \left[\frac{n}{2}\right]+3 \right). \tag{5.31} \]

From Lemma 2.9 of [2] and inequalities (5.25) and (5.27) there follows the estimate

\[ \|u\|_g \leq B\varepsilon^2, \tag{5.32} \]

where \(B\) is a constant that depends only on the domain \(g\).

Applying the embedding theorem (see [4]), and also estimates (5.30), (5.31), and (5.32), we obtain

\[ \|u\|_{C(g)} \leq c_g \varepsilon^2, \tag{5.33} \]

where the constant \(c_g\) also depends only on the domain \(g\). An analogous estimate for sufficiently large \(|x|\) follows from (5.25) by virtue of (2.10) [1]. Together with (5.33) this leads to

\[ \|u\|_{C(R_n \times G)} \leq \mu \varepsilon^2, \tag{5.34} \]

where the constant \(\mu\) does not depend on \(\varepsilon\) and \(u\).

Taking into account the conditions imposed on \(A_{ik}\) and \(f\), we can prove that \(u_2\) belongs to the space \(C^2(G)\). Hence, from (5.34) and the fact that the jumps of \(u_2\) on \(\Lambda\) have order \(\varepsilon^2\), on the basis of the maximum principle it follows that

\[ \|u_2\|_{C(G)} \leq \bar{\mu}\varepsilon^2,\qquad \|u_2\|_{C(R_n)} \leq \bar{\bar{\mu}}\varepsilon^2. \]

Now we can estimate the solution \(u\) of the variational problem (1.5) [1]. Using (5.6), (5.7), and (5.15), we shall have

\[ \|u\|_{C(R_n)} \leq \lambda \varepsilon^2, \]

where

\[ \lambda = \mu+\|f\|_{C(G)} + \frac{1}{q} \left\| \frac{\partial f}{\partial \nu} \right\|_{C(\Lambda)} + \|f_1\|_{C(G)}. \]

5.9. This estimate is sharp for the entire class of equations under consideration, in the sense of order with respect to \(\varepsilon\). This is easily verified by considering the following example (\(n=3\)):

\[ \begin{gathered} \Delta u-\frac{1}{\varepsilon^{2}}u=1,\quad \text{if } x\in G,\\ \Delta u=0,\quad \text{if } x\in \overline{G},\\ u_{+}-u_{-}=0,\quad \text{if } x\in \Lambda,\\ \left(\frac{\partial u}{\partial n}\right)_{+} -\left(\frac{\partial u}{\partial n}\right)_{-}=0,\quad \text{if } x\in \Lambda. \end{gathered} \tag{5.35} \]

Here \(G\) is the sphere of radius 1 with center at the origin. Let us rewrite equation (5.35) in the spherical coordinate system, taking into account the resulting symmetry:

\[ \frac{1}{r^{2}}\frac{\partial}{\partial r} \left(r^{2}\frac{\partial u}{\partial r}\right) -\frac{1}{\varepsilon^{2}}u=1,\quad \text{if } r<1, \]

\[ \frac{1}{r^{2}}\frac{\partial}{\partial r} \left(r^{2}\frac{\partial u}{\partial r}\right)=0,\quad \text{if } r>1. \]

The solution can be written in explicit form:

\[ u=-\varepsilon^{2} -\frac{\varepsilon^{3}\left(\exp\left(-\frac{r}{\varepsilon}\right)-\exp\frac{r}{\varepsilon}\right)} {r\left(\exp\left(-\frac{1}{\varepsilon}\right)+\exp\frac{1}{\varepsilon}\right)}, \quad \text{if } r<1, \]

\[ u=\left[ -\varepsilon^{2} -\frac{\varepsilon^{3}\left(\exp\left(-\frac{1}{\varepsilon}\right)-\exp\frac{1}{\varepsilon}\right)} {\exp\left(-\frac{1}{\varepsilon}\right)+\exp\frac{1}{\varepsilon}} \right]\frac{1}{r}, \quad \text{if } r>1, \]

which proves our assertion.

APPENDIX

We give two examples of problems from mathematical physics that reduce to differential equations (0.1) with boundary conditions (0.3), (0.4) (see [1]). We shall assume that \(n=3\).

First example. Suppose that in a bounded domain \(G\) there are stationary sources, characterized by a function \(f(x)\), and at the same time absorption proportional to the concentration \(u\) with coefficient \(\frac{1}{\varepsilon^{2}}\), while through the boundary \(\Lambda\) of the domain \(G\) free exchange takes place. Then the concentration \(u\) is the solution of a differential problem of the form (0.1), (0.3), (0.4) [1]:

\[ \Delta u-\frac{1}{\varepsilon^{2}}u=f\quad (x\in G'), \]

\[ \Delta u=0\quad (x\in \overline{G}), \]

\[ u_{+}-u_{-}=0\quad (x\in \Lambda), \]

\[ \left(\frac{\partial u}{\partial n}\right)_{+} -\left(\frac{\partial u}{\partial n}\right)_{-}=0\quad (x\in \Lambda), \]

\[ \lim_{|x|\to\infty} u=0. \]

Second example. Let there be a stationary source at the point \(x_0 \in \overline{G}\), and let \(G\) be a region of absorption proportional to the concentration \(v\) with coefficient \(\dfrac{1}{\varepsilon^2}\).

The concentration \(v\) is the solution of the following differential problem:

\[ \Delta v-\frac{1}{\varepsilon^2}v=0 \qquad (x\in G), \]

\[ \Delta v=\hat{\delta}(x-x_0) \qquad (x\in \overline{\overline{G}}); \]

\[ v_+ - v_- = 0 \qquad (x\in \Lambda), \]

\[ \left(\frac{\partial v}{\partial n}\right)_+ - \left(\frac{\partial v}{\partial n}\right)_- =0 \qquad (x\in \Lambda), \]

\[ \lim_{|x|\to\infty} v=0, \]

where \(\hat{\delta}(x-x_0)\) is the Dirac delta function.

We write the solution \(v\) in the form

\[ v=u+\frac{1}{4\pi |x-x_0|}. \]

Then \(u\) is the solution of the differential problem (0.1), (0.3), (0.4) from [1]; moreover, the parameter \(\varepsilon\) enters into the expression for \(f(x)\):

\[ f(x)=\frac{1}{4\pi \varepsilon^2 |x-x_0|} \qquad (x\in \overline{G}). \]

In this case we can prove the boundedness of the solution \(u\):

\[ \|u\|_{C(R_n)} \leq c. \]

References

1. Kopchenov V. D. Differential Equations, 2, No. 6, 1966.
2. Nikolsky S. M. Izvestiya AN SSSR, ser. matem., 22, No. 5, 599, 1958.
3. Uspensky S. V. DAN SSSR, 127, No. 3, 526, 1959.
4. Sobolev S. L. Some applications of functional analysis to mathematical physics. L., 1950.
5. Focht A. S. DAN SSSR, 154, No. 6, 1235—1239, 1964.
6. Picone M. Rend. Acc. Lincei, 28, 331—338, 1938.

Received by the editors
February 12, 1966

Moscow Forestry Engineering
Institute