UDC 517.946:519.3

ON A VARIATIONAL PROBLEM WITH A SMALL PARAMETER. I

V. D. KOPCHENOV

In this paper the following problem is considered. In the \(n\)-dimensional space \(R_n\) of points \(x=(x_1,\ldots,x_n)\) there is given a bounded domain \(G\) with boundary \(\Lambda\) of class \(C^{\left[\frac n2\right]+3}\) and a differential equation

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

\[ L(u)=0\quad (x\in R_n\setminus \overline G), \tag{0.1} \]

\[ L(u)=\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(A_{ik}\frac{\partial u}{\partial x_k}\right)\quad (n\geqslant 3), \]

where \(\varepsilon>0\) is a small parameter.

It is assumed that the coefficients \(A_{ik}=A_{ik}(x)\) are bounded and continuous on \(R_n\), together with their derivatives up to order \(\left[\frac n2\right]+2\) inclusive, and that the ellipticity condition is satisfied

\[ \sum_{i,k=1}^{n} A_{ik}\xi_i\xi_k\geqslant \beta\sum_{i=1}^{n}\xi_i^2, \tag{0.2} \]

where the constant \(\beta>0\) does not depend on \(x\in R_n\) and \(\xi=(\xi_1,\ldots,\xi_n)\). The function \(f\in C^{\left[\frac n2\right]+4}(\overline G)\). It is proved that on \(R_n\) there exists a unique function \(u=u_\varepsilon\) possessing the following properties:

1) \(u\) is continuous, together with its first-order partial derivatives, on \(R_n\). Thus, in particular,

\[ u_+\big|_{\Lambda}=u_-\big|_{\Lambda},\qquad \frac{\partial u}{\partial \nu_+}\bigg|_{\Lambda} = \frac{\partial u}{\partial \nu_-}\bigg|_{\Lambda}, \tag{0.3} \]

where \(u_+\), \(u_-\) denote the limiting values of \(u\) on \(\Lambda\), respectively from inside and from outside, and \(\dfrac{\partial u}{\partial \nu}\) is the derivative along the conormal to \(\Lambda\).

2)

\[ \lim_{|x|\to\infty}u(x)=0 \tag{0.4} \]

under the following additional restriction imposed on the coefficients \(A_{ik}\) \((i,k=1,\ldots,n)\) (see [7]):

\[ \left|\frac{\partial^l A_{ik}}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}}\right| < \frac{c}{|x|^{1/2}} \quad (\alpha_1+\cdots+\alpha_n=l;\; 1\leq l\leq [n/2]). \]

Here \(|x|>R>1\), and \(c\) is a constant independent of \(|x|\).

3) \(u\) has second-order derivatives, continuous on \(\overline G\) and on \(R_n\setminus G\), and satisfies the differential equation (0.1) on \(R_n\setminus \Lambda\).

4) The integral

\[ D(u)=\int_{R_n}\sum_{i=1}^n\left(\frac{\partial u}{\partial x_i}\right)^2\,dx<\infty \tag{0.5} \]

is finite. In this case the estimate holds (see [8])

\[ |u_\varepsilon(x)|\leq c\varepsilon^2 \qquad (0<\varepsilon<\varepsilon_0), \tag{0.6} \]

where the constant \(c\) depends on the data of the problem, but does not depend on \(\varepsilon\).

A problem analogous in a certain sense was considered by V. K. Saul’ev in [9]. He obtained the estimate \(|u_\varepsilon(x)|\leq c\varepsilon\).

M. I. Vishik and L. A. Lyusternik in [10] considered certain problems with a small parameter, an essential feature of which is the presence of a boundary layer near the surface \(\Lambda\), on which the coefficients of equation (0.1) have a discontinuity. This means that the solution is different from zero in a thin layer \(\Omega\subset R_n\setminus G\), adjacent to \(\Lambda\), and is negligibly small in the remaining part of the space \(R_n\setminus G\). The problems considered by these authors lead to solutions whose squares are integrable over the whole space \(R_n\).

The formulation of our problem is somewhat different; it leads to a solution whose square is not integrable on \(R_n\) in general.

In this Part I of the work, the variational problem corresponding to the differential problem (0.1), (0.3), (0.4) is considered. The boundedness below of our functional in the corresponding \((M)\) class of functions is proved, as well as the uniqueness of the solution \(u\) of the variational problem and its membership in this same class of functions. Then the differential properties of the solution obtained are studied. Estimates are found for the derivatives of \(u\) up to order \([n/2]+3\) in the \(L_2\) metric in closed domains not containing points of the surface \(\Lambda\). Further, for the solution of another variational problem considered by S. M. Nikol’skii in [1], analogous estimates are found for the derivatives along \(\Lambda\), already right up to the surface \(\Lambda\).

In the second paragraph, for the solution \(u\) of the original variational problem, property (0.4) is proved.

In Part II of the work a method for obtaining the estimate (0.6) is presented; it is shown there that the estimate is sharp in order with respect to \(\varepsilon\).

The results of the present work are obtained by the variational method.

§ 1. FORMULATION OF THE PROBLEM. BASIC LEMMAS

1.1. In the present work we study the properties of the solution of a certain variational problem, to which there corresponds a functional with a discontinuity of the first kind in the coefficient of the square of the admissible function on a given closed sufficiently smooth surface. The principal results concerning this problem, called the Hilbert variational problem, were obtained in the case of smooth coefficients by S. M. Nikol’skii [1] and generalized by S. V. Uspenskii [4].

We consider the class \(\mathbf{M}\) of functions \(g\), each of which satisfies the following conditions. The generalized derivatives \(\dfrac{\partial g}{\partial x_i}\) \((i=1,\ldots,n)\) exist in \(G\) and in \(R_n\setminus \overline{G}\),

\[ \Psi(g)=\int_{R_n\setminus \Lambda} \left( \sum_{i,k=1}^{n} A_{ik}(P)\frac{\partial g}{\partial x_i}\frac{\partial g}{\partial x_k} + B(P)g^2 \right)dR_n<\infty, \tag{1.1} \]

where \(B(P)=\dfrac{1}{\varepsilon^2}\), if \(P\in G\), \(B(P)=0\), if \(P\in \overline{G}\),

\[ g_+-g_-=0 \quad \text{on } \Lambda, \tag{1.2} \]

\[ g|_\infty=\lim_{\rho\to\infty} g(Q,\rho)=0, \tag{1.3} \]

where \(\rho\) is the distance of the point \(P(Q,\rho)\) from the origin and \(Q\) is the point of intersection of the radius vector of the point \(P\) with the unit sphere centered at the origin. The limit (1.3) is understood in the sense of convergence for almost all \(Q\). Here \(g_+\), \(g_-\) denote the limiting values of \(g\) on \(\Lambda\) (almost everywhere or in the mean), respectively from inside and outside \(\Lambda\) in the direction of the conormals to \(\Lambda\); \(\varepsilon>0\) is a small parameter.

1.2. From (0.2) and (1.1) it follows that \(D_{R_n\setminus \Lambda}(g)<\infty\), but since (1.2) is satisfied, as is known and (see [2, 3]),

\[ D_{R_n}(g)=\int_{R_n}\sum_{i=1}^{n} \left(\frac{\partial g}{\partial x_i}\right)^2 dx<\infty. \tag{1.4} \]

Let us note that, conversely, the finiteness of the Dirichlet integral implies the existence of \(g_+\) and \(g_-\) and their equality.

As S. V. Uspenskii showed, from (1.4), for \(n\ge 3\), there follows the existence for almost all \(Q\) of the limit \(\lim_{\rho\to\infty} g(Q,\rho)=A\), independent of \(Q\). Thus, from (1.1) and (1.2), for \(n\ge 3\) the existence of this limit already follows. Condition (1.3) adds only that it is equal to zero.

1.3. Let \(E\) be any open set in \(n\)-dimensional space \(R_n\) (\(E\) may coincide with \(R_n\)). By the symbol \((v,w)_E\) we shall, as usual, denote the scalar product

\[ (v,w)_E=\int_E vw\,dE,\quad v,w\in L_2(E). \]

Put \(\|v\|_E=\sqrt{(v,v)_E}\).

We denote by \(W_2^{(1)}(E)\) the space of functions having on \(E\) generalized partial derivatives of first order which, like the function itself, belong to the space \(L_2(E)\). As the norm of a function \(v\in W_2^{(1)}(E)\) we take

\[ \|v\|_{W_2^{(1)}(E)} = \|v\|_E+\sum_{i=1}^{n} \left\|\frac{\partial v}{\partial x_i}\right\|_E . \]

Further, for \(v\in W_2^{(1)}(E)\) put

\[ D_E(v)=\int_E\sum_{i=1}^{n} \left(\frac{\partial v}{\partial x_i}\right)^2 dE. \]

It is easy to see that, for every function \(v\in W_2^{(1)}(E)\), the inequality holds

\[ D_E^{1/2}(v) \le \sum_{i=1}^{n} \left\|\frac{\partial v}{\partial x_i}\right\|_E \le n^{1/2}D_E^{1/2}(v). \]

(the right-hand side of the inequality follows from Hölder’s inequality for sums).

Let \(\theta(g)=\psi(g)-2(f,g)_G\), where \(f\in C^{\left[\frac n2\right]+4}(\overline G)\).
The question is posed of the existence and properties of a function for which the minimum of the variational problem

\[ \min_{g\in \mathbf M}\,[\psi(g)-2(f,g)_G]. \tag{1.5} \]

1.4. Lemma 1. The functional \(\theta(g)\) is bounded below on the class \(g\in \mathbf M\).

Proof. Indeed,
\[ \theta(g)=\psi(g)-2(f,g)_G \ge \psi(g)-2|(f,g)_G| \ge \psi(g)-\left(\frac{1}{2\varepsilon^2}(g,g)_G+2\varepsilon^2(f,f)_G\right) \]
\[ \ge \psi(g)-\left(\frac12\psi(g)+2\varepsilon^2(f,f)_G\right) =\frac12\psi(g)-2\varepsilon^2(f,f)_G \ge -2\varepsilon^2(f,f)_G . \]
The lemma is proved. From it follows the existence of a finite exact lower bound
\[ d=\inf_{g\in \mathbf M}\theta(g). \]

From the set \(\mathbf M\) of functions \(g\) one can extract a minimizing sequence \(\{g_k\}\) for which
\[ \lim_{k\to\infty}\theta(g_k)=d. \]

Let \(\delta>0\) be given. There exists \(N>0\) such that \(\theta(g_k)<d+\delta\), if \(k>N\). Let \(k\) and \(m>N\). Clearly,
\[ \frac{g_k+g_m}{2}\in \mathbf M, \]
and therefore
\[ \theta\left(\frac{g_k+g_m}{2}\right)\ge d. \]

From the obvious equality (here \(v\) and \(w\in \mathbf M\))
\[ \psi\left(\frac{v-w}{2}\right) =\frac12\theta(v)+\frac12\theta(w)-\theta\left(\frac{v+w}{2}\right) \]
there follows the inequality
\[ \psi\left(\frac{g_k-g_m}{2}\right) =\frac12\theta(g_k)+\frac12\theta(g_m) -\theta\left(\frac{g_k+g_m}{2}\right) \le \frac{d+\delta}{2}+\frac{d+\delta}{2}-d=\delta, \]
or
\[ \psi(g_k-g_m)\le 4\delta, \]
i.e.,
\[ \psi(g_k-g_m)\to 0 \quad (k,m\to\infty). \]

On the basis of Lemma 2 of [4], on any open set \(E\), by virtue of (1.3) it is true that

\[ \|g\|_E\le c(E)\sqrt{\overline{D_{R_n}(g)}}, \tag{1.6} \]

where \(g\in \mathbf M,\ E\subset R_n,\ n\ge 3\); hence
\[ \|g_k-g_m\|_E\le c(E)\sqrt{\overline{D_{R_n}(g_k-g_m)}} \le c_1(E)\sqrt{\psi(g_k-g_m)}\to 0 \quad (k,m\to\infty), \]
and, correspondingly, there exists on \(R_n\) a function \(u\) such that on any ball \(\omega\)
\[ u\in L_2(\omega)\quad\text{and}\quad \|g_n-u\|_\omega\to 0 \quad (n\to\infty), \]
and also, by virtue of the closedness of the operator of generalized differentiation (see [5]),
\[ D_\omega(g_n-u)\to 0 \quad (n\to\infty). \]
We have further (see (0.2))
\[ D_{R_n}(g_k-g_m)\le \frac1\beta\,\psi(g_k-g_m). \]
From this inequality and from the fact that
\[ \psi(g_k-g_m)\to 0 \quad (k,m\to\infty), \]
it follows that
\[ D_\omega(g_n-u)\le \frac1\beta\,\psi(g_n-u)\quad (n\to\infty), \]

and hence

\[ D_{R_n}(g_n-u)\leqslant {1\over \beta}\,\psi(g_n-u)\quad (n\to\infty). \tag{1.7} \]

It follows that \(\psi(u)<\infty,\ \theta(u)<\infty\).

Introduce the notation

\[ \psi(u,v)=\int_{R_n}\left(\sum_{i,k=1}^n A_{ik}(P)\,{\partial u\over \partial x_i}\,{\partial v\over \partial x_k}+B(P)uv\right)\,dR_n . \]

\(\psi(u,v)\) is a symmetric and bilinear functional in \(u\) and \(v\); therefore, as is known, the inequalities

\[ |\psi(u,v)|\leqslant \psi^{1/2}(u)\psi^{1/2}(v),\qquad \psi^{1/2}(u+v)\leqslant \psi^{1/2}(u)+\psi^{1/2}(v) \]

hold for it, and hence

\[ |\psi^{1/2}(u)-\psi^{1/2}(v)|\leqslant \psi^{1/2}(u-v), \tag{1.8} \]

where \(u,v\in W_2^{(1)}(R_n)\).

We shall prove that \(\theta(u)=d\). Using inequality (1.8), we have

\[ |\theta(g_m)-\theta(u)|\leqslant |\psi(g_m)-\psi(u)|+2|(f,g_m-u)_G|\leqslant \]

\[ \leqslant |\psi^{1/2}(g_m)-\psi^{1/2}(u)|\bigl(\psi^{1/2}(g_m)+\psi^{1/2}(u)\bigr) +2\|f\|_G\|g_m-u\|_G\leqslant \]

\[ \leqslant \psi^{1/2}(g_m-u)\bigl(\psi^{1/2}(g_m)+\psi^{1/2}(u)\bigr) +2\|f\|_G\|g_m-u\|_G . \tag{1.9} \]

Each term in the last part of inequality (1.9) tends to zero, whence it follows that

\[ d=\lim_{m\to\infty}\theta(g_m)=\theta(u). \]

1.5. We shall show that \(u\in \mathbf{M}\). For this it is necessary to prove that

\[ u_+-u_-=0\quad \text{on } \Lambda, \tag{1.10} \]

\[ u|_\infty=\lim_{\rho\to\infty}u(Q,\rho)=0. \tag{1.11} \]

These equalities are understood in the same sense as the analogous equalities (1.2) and (1.3).

S. V. Uspenskii showed in [4] that, if \(D_{R_n}(v)<\infty\), then

\[ \lim_{\rho\to\infty}\int_\sigma v^2(Q,\rho)\,dQ=c, \]

where \(\sigma\) is the unit sphere with center at the origin of the coordinates, \(c\geqslant 0,\ n\geqslant 3\).

Since in our case \(D_{R_n}(u)\leqslant {1\over \beta}\psi(u)\), there exists

\[ \lim_{\rho\to\infty}\int_\sigma u^2(Q,\rho)\,dQ=N. \tag{1.12} \]

We shall show that \(N=0\). We have

\[ \int_\sigma u^2(Q,\rho)\,dQ =\int_\sigma \bigl(u^2(Q,\rho)-g_k^2(Q,\rho)\bigr)\,dQ +\int_\sigma g_k^2(Q,\rho)\,dQ\leqslant \]

\[ \leqslant {1\over \rho^{\,{n-1\over 2}}} \left(\int_{\sigma_\rho} (u(P)-g_k(P))^2\,d\sigma_\rho \int_\sigma (u(Q,\rho)+g_k(Q,\rho))^2\,dQ\right)^{1/2} + \]

\[ +\int_\sigma g_k^2(Q,\rho)\,dQ, \tag{1.13} \]

where \(\{g_k\}\) is our minimizing sequence, and \(\sigma_\rho\) is the sphere of radius \(\rho\) with center at the origin.

S. V. Uspenskii in his work [4] obtained the inequality

\[ \int_\sigma g^2(Q,\rho)\,dQ \leq c \rho^{2-n} D_{\Omega_\rho}(g), \tag{1.14} \]

where \(g \in M\), \(\Omega_\rho\) is the exterior of the sphere \(\sigma_\rho\), and \(c\) does not depend on \(\rho\) or \(g\). Therefore, for the functions \(g_k\) of the minimizing sequence, the estimate

\[ \int_\sigma g_k^2(Q,\rho)\,dQ \leq c \rho^{2-n} D_{R_n}(g_k) \leq c \rho^{2-n} M \tag{1.15} \]

holds.

Here \(M \geq D_{R_n}(g_k)\) \((k=1,2,\ldots)\), since \(D_{R_n}(g_k) \to D_{R_n}(u) < \infty\) as \(k \to \infty\). The right-hand side of inequality (1.15) does not depend on \(k\), and consequently the estimate is uniform in \(k\).

Let \(\rho_0\) be chosen so that

\[ \int_\sigma (u(Q,\rho)+g_k(Q,\rho))^2\,dQ < A \tag{1.16} \]

for \(\rho>\rho_0\), where \(A\) is a constant independent of \(\rho\). This can always be done, since (1.12) and (1.15) hold.

Using Lemma 2.9 of [1], we write the inequality

\[ \frac{1}{\rho^{\frac{n-1}{2}}}\|u-g_k\|_{\sigma_\rho} \leq c_1(\rho)\left(\|u-g_k\|_{\omega_\rho} +\sqrt{D_{\omega_\rho}(u-g_k)}\right), \tag{1.17} \]

where \(\omega_\rho\) and \(\sigma_\rho\) are respectively the ball and the sphere of radius \(\rho\).

On the basis of the inequality (see (1.6))

\[ \|u-g_k\|_{\omega_\rho} \leq c(\omega_\rho)\sqrt{D_{R_n}(u-g_k)} \]

we estimate the first term on the right-hand side of (1.17):

\[ \frac{1}{\rho^{\frac{n-1}{2}}}\|u-g_k\|_{\sigma_\rho} \leq c_2(\rho)\sqrt{D_{R_n}(u-g_k)}. \tag{1.18} \]

Applying to the right-hand side of inequality (1.13) the estimates (1.14), (1.16), (1.18), we obtain the inequality

\[ \int_\sigma u^2(Q,\rho)\,dQ \leq c_3(\rho)\sqrt{D_{R_n}(u-g_k)} + c \rho^{2-n} D_{\Omega_\rho}(g_k). \tag{1.19} \]

Next, if we let \(k \to \infty\) and take into account condition (1.7), we obtain the inequality

\[ \int_\sigma u^2(Q,\rho)\,dQ \leq c \rho^{2-n} D_{\Omega_\rho}(u), \tag{1.20} \]

valid for any fixed \(\rho\) (\(c\) is a constant independent of \(\rho\)). Let \(\rho \to \infty\); then the right-hand side of inequality (1.20) tends to zero. Hence

\[ N=\lim_{\rho\to\infty}\int_\sigma u^2(Q,\rho)\,dQ=0. \]

In [4] it is proved that from this equality and from (0.5) it follows that

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

for almost all \(Q\). Condition (1.11) is proved.

Condition (1.10) is a consequence of the fact that the Dirichlet integral is finite (see (1.7)):

\[ D_{R_n}(u)=\int_{R_n}\sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_i}\right)^2 dx < \infty. \]

Thus, (1.10) and (1.11) have been established, which means that the function \(u\) giving the minimum of the functional \(\theta(g)\), \(g \in \mathbf M\), belongs to the space \(\mathbf M\).

1.6. It is easy to prove the uniqueness of the solution of our variational problem. Suppose that there exist two solutions \(u\) and \(\bar u\). Since \(\dfrac{u+\bar u}{2}\in \mathbf M\) and

\[ \theta\left(\frac{u+\bar u}{2}\right)=\lambda\ge d, \]

we have

\[ 0 \le \|u-\bar u\|_\omega^2 < c(\omega)D_{R_n}(u-\bar u)\le c_1(\omega)\psi\left(\frac{u-\bar u}{2}\right)= \]

\[ = c_1(\omega)\left[2\theta(u)+2\theta(\bar u)-4\theta\left(\frac{u+\bar u}{2}\right)\right] = c_1(\omega)(2d+2d-4\lambda)\le 0 \]

and, consequently, \(u=\bar u\) on any ball \(\omega\), and hence on \(R_n\). We have proved the following theorem.

Theorem 1. Among the functions \(g\) of the class \(\mathbf M\) there exists a unique function \(u\), and moreover a unique one, which realizes the minimum of the variational problem (1.5), i.e.

\[ \min_{g\in \mathbf M}\theta(g)=\theta(u). \]

1.7. Introduce an \(n\)-dimensional closed cube \(\Delta\), with edges parallel to the coordinate axes, belonging to the exterior of \(\bar G\). By \(\Delta_\delta\) \((\delta>0)\) we shall denote a cube containing \(\Delta\), with edges parallel to the coordinate axes, such that the corresponding faces of \(\Delta\) and \(\Delta_\delta\) are at distance \(\delta\).

In this and the following subsections we shall estimate all generalized derivatives of the solution \(u\) of the variational problem (1.5) up to order

\[ l \le \left[\frac n2\right]+3. \]

First we obtain estimates in the space \(L_2(\Delta)\), where \(\Delta\subset R_n\setminus \bar G\), via the Dirichlet integral \(D_{\Delta_{2l\delta}}(u)\), \(\Delta\subset \Delta_{2l\delta}\subset R_n\setminus \bar G\). Then the same will be done for any domain \(R_n\setminus \bar G'\), where \(\bar G\subset G'\), via the Dirichlet integral taken over the domain \(R_n\setminus \bar G\). In doing so we shall keep track of how the constants obtained in the estimates depend on the parameters.

Let us estimate the first difference with respect to the coordinate \(x_j\) of

\[ \frac{\partial u}{\partial x_i}\quad (i,j=1,\ldots,n) \]

in the cube \(\Delta\). Write the variational equation:

\[ \psi(u,v)=(f,v)_G, \tag{1.21} \]

where \(v\) is a function such that \(v\in \mathbf M\). In particular, for the function \(v_0\), equal to zero outside \(\Delta_\delta\), \(\Delta_{2\delta}\subset R_n\setminus \bar G\), equation (1.21) becomes

\[ \int_{\Delta_\delta}\sum_{i,k=1}^n A_{ik}\frac{\partial u}{\partial x_i}\frac{\partial v_0}{\partial x_k}\,dx=0. \tag{1.22} \]

Introduce a new function

\[ v_0(x-\bar h_j)=v_0(x_1,\ldots,x_{j-1},x_j-h_j,x_{j+1},\ldots,x_n)\quad (|\bar h_j|\le \delta), \]

where \(\bar h_j\) is a vector of length \(h_j\), directed along the axis \(x_j\). For it the variational equation has the form

\[ \int_{R_n\setminus \overline{G}} \sum_{i,k=1}^{n} A_{ik}\, \frac{\partial u}{\partial x_i}\, \frac{\partial v_0(x-\overline{h}_j)}{\partial x_k}\,dx = \int_{\Delta_\delta}\sum_{i,k=1}^{n} A_{ik}(x+\overline{h}_j)\, \frac{\partial u(x+\overline{h}_j)}{\partial x_i}\, \frac{\partial v_0}{\partial x_k}\,dx=0. \tag{1.23} \]

Introduce the notation:
\[ u_{h_j}=\frac{u(x+\overline{h}_j)-u(x)}{h_j},\qquad A_{ikh_j}=\frac{A_{ik}(x+\overline{h}_j)-A_{ik}(x)}{h_j}. \]

Subtracting (1.22) from (1.23), after first dividing by \(h_j\), we obtain
\[ \int_{\Delta_\delta}\sum_{k=1}^{n}\frac{\partial v_0}{\partial x_k} \left( \sum_{i=1}^{n}A_{ik}\frac{\partial u_{h_j}}{\partial x_i} + \frac{\partial u(x+\overline{h}_j)}{\partial x_i}\,A_{ikh_j} \right)\,dx=0. \tag{1.24} \]

Now take the function \(v_0\) in the form
\[ v_0=\eta u_{h_j}, \tag{1.25} \]
where \(\eta\) is a nonnegative function continuous on \(R_n\), such that
\[ \eta= \begin{cases} 1, & \text{if } x\in \Delta,\\ 0, & \text{if } x\in \overline{\Delta_\delta}, \end{cases} \tag{1.26} \]
and the conditions
\[ 0\leqslant \eta \leqslant 1,\qquad \left(\frac{\partial \eta}{\partial x_i}\right)^2\leqslant c(\delta)\eta \quad (i=1,\ldots,n) \tag{1.27} \]
are satisfied, where \(c(\delta)\) does not depend on \(x_i\). Such a function can be constructed (see [6]). Write equation (1.24) for the function (1.25):
\[ \begin{aligned} &\int_{\Delta_\delta}\eta\sum_{i,k=1}^{n}A_{ik} \frac{\partial u_{h_j}}{\partial x_i} \frac{\partial u_{h_j}}{\partial x_k}\,dx + \int_{\Delta_\delta}\eta\sum_{i,k=1}^{n}A_{ikh_j} \frac{\partial u(x+\overline{h}_j)}{\partial x_i} \frac{\partial u_{h_j}}{\partial x_k}\,dx \\ &\quad+ \int_{\Delta_\delta}u_{h_j}\sum_{i,k=1}^{n}A_{ik} \frac{\partial u_{h_j}}{\partial x_i} \frac{\partial \eta}{\partial x_k}\,dx + \int_{\Delta_\delta}u_{h_j}\sum_{i,k=1}^{n}A_{ikh_j} \frac{\partial \eta}{\partial x_k} \frac{\partial u(x+\overline{h}_j)}{\partial x_i}\,dx \\ &=J_1+J_2+J_3+J_4=0. \end{aligned} \tag{1.28} \]

1.8. The following inequality holds (see Lemma 3, [7]):
\[ \int_{\Delta_\delta}u_{h_j}^{2}\,dx \leqslant \int_{\Delta_{2\delta}}\left(\frac{\partial u}{\partial x_j}\right)^2 dx \qquad (|\overline{h}_j|\leqslant \delta). \tag{1.29} \]

Introduce the notation
\[ M=\max_{\substack{i,k\\ x\in R_n}} \left\{ |A_{ik}|,\ldots, \left| \frac{\partial^{\,l}A_{ik}} {\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right| \right\} \tag{1.30} \]
\[ \left( \alpha_1+\cdots+\alpha_n=l;\quad l\leqslant \left[\frac n2\right]+2 \right). \]

Estimate the integral \(J_1\) from below, and the integrals \(J_2,J_3,J_4\) from above:
\[ J_1\geqslant \beta \int_{\Delta_\delta}\eta\sum_{i=1}^{n} \left(\frac{\partial u_{h_j}}{\partial x_i}\right)^2 dx = \beta A_{h_j}^{2}, \]

\[ |J_2| \leq n^{3/2} M \sqrt{\int_{\dot\Delta_\delta} \eta \sum_{i=1}^n \left(\frac{\partial u(x+\bar h_j)}{\partial x_i}\right)^2 dx} \leq A_1 \sqrt{D_{\Delta_{2\delta}}(u)}\,\Lambda h_j , \]

where \(A_1\) and, subsequently, \(A_2, A_3,\ldots\) are constants independent of \(h_j\),

\[ |J_3| \leq c_1(\delta)n^{3/2}M \sqrt{\int_{\Delta_\delta} u_{h_j}^2\,dx\,\Lambda h_j} \leq A_2 \sqrt{D_{\Delta_{2\delta}}(u)}\,\Lambda h_j , \]

where \(c_1(\delta)=\sqrt{c(\delta)}\) (see (1.27)),

\[ |J_4| \leq c_1(\delta)n^{3/2}M \sqrt{\int_{\Delta_\delta} u_{h_j}^2\,dx}\times \]

\[ {}\times \sqrt{\int_{\dot\Delta_\delta}\eta \sum_{i=1}^n \left(\frac{\partial u(x+\bar h_j)}{\partial x_i}\right)^2 dx} \leq A_3 D_{\Delta_{2\delta}}(u). \]

From (1.28) it then follows that
\[ \beta \Lambda h_j^2 \leq A_4 \sqrt{D_{\Delta_{2\delta}}(u)}\,\Lambda h_j + A_3 D_{\Delta_{2\delta}}(u), \]
where \(A_3, A_4, \beta\) do not depend on \(h_j\), and finally

\[ \left\|\frac{\partial u_{h_j}}{\partial x_i}\right\|_{\Delta} \leq A_5 \sqrt{D_{\Delta_{2\delta}}(u)} \quad (i,j=1,\ldots,n), \tag{1.31} \]

where \(A_5\) is a constant which depends on \(\delta,n,M,\beta\), but does not depend on the position of \(\Delta\) or on the magnitude of \(h_j\). Using Lemma 1 [6], we obtain from (1.31)

\[ \left\|\frac{\partial^2 u}{\partial x_i\partial x_j}\right\|_{\Delta} \leq A_5 D_{\Delta_{2\delta}}^{1/2}(u) \leq A_6\sum_{i=1}^n \left\|\frac{\partial u}{\partial x_i}\right\|_{\Delta_{2\delta}} \quad (i,j=1,\ldots,n). \]

Applying the method of induction, it is easy to prove the following lemma.

Lemma 2. The derivatives
\[ \frac{\partial^{\,l+1}u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \]
of the solution \(u\) of the variational problem (1.5) belong to \(L_2(\Delta)\), and the inequality

\[ \left\| \frac{\partial^{\,l+1}u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right\|_{\Delta} \leq A D_{\Delta_{2l\delta}}^{1/2}(u) \leq A'\sum_{i=1}^n \left\|\frac{\partial u}{\partial x_i}\right\|_{\Delta_{2l\delta}} \]

\[ \left( \alpha_1+\cdots+\alpha_n=l+1;\quad l\leq \left[\frac n2\right]+2;\quad \Delta_{2l\delta}\subset R_n\setminus \bar G;\quad |\bar h_j|\leq \delta \right). \tag{1.32} \]

Here \(A,A'\) are constants which depend on \(n,\beta,\delta,M\), but do not depend on the position of the cube \(\Delta\). Hence it follows easily that if \(G'\) is an arbitrary open domain containing \(\bar G\), then the inequality

\[ \left\| \frac{\partial^{\,l+1}u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right\|_{R_n\setminus G'} \leq A(G')\sum_{i=1}^n \left\|\frac{\partial u}{\partial x_i}\right\|_{R_n\setminus \bar G} \tag{1.33} \]

holds.

1.9. Introduce the notation

\[ N=\max_{x\in \bar G} \left\{ |f|,\ldots, \left| \frac{\partial^l f}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right| \right\} \left( \alpha_1+\cdots+\alpha_n=l;\quad l\leq \left[\frac n2\right]+4 \right). \]

If the \(n\)-dimensional cube \(\Delta \subset \Delta_{2l\delta}\) lies inside the domain \(G\), so that \(\Delta_{2l\delta}\subset G\), then for the derivatives
\(\dfrac{\partial^{l+1}u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}}\) of the solution \(u\) of the variational problem (1.5) one can write an inequality obtained by analogy with how inequality (1.32) was obtained, but somewhat different from it:

\[ \left\| \frac{\partial^{l+1}u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right\|_{\Delta} \leq B\sum_{i=1}^{n}\left\|\frac{\partial u}{\partial x_i}\right\|_{\Delta_{2l\delta}} + C\sqrt{\sum_{i=1}^{n}\left\|\frac{\partial u}{\partial x_i}\right\|_{\Delta_{2l\delta}}}, \tag{1.34} \]

where \(B, C\) are constants which depend on \(n,\beta,\delta,N,M\), but do not depend on the position of the cube \(\Delta\subset G\).

1.10. Fix a point \(Q\in\Lambda\). From it, as from a center, describe a sphere of sufficiently small radius \(r\) so that the piece \(\sigma\) cut out by it from \(\Lambda\) (an open set) is projected one-to-one onto one of the \((n-1)\)-dimensional coordinate planes. We shall assume, for definiteness, that this is the plane \((x_1,\ldots,x_{n-1})\). Then \(\sigma\) is defined by the equation

\[ x_n=\theta(x_1,\ldots,x_{n-1}),\qquad (x_1,\ldots,x_{n-1})\in\sigma', \]

where \(\sigma'\) is an \((n-1)\)-dimensional (open) domain (the projection of \(\sigma\) onto the plane \(x_n=0\)), and the function \(\theta\) is of class \(C^{\left[\frac n2\right]+3}(\overline{\sigma'})\).

Introduce inside the sphere local coordinates \((y_1,\ldots,y_n)\), related to the old ones by the equalities:

\[ y_1=x_1, \]

\[ \cdots \]

\[ y_{n-1}=x_{n-1}, \]

\[ y_n=x_n-\theta(x_1,\ldots,x_{n-1}). \tag{1.35} \]

Let \(H_\rho\) be a circular cylinder of radius \(\rho=\dfrac r2\), such that the plane \(y_n=0\) is perpendicular to its axis, and the axis passes through the point \(Q\). By \(H_{\rho,a_1,a_2}\) \((a_1<a_2)\) denote the part of \(H_\rho\) lying between the planes \(y_n=a_1,\ y_n=a_2\). Owing to the smoothness of the boundary \(\Lambda\), there exists a sufficiently small positive number \(a\) such that one of the finite cylinders \(H_{\rho,0,a}, H_{\rho,-a,0}\) belongs entirely to the domain \(G\), and the other to \(R_n\setminus \overline G\). For definiteness, we shall assume that the first cylinder belongs to \(G\). Its points have coordinate \(y_n>0\). Note that the transformation (1.35) has Jacobian equal to one.

1.11. Consider the class \(\overline M\) of functions \(g\), each of which is subject to the following conditions.

The generalized derivatives \(\dfrac{\partial g}{\partial x_i}\) \((i=1,\ldots,n)\) exist in \(G\) and in \(R_n\setminus\overline G\),

\[ \overline{\psi}(g) = \int_{R_n\setminus\Lambda} \left( \sum_{i,k=1}^{n} A_{ik}(P)\, \frac{\partial g}{\partial x_i} \frac{\partial g}{\partial x_k} + B(P)g^2 \right)\,dR_n <\infty, \]

where \(B(P)=\dfrac{1}{\varepsilon^2}\), if \(P\in G\), \(B(P)=0\), if \(P\notin\overline G\),

\[ g_+-g_-=\varepsilon^2\omega \quad \text{on } \Lambda, \tag{1.36} \]

\[ g|_\infty=\lim_{\rho\to\infty} g(Q,\rho)=0. \tag{1.37} \]

The function \(\omega(x)\) is defined in some \(n\)-dimensional neighborhood \(O\) of the surface \(\Lambda\), with \(\omega(x)=0\) for \(G\cap O\), \(\omega(x)\in W_2^{\left[\frac n2\right]+3}(O\setminus G)\). The limits (1.36) and (1.37) are understood in the same sense as the limits (1.2) and (1.3). The variational problem of finding

\[ \min_{g\in \overline{M}}\,[\psi(\bar g)] \tag{1.38} \]

was considered by S. M. Nikol’skii in [1]. There it is proved that the solution \(u\) of the variational problem (1.38) exists, is unique, and belongs to the class \(\overline{M}\). We note that the function

\[ \tilde u(x)=u(x)-\varepsilon^2\omega(x) \tag{1.39} \]

has first generalized derivatives everywhere in \(O\) (see § 1.2).

Let us obtain some estimates for the generalized derivatives of the solution of the variational problem (1.38) in \(O\setminus\Lambda\). For this purpose we write the variational equation in the cylinder \(H_{\rho-\delta,-a+\delta,a-\delta}\subset O\) \((0<\delta<a,\rho)\) in the coordinates \((y_1,\ldots,y_n)\) (see § 1.10), assuming that \(v\in \overline{M}_0\) (see [1]) is equal to zero outside the cylinder \(H_{\rho-\delta,-a+\delta,a-\delta}\),

\[ \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \sum_{i,k=1}^n B_{ik}\frac{\partial u}{\partial y_i}\frac{\partial v}{\partial y_k}\,dy +\frac{1}{\varepsilon^2} \int_{H_{\rho-\delta,0,a-\delta}}uv\,dy=0, \tag{1.40} \]

where

\[ B_{ik}=\sum_{j,l=1}^n A_{jl}\frac{\partial y_i}{\partial x_j}\frac{\partial y_k}{\partial x_l} \quad\text{and}\quad \sum_{i,k=1}^n B_{ik}\xi_i\xi_k\ge \beta_1\sum_{i=1}^n \xi_i^2. \tag{1.41} \]

We perform with equation (1.40) all the operations that we performed with equation (1.22). Then we obtain an equation analogous to (1.24):

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

\[ +\frac{1}{\varepsilon^2} \int_{H_{\rho-\delta,0,a-\delta}}u_{h_j}v\,dy=0. \tag{1.42} \]

Here

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

\[ (j=1,\ldots,n-1;\ |\bar h_j|<\delta), \]

the remaining notation is analogous to the notation of § 1.7.

Take as \(v\) the function (see (1.39)) \(v=\eta \tilde u_{h_j}=\eta(u_{h_j}-\varepsilon^2\omega_{h_j})\), where

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

\(\eta\) is a continuous, nonnegative function on \(H_{\rho-\delta,-a+\delta,-a-\delta}\), such that

\[ \eta= \begin{cases} 1, & \text{if } y\in H_{\rho-2\delta,-a+2\delta,a-2\delta},\\ 0, & \text{if } y\in H_{\rho-\delta,-a+\delta,a-\delta}. \end{cases} \tag{1.43} \]

and the condition is satisfied

\[ \left(\frac{\partial \eta}{\partial y_i}\right)^2 < c(\delta)\eta \quad (i=1,\ldots,n). \tag{1.44} \]

Then equation (1.42) will take the form

\[ \begin{aligned} &\int_{H_{\rho-\delta,-a+\delta,a-\delta}} \eta \sum_{i,k=1}^{n} B_{ik} \frac{\partial u_{h_j}}{\partial y_i} \frac{\partial u_{h_j}}{\partial y_k}\,dy + \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \eta \sum_{i,k=1}^{n} B_{ik h_j}\times \\ &\qquad \times \frac{\partial u_{h_j}}{\partial y_k} \frac{\partial u(y+\bar h_j)}{\partial y_i}\,dy + \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \sum_{i,k=1}^{n} B_{ik} \frac{\partial \eta}{\partial y_k} \frac{\partial u_{h_j}}{\partial y_i} u_{h_j}\,dy + \\ &\quad +\varepsilon^2 \int_{H_{\rho-\delta,-a+\delta,0}} \sum_{i,k=1}^{n} B_{ik} \frac{\partial \omega_{h_j}}{\partial y_k} \frac{\partial u_{h_j}}{\partial y_i}\,dy + \varepsilon^2 \int_{H_{\rho-\delta,-a+\delta,0}} \sum_{i,k=1}^{n} B_{ik}\omega_{h_j}\times \\ &\qquad \times \frac{\partial \eta}{\partial y_k} \frac{\partial u_{h_j}}{\partial y_i}\,dy + \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \sum_{i,k=1}^{n} B_{ik h_j} \frac{\partial \eta}{\partial y_k} \frac{\partial u(y+\bar h_j)}{\partial y_i} u_{h_j}\,dy + \\ &\quad +\varepsilon^2 \int_{H_{\rho-\delta,-a+\delta,0}} \sum_{i,k=1}^{n} B_{ik h_j} \frac{\partial \omega_{h_j}}{\partial y_k} \frac{\partial u(y+\bar h_j)}{\partial y_i}\,dy + \varepsilon^2 \int_{H_{\rho-\delta,-a+\delta,0}} \sum_{i,k=1}^{n} B_{ik h_j}\times \\ &\qquad \times \frac{\partial \eta}{\partial y_k} \omega_{h_j} \frac{\partial u(y+\bar h_j)}{\partial y_i}\,dy + \\ &\quad +\frac{1}{\varepsilon^2} \int_{H_{\rho-\delta,0,a-\delta}} \eta\cdot u_{h_j}^{2}\,dy = J_1+\cdots+J_9=0. \end{aligned} \tag{1.45} \]

The integrals \(J_4,J_5,J_7,J_8\) are taken over the cylinder \(H_{\rho-\delta,-a+\delta,0}\) by virtue of the definition of the function \(\omega(x)\) \((\omega(x)=0\) if \(x\in G)\).

1.12. Let us estimate the integrals \(J_1\) and \(J_9\) from below, and the integrals \(J_2,\ldots,J_8\) from above:

\[ J_1 \geq \beta_1 \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \eta \sum_{i=1}^{n} \left(\frac{\partial u_{h_j}}{\partial y_i}\right)^2\,dy = \beta_1 \bar{\Lambda}_{h_j}^{\,2}. \]

Introduce the notation

\[ \bar M = \max_{\substack{y\in H_{\rho,-a,a}\\ i,k}} \left\{ |B_{ik}|,\ldots, \left| \frac{\partial^l B_{ik}} {\partial y_1^{\alpha_1}\cdots \partial y_n^{\alpha_n}} \right| \right\} \]

\[ \left( \alpha_1+\cdots+\alpha_n=l;\quad l\leq \left[\frac{n}{2}\right]+2 \right), \]

further,

\[ |J_2| \leq n^{3/2}\bar M \sqrt{ \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \eta \sum_{i=1}^{n} \left( \frac{\partial u(y+\bar h_j)}{\partial y_i} \right)^2 \,dy\, \bar{\Lambda}_{h_j} } \leq \bar A_1 \sqrt{\bar D_{H_{\rho,-a,a}}(u)\Lambda_{h_j}}, \]

where \(\bar A_1\) and, in what follows, \(\bar A_2,\bar A_3\ldots\) are constants independent of \(h_j\),

\[ |J_3|\le c_1(\delta)n^{3/2}\bar M \sqrt{ \int_{H_{\rho-\delta,-a+\delta,a-\delta}} u_{h_j}^2\,dy\,\bar\Lambda_{h_j} } \le \bar A_2\sqrt{D_{H_{\rho,-a,a}}(u)\,\bar\Lambda_{h_j}}, \]

where \(c_1(\delta)=\sqrt{c(\delta)}\) (see (1.44)),

\[ |J_4|\le \varepsilon^2 n^{3/2}\bar M \sqrt{ \int_{H_{\rho-\delta,-a+\delta,0}} \eta\sum_{i,k=1}^{n} \left(\frac{\partial^2\omega}{\partial y_j\partial y_k}\right)^2 dy\,\bar\Lambda_{h_j} } \le \bar A_3\varepsilon^2\bar\Lambda_{h_j}, \]

\[ |J_5|\le \varepsilon^2 c_1(\delta)n^{3/2}\bar M \sqrt{ \int_{H_{\rho-\delta,-a+\delta,0}} \sum_{i=1}^{n} \left(\frac{\partial\omega}{\partial y_i}\right)^2 dy\,\bar\Lambda_{h_j} } \le \bar A_4\varepsilon^2\bar\Lambda_{h_j}, \]

\[ |J_6|\le c_1(\delta)n^{3/2}\bar M \sqrt{ \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \sum_{i=1}^{n} \left(\frac{\partial u(y+\bar h_j)}{\partial y_i}\right)^2dy } \times \]

\[ \times \sqrt{ \int_{H_{\rho-\delta,-a+\delta,a-\delta}} u_{h_j}^2\,dy } \le \bar A_5D_{H_{\rho,-a,a}}(u), \]

\[ |J_7|\le \varepsilon^2 n^{3/2}\bar M \sqrt{ \int_{H_{\rho-\delta,-a+\delta,0}} \sum_{i,k=1}^{n} \left(\frac{\partial^2\omega}{\partial y_i\partial y_k}\right)^2dy } \times \]

\[ \times \sqrt{ \int_{H_{\rho-\delta,-a+\delta,0}} \sum_{i=1}^{n} \left(\frac{\partial u(y+\bar h_j)}{\partial y_i}\right)^2dy } \le \bar A_6\varepsilon^2\sqrt{D_{H_{\rho,-a,a}}(u)}, \]

\[ |J_8|\le \varepsilon^2 c_1(\delta)n^{3/2}\bar M \sqrt{ \int_{H_{\rho-\delta,-a+\delta,0}} \omega_{h_j}^{2}\,dy } \times \]

\[ \times \sqrt{ \int_{H_{\rho-\delta,-a+\delta,a-\delta}} \sum_{i=1}^{n} \left(\frac{\partial u(y+\bar h_j)}{\partial y_i}\right)^2dy } \le \bar A_7\varepsilon^2\sqrt{D_{H_{\rho,-a,a}}(u)}, \]

\[ J_9>0. \]

From (1.45) it then follows that

\[ \beta_1\bar\Lambda_{h_j}^{\,2} \le \bar\Lambda_{h_j}\left(\bar A_9\sqrt{D_{H_{\rho,-a,a}}(u)}+\bar A_{10}\varepsilon^2\right) + \bar A_{11}D_{H_{\rho,-a,a}}(u) + \bar A_{12}\varepsilon^2\sqrt{D_{H_{\rho,-a,a}}(u)} \]

and, finally,

\[ \left\| \frac{\partial u_{h_j}}{\partial y_i} \right\|_{H_{\rho-2\delta,-a+2\delta,a-2\delta}} < \bar A_{13}\varepsilon^2 \left( \frac{1}{\varepsilon^2}\sqrt{D_{H_{\rho,-a,a}}(u)} + \frac{1}{\varepsilon}\sqrt[4]{D_{H_{\rho,-a,a}}(u)} + 1 \right) < \bar A_{14}\varepsilon^2 \left( \frac{1}{\varepsilon^2}\sum_{i=1}^{n} \left\| \frac{\partial u}{\partial y_i} \right\|_{H_{\rho,-a,a}} + 1 \right). \tag{1.46} \]

Using Lemma 1 [6], we obtain from (1.46)

\[ \left\|\frac{\partial^2 u}{\partial y_i \partial y_j}\right\|_{H_{\rho-2\delta,-a+2\delta,a-2\delta}} < \bar A_{14}\left(\sum_{i=1}^n \left\|\frac{\partial u}{\partial y_i}\right\|_{H_{\rho,-a,a}} +\varepsilon^2\right) \]

for \(i=1,\ldots,n;\ j=1,\ldots,n-1\).

1.13. Denote by \(H_{\rho_0,-a_0,a_0}\) the cylinder lying strictly inside the cylinder \(H_{\rho,-a,a}\) and having a common axis with it.

Let \(\nu=\min\{\rho-\rho_0,\ a-a_0\}\). Applying the method of induction, it is easy to prove the following inequality:

\[ \left\|\frac{\partial^l u}{\partial y_1^{\alpha_1},\ldots,\partial y_n^{\alpha_n}}\right\|_{H_{\rho_0,-a_0,a_0}} \leq \bar A(\nu)\left(\sum_{i=1}^n \left\|\frac{\partial u}{\partial y_i}\right\|_{H_{\rho,-a,a}} +\varepsilon^2\right), \tag{1.47} \]

where \(\alpha_1+\cdots+\alpha_n=l;\ \alpha_n=0,1;\ 2\leq l\leq \left[\frac n2\right]+3\), and \(\bar A(\nu)\) does not depend on the factor standing beside it.

Analogous estimates for higher derivatives of \(u\) can be obtained in a neighborhood of any point \(Q\in\Lambda\), by constructing the corresponding cylinder \(H_{\rho,-a,a}\), where \(a\) and \(\rho\) depend on the local properties of the surface \(\Lambda\).

§ 2. PROPERTIES AT INFINITY OF THE SOLUTION OF THE VARIATIONAL PROBLEM (1.5)

2.1. We introduce additional conditions on \(A_{ik}\). Suppose that there exists a sufficiently large number \(R\) such that, for \(|x|=r\geq R\), the condition

\[ \left| \frac{\partial^l A_{ik}}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right| <\frac{c}{r^{1/2}} \left(\alpha_1+\cdots+\alpha_n=l;\ 1\leq l\leq \left[\frac n2\right]\right), \tag{2.1} \]

is satisfied, where \(c\) does not depend on \(r>1\).

Denote by \(\Omega_R\) the exterior of the ball \(\omega_R\) of radius \(R\). Write equation (1.22), assuming that the function \(v\) is equal to zero on \(\omega_R\):

\[ \int_{\Omega_R}\sum_{i,k=1}^n A_{ik} \frac{\partial u}{\partial x_i} \frac{\partial v}{\partial x_k}\,dx=0. \tag{2.2} \]

Construct the function \(v=\eta u_h\), where

\[ \eta= \begin{cases} 0, & \text{if } r\leq R,\\[4pt] \left(\dfrac{r-R}{R}\right)^2, & \text{if } R<r\leq 2R,\\[6pt] 1, & \text{if } r>2R. \end{cases} \tag{2.3} \]

Hence, for \(R<r\leq 2R\), we obtain
\[ \frac{\partial \eta}{\partial x_k} = 2\,\frac{r-R}{R}\,\frac{1}{R}\cos\widehat{(x_k,r)} \]
and, consequently,
\[ \left(\frac{\partial \eta}{\partial x_k}\right)^2 < \frac{4}{R^2}\left(\frac{r-R}{R}\right)^2. \]
In view of this, throughout the whole space the inequality

\[ \left(\frac{\partial \eta}{\partial x_k}\right)^2 \leq \frac{4}{R^2}\eta\quad (k=1,\ldots,n) \tag{2.4} \]

holds.

Let us carry out, for equation (2.2), calculations analogous to those which we made in §§ 1.7, 1.8 for equation (1.22). We again obtain equation (1.28), where now \(\eta\) has the meaning defined in (2.3). From the estimates of the integrals \(J_k\) in (1.28), taking (2.4) and (2.1) into account, we have

\[ J_1 \geqslant \beta \int_{\mathring{\Omega}_R} \eta \sum_{i=1}^{n}\left(\frac{\partial u_{h_j}}{\partial x_i}\right)^2 dx = \beta Y_{h_j}^{2}, \]

\[ |J_2| \leqslant \frac{c n^{3/2}}{R^{1/2}} \sqrt{ \int_{\mathring{\Omega}_R} \eta \sum_{i=1}^{n} \left(\frac{\partial u(x+\bar h_j)}{\partial x_i}\right)^2 dx }\,Y_{h_j} \leqslant \frac{c_1}{R^{1/2}}\sqrt{D_{\Omega_{R-h_j}}(u)}\,Y_{h_j}, \]

where \(c\) is the same constant as in (2.1),

\[ |J_3| \leqslant \frac{2Mn^{3/2}}{R} \sqrt{ \int_{\mathring{\Omega}_R} u_{h_j}^{2}\,dx }\,Y_{h_j} \leqslant \frac{c_2}{R}\sqrt{D_{\Omega_{R-h_j}}(u)}\,Y_{h_j}, \]

where \(M\) is the same as in (1.30),

\[ |J_4| \leqslant \frac{2c n^{3/2}}{R^{3/2}} \sqrt{ \int_{\mathring{\Omega}_R} u_{h_j}^{2}\,dx } \times \]

\[ \times \sqrt{ \int_{\mathring{\Omega}_R} \eta \sum_{i=1}^{n} \left(\frac{\partial u(x+\bar h_j)}{\partial x_i}\right)^2 dx } \leqslant \frac{c_3}{R^{3/2}}D_{\Omega_{R-h_j}}(u). \]

It now follows from (1.28) that

\[ Y_{h_j}^{2} - \frac{2c_4}{R^{1/2}}\sqrt{D_{\Omega_{R-h_j}}(u)}\,Y_{h_j} - \frac{c_3}{R^{3/2}}D_{\Omega_{R-h_j}}(u) \leqslant 0 \]

or

\[ \left( Y_{h_j} - \frac{c_4}{R^{1/2}}\sqrt{D_{\Omega_{R-h_j}}(u)} \right)^2 \leqslant \frac{c_4^2}{R}D_{\Omega_{R-h_j}}(u) + \frac{c_3}{R^{3/2}}D_{\Omega_{R-h_j}}(u), \]

so that

\[ Y_{h_j} \leqslant \frac{c_5}{R^{1/2}}\sqrt{D_{\Omega_{R-h_j}}(u)} \qquad (R>1), \]

whence

\[ \int_{\Omega_{2R}} \sum_{i=1}^{n} \left(\frac{\partial u_{h_j}}{\partial x_i}\right)^2 dx < \int_{\mathring{\Omega}_R} \eta \sum_{i=1}^{n} \left(\frac{\partial u_{h_j}}{\partial x_i}\right)^2 dx < Y_{h_j}^{2} < \frac{c_6}{R}D_{\Omega_{R-\delta}}(u), \]

where the constant \(c_6=c_5^2\) does not depend on the nearby multiplier, and \(\delta>|\bar h_j|>0\) is so small that \(\Omega_{R-\delta}\cap G=0\). Finally, replacing \(2R\) by \(R\), we obtain

\[ \int_{\Omega_R} \sum_{i=1}^{n} \left(\frac{\partial u_{h_j}}{\partial x_i}\right)^2 dx < \frac{c_7}{R}D_{\Omega_{\frac{R}{2}-\delta}}(u), \]

where \(c_7\) does not depend on the nearby multiplier. Hence it follows directly (see Lemma 1, [6]) that the generalized derivative

\[ \frac{\partial^2 u}{\partial x_i\,\partial x_j} \]

exists, and the inequality

\[ \int_{\Omega_R} \left( \frac{\partial^2 u}{\partial x_i \partial x_j} \right)^2 dx < \frac{c_7}{R} D_{R_n\setminus G}(u) \quad (i,j=1,\ldots,n), \]

where \(c_7\) does not depend on \(R\) and \(D_{R_n\setminus G}(u)\). Estimating differences of higher order by induction, we obtain, for sufficiently large \(R\),

\[ \int_{\Omega_R} \left( \frac{\partial^l u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right)^2 dx < \frac{c}{R} D_{R_n\setminus G}(u) \tag{2.5} \]

\[ \left( \alpha_1+\cdots+\alpha_n=l;\ 2\le l\le \left[\frac n2\right]+1 \right). \]

Remark. Arguing in an analogous way, in the case of constant \(A_{ik}\), for the same \(R\) and \(l\) we obtain the inequality

\[ \int_{\Omega_R} \left( \frac{\partial^l u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right)^2 dx < \frac{c}{R^{2l-2}}D_{\Omega_{R/2^{\,l-1}}}, \tag{2.6} \]

where \(c\) in (2.5) and (2.6) is a constant independent of \(R\).

2.2. Integrate inequality (1.20) over the spherical layer \((R<\rho<R+2r)\), where \(R\) is such that inequality (2.5) holds:

\[ \int_R^{R+2r} \rho^{n-1}\int_\sigma |u(Q,\rho)|^2\,dQ\,d\rho \le c\int_R^{R+2r} \rho D_{\Omega_\rho}(u)\,d\rho < 2cr(R+r)D_{\Omega_R}(u). \tag{2.7} \]

Let \(\omega_r\) denote the ball of radius \(r\) (\(r>1\)) with center at the origin. Introduce new coordinates:
\(x_1=\xi_1 r,\ldots,x_n=\xi_n r\), where, thus,
\(\xi=(\xi_1,\ldots,\xi_n)\in\omega=\omega_1\), if \(x\in\omega_r\).

Let
\[ u(x_1,\ldots,x_n)=u(\xi_1 r,\ldots,\xi_n r) = u_r(\xi_1,\ldots,\xi_n). \]
For \(l>\dfrac n2\) the embedding theorem holds (see [5]):

\[ \|u_r(\xi_1,\ldots,\xi_n)\|^2_{C(\omega)} < K\left( \int_\omega u_r^2\,d\xi + \sum_{\alpha_1+\cdots+\alpha_n=l} \int_\omega \left( \frac{\partial^l u}{\partial \xi_1^{\alpha_1},\ldots,\partial \xi_n^{\alpha_n}} \right)^2 d\xi \right). \]

In this inequality, applied to \(u_r(\xi_1,\ldots,\xi_n)\), we pass to the variables \((x_1,\ldots,x_n)\):

\[ \|u(x_1,\ldots,x_n)\|^2_{C(\omega_r)} < K\left( \frac{1}{r^n}\int_{\omega_r} u^2\,dx + r^{2l-n} \sum_{\alpha_1+\cdots+\alpha_n=l} \int_\omega \left( \frac{\partial^l u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right)^2 dx \right). \tag{2.8} \]

Obviously, inequality (2.8) remains valid for the ball \(\omega_r\) of radius \(r\) with center at an arbitrary point. We shall assume that it is located at the point \((Q_0,R+r)\), where \(Q_0\) is an arbitrary direction. We replace the integral in the first term on the right-hand side of (2.8), using inequality (2.7). Then we obtain

\[ \|u(x_1,\ldots,x_n)\|^2_{C(\omega_r,Q_0,R+r)} < 2cK\frac{R+r}{r^{n-1}}D_{\Omega_R}(u) + Kr^{2l-n}\times \]

\[ \times \sum_{\alpha_1+\cdots+\alpha_n=l}\int_{(\omega_r,Q_0,R+r)} \left( \frac{\partial^l u}{\partial x_1^{\alpha_1},\ldots,\partial x_n^{\alpha_n}} \right)^2 dx, \tag{2.9} \]

where \((\omega_r,Q_0,R+r)\) is the ball of radius \(r\) with center at the point \((Q_0,R+r)\).

Using inequality (2.5), from (2.9) we obtain

\[ \|u(x_1,\ldots,x_n)\|_{C(\omega_r,Q_0,R+r)}^2 \le 2cK\,\frac{R+r}{r^{\,n-1}}\,D_{\Omega_R}(u) + \frac{c_1Kr^{2l-n}}{R}\,D_{R^n\setminus G}(u). \]

Since the right-hand side of the last inequality does not depend on \(Q_0\), it follows that

\[ \|u(x_1,\ldots,x_n)\|_{C(R\le |x|\le R+2r)}^2 \le 2cK\,\frac{R+r}{r^{\,n-1}}\,D_{\Omega_R}(u) + \frac{c_1Kr^{2l-n}}{R}\,D_{R^n\setminus G}(u). \tag{2.10} \]

If we assume that \(R\to\infty\), \(r\to\infty\) in such a way that

\[ \frac{R+r}{r^{\,n-1}}<\text{const}, \]

and

\[ \frac{r^{2l-n}}{R}\to 0, \]

for example, if we put \(R=ar^{\,n-1}\to\infty\) when \(n-\frac12>l>\frac n2\), then we obtain the lemma.

Lemma 3. The solution of the variational problem (1.5) under condition (2.1) tends to zero as \(x\to\infty\).

In conclusion I should like to express my deep gratitude to my supervisor, Professor S. M. Nikol’skii, for his constant attention and valuable advice. I also express my gratitude to O. V. Besov, who carefully read the work and made a number of valuable comments, and to V. P. Mikhailov for taking part in the discussion of the results.

References

1. S. M. Nikol’skii, Izv. Akad. Nauk SSSR, Ser. Mat., 22, No. 5, 599, 1958.
2. S. M. Nikol’skii, Trudy Mat. Inst. im. V. A. Steklova Akad. Nauk SSSR, 38, 1951, pp. 244—278.
3. S. M. Nikol’skii, Mat. Sb., 33 (75): 2, 1953, pp. 261—326.
4. S. V. Uspenskii, Dokl. Akad. Nauk SSSR, 127, No. 3, 526, 1959.
5. S. L. Sobolev, Some Applications of Functional Analysis to Mathematical Physics. L., 1950.
6. A. S. Fokht, Dokl. Akad. Nauk SSSR, 154, No. 6, 1235—1239, 1964.
7. V. D. Kopchenov, Dopovidi Akad. Nauk Ukr. RSR, No. 2, 153, 1965.
8. V. D. Kopchenov, Collection of Reports of the Scientific-Technical Conference of MLTI, 1965.
9. V. K. Saul’ev, Dokl. Akad. Nauk SSSR, 147, No. 2, 303—305, 1962.
10. M. I. Vishik, L. A. Lyusternik, Uspekhi Mat. Nauk, 15, issue 4 (94), 27—95, 1960.

Received by the editors
February 12, 1966

Moscow Forestry Engineering Institute