s_1\,$, the conclusions of the lemma follow from
Poincar\'e's inequality and Rellich's compactness theorem.
\medskip
Let $\zeta_s$ be an arbitrary sequence in $R^1$ such that
$$
\underset{s\to\infty}\to\lim\,\zeta_s = 0.
\tag{3.17}
$$
Let us define the sets $I^\pri_{1,s}$, $I^\pri_{2,s}$ of
multi-indices by
$$
I^\pri_{1,s}=\big\{\al\in I^\pri_s:\zeta_s |q_\al^{(s)}| ^{m-1}\, \< 1\big\},
\quad
I^\pri_{2,s}=\big\{\al\in I^\pri_s:\zeta_s |q_\al^{(s)}| ^{m-1}\, > 1\big\},
\tag{3.18}
$$
and denote
$$
r_{i,s}^\pri (x) \, = \,\underset{\al\in I_{i,s}^\pri}\to\sum\,\,
v_\al^{(s)} (x, q_\al^{(s)}) \,\,\varphi_\al^{(s)} (x),\quad i=1,2.
\tag{3.19}
$$
\medskip
{\bf Lemma 3.4.} {\it Assume that the conditions of Lemma 3.3 are satisfied
and let $\zeta_s$ be an arbitrary sequence in $R^1$ satisfying (3.17).
Then the sequence $r_{2,s}^\pri (x)$ defined by (3.19) converges strongly
to zero in $W_m^1 (\Om)$.}
\medskip
{\bf Proof.} % Denote by $|I_{2,s}^\pri|$ the number
% of multiindeces in the set $I_{2,s}^\pri$ defined by (3.18) and
Define
$$
Q_s = \bigcup_{\al\in I_{2,s}^\pri} K_s (\al)\,.
\tag{3.20}
$$
{}From (3.14) and from
$\displaystyle
\,\,\zeta_s^{- \frac m{m-1}} \text {\rm meas}\, Q_s\,\<\,C_{12}\,
\underset{\al\in I_{2,s}^\pri}\to\sum\,\,|q_\al^{(s)}|^m\,\,[\lm_s\rho_s]^n
\,\,$
we get
$$
\text{\rm meas}\,Q_s\,\,\<\,\,C_{12}\,\,\zeta_s^{\frac m{m-1}}\,\,
\underset\Om\to\int\,\,|q_s(x)|^m\,\,dx.
\tag{3.21}
$$
As in the proof of inequality (3.16), we obtain
$$
\underset\Om\to\int \bigg|\frac{\pr r^\pri_{2,s}(x)}{\pr x}\bigg|^m\,dx
\,\<\,C_{13}\, \underset{Q_s}\to\int\,\big(|q_s(x)|^m +
1\big)\,\,dx\,,
$$
and the convergence to zero of the right-hand side of the last inequality
follows from (3.17), (3.21), and the assumption on the sequence $q_s(x)$.
The proof of the lemma is complete.
\medskip
{\bf Lemma 3.5.} {\it Assume that the conditions of Lemma 3.3 are satisfied.
Then the sequence
$$
r_s^{\pri\pri} (x) = \underset{\al\in I_s^{\pri\pri}}\to\sum\,\,
v_\al^{(s)} (x, q_\al^{(s)})\, \varphi_\al^{(s)} (x)
\tag{3.22}
$$
converges strongly to zero in $W_m^1 (\Om)$.}
\medskip
The proof follows immediately from the estimate
$$
\underset\Om\to\int \bigg|\frac{\pr r^{\pri\pri}_s (x)}{\pr x}\bigg|^m\,dx
\,\,\<\,\,C_{14}\,\underset{\al\in I_s^{\pri\pri}}\to\sum\,
\big(\mu_s^m + [\lm_s\rho_s]^m \big)\, [\lm_s \rho_s]^n
\,\,\<\,\,C_{14} \big(\mu_s^m + [\lm_s\rho_s]^m \big)\,
\text{\rm meas}\,\Omega\,,
$$
that is obtained as in (3.13), using the definition of the
set $I_s^{\pri\pri}$ in (3.3).
\medskip
{\bf Proof of Theorem 1.1.} Define the sequence $\zeta_s$ by
$$
\zeta_s = \max \big\{||z_s (x)||_{L_m(\Om)},\,\,\lm_s\rho_s\big\}\,,
\tag{3.23}
$$
where $z_s (x)$ is the sequence introduced in the statement of
Theorem 1.1. Then $\zeta_s$ tends to zero as $s\to \infty$.
Let $r_{1,s}^\pri (x)$, $r_{2,s}^\pri (x)$ be the sequences defined by
(3.19) for this choice of $\zeta_s$.
Using condition A.2, Lemmas 3.3--3.5, and the assumptions on $z_s(x)$
we obtain
$$
\underset{s\to\infty}\to\lim\,\underset{j=1}\to{\overset n\to\sum}\,
\underset\Om\to\int\,\bigg[a_j\big(x, \frac{\pr r_s (x)}{\pr x}\big) -
a_j\big(x, \frac{\pr r_{1,s}^\pri (x)}{\pr x}\big) \bigg]\,\,
\frac {\pr z_s (x)}{\pr x_j}\,\,dx = 0\,,
\tag{3.24}
$$
and it is sufficient to study the behaviour of the term
$$
J_s = \underset{j=1}\to{\overset n\to\sum}\,
\underset\Om\to\int\, a_j\big(x, \frac{\pr r_{1,s}^\pri (x)}{\pr x}\big)
\,\,\frac {\pr z_s (x)}{\pr x_j}\,\,dx .
\tag{3.25}
$$
Let $\eta_\al^{(s)} (x)$ be a function of class $C_0^\infty (\Om_0)$,
which is equal to one on $K\bigg(x_\al^{(s)},\,\frac{3 \lm_s\rho_s}2\bigg)$,
to zero outside $K (x_\al^{(s)},\,2 \lm_s\rho_s)$, and such that
$\bigg|\frac{\pr \eta_\al^{(s)} (x)}{\pr x}\bigg| \< \frac 4{\lm_s \rho_s}$.
We rewrite $J_s$ in the form
$$
J_s = \underset{j=1}\to{\overset 3\to\sum}\, J_s^{(i)}\,,
\tag{3.26}
$$
where
$$
\aligned
& J_s^{(1)} = \underset{\al\in I_{1,s}^\pri}\to\sum\,
\underset{j=1}\to{\overset n\to\sum}\,\underset{\tl K_s (\al)}\to\int\,
\bigg[a_j\big(x, \frac \pr{\pr x} (v_\al^{(s)} \varphi_\al^{(s)})\big) -
a_j\big(x, \frac {\pr v_\al^{(s)}}{\pr x}\big)\bigg]\,\,
\frac {\pr z_s (x)}{\pr x_j}\,\,dx ,\\
& J_s^{(2)} = \underset{\al\in I_{1,s}^\pri}\to\sum\,
\underset{j=1}\to{\overset n\to\sum}\,\underset{\tl K_s (\al)}\to\int\,
a_j\big(x, \frac {\pr v_\al^{(s)}}{\pr x}\big)\,\frac \pr {\pr x_j}\,
\big[\eta_\al^{(s)} (x) z_s (x)\big]\,dx, \\
& J_s^{(3)} = \underset{\al\in I_{1,s}^\pri}\to\sum\,
\underset{j=1}\to{\overset n\to\sum}\,\underset{\tl K_s (\al)}\to\int\,
a_j\big(x, \frac {\pr v_\al^{(s)}}{\pr x}\big)\,\,
\frac \pr{\pr x_j} \big[(1 - \eta_\al^{(s)} (x)) z_s (x)\big]\,\,dx;
\endaligned
\tag{3.27}
$$
here $v_\al^{(s)} = v_\al^{(s)} (x, q_\al^{(s)})$ and
$\tl K_s (\al) = K (x_\al^{(s)}, 2 \lm_s\rho_s)$.
Define
$
E_\al^{(s)} (\mu) = \big\{ x\in \tl K_s (\al):|v_\al^{(s)}
(x, q_\al^{(s)})| \< \mu \big\}$.
The function $\varphi_\al^{(s)} (x)$ is equal to one if
$|v_\al^{(s)}(x, q_\al^{(s)})| \> \mu_\al^{(s)}$, $\al\in I_s^\pri$,
and using (1.3) and H\"older's inequality we obtain the estimate
$$
\aligned
& |J_s^{(1)}| \< C_{15}\,\bigg\{\underset{\al\in I^\pri_{1,s}}\to\sum\,\,
\underset{E_\al^{(s)} (\mu_\al^{(s)})}\to\int\,\,
\bigg[1 + \big|\frac \pr{\pr x} (v_\al^{(s)} \varphi_\al^{(s)})\big| +
\big|\frac {\pr v_\al^{(s)}}{\pr x}\big|\bigg]^m\,\,dx
\bigg\}^{\frac{m-2}m} \cdot \\
&\cdot\bigg\{\underset{\al\in I^\pri_{1,s}}\to\sum\,\,
\underset{E_\al^{(s)} (\mu_\al^{(s)})}\to\int\,\,
\big|\frac \pr{\pr x} \big[v_\al^{(s)} (1 - \varphi_\al^{(s)})\big]\big|^m\,
dx \bigg\}^{\frac 1m} \cdot \bigg\{\underset\Om\to\int
\big|\frac {\pr z_s (x)}{\pr x}\big|^m\,\,dx\bigg\}^{\frac 1m}.
\endaligned
\tag{3.28}
$$
The first factor in the right-hand side of the last inequality can be
estimated from above by a constant independent of $s$. This can be
obtained as in the proof of inequality (3.16).
We assume now that $s$ is large enough so that inequality (3.11)
is satisfied. The second factor in the right-hand side of (3.28) is
estimated using inequalities (2.2), (3.14), and
condition B. We obtain
$$
\aligned
&\underset{\al\in I^\pri_{1,s}}\to\sum\,\,
\underset{E_\al^{(s)} (\mu_\al^{(s)})}\to\int\,\,
\big|\frac \pr{\pr x} \big[v_\al^{(s)} (1 - \varphi_\al^{(s)})\big]
\big|^m\,dx \< \\
\< C_{16}& \,\,\mu_s \underset{\al\in I^\pri_{1,s}}\to\sum\,\,
|q_\al^{(s)}|^m \,[\lm_s \rho_s]^n \<
C_{16}\,\, \mu_s \underset\Om\to\int\,\, |q_s (x)|^m \,dx\,,
\endaligned
$$
and the right-hand side of the last inequality tends to zero as $s\to\infty$.
Taking the assumption on $z_s (x)$ into account we obtain
$$
\underset{s\to\infty}\to\lim\,\,J_s^{(1)} = 0.
\tag{3.29}
$$
The equality
$$
J_s^{(2)} = 0
\tag{3.30}
$$
follows from the definition of the functions $v_\al^{(s)} (x, q_\al^{(s)})$
(see (1.11) and (1.12)) and from the properties of $\eta_\al^{(s)} (x)$
and $z_s (x)$.
In order to estimate $J_s^{(3)}$ we remark that the inequality
$$
|v_\al^{(s)} (x, q_\al^{(s)})| \< \ov\mu_\al^{(s)},\quad
\ov\mu_\al^{(s)} = C_{17} [\lm_s \rho_s]^2\,|q_\al^{(s)}|^{m-1}
\tag{3.31}
$$
holds for $\al\in I^\pri_{1,s}$ and
$x\in\tl K_s (\al)\setminus K\big(x_\al^{(s)}, \frac{3 \lm_s\rho_s}2\big)$.
We obtain this estimate using Lemma 2.3 and (3.11), taking into
account that $|q_\al^{(s)}|^{m-1}\cdot \lm_s\rho_s \< 1$ for
$\al\in I_{1,s}^\pri$, which implies that the second condition in (2.4) is satisfied.
By condition A.2 and H\"older's inequality we obtain the estimate for
$J_s^{(3)}$:
$$
|J_s^{(3)}| \< C_{18}\,\frac 1{\lm_s\rho_s}\,\, \bigg\{
\underset{\al\in I^\pri_{1,s}}\to\sum\,\,
\underset{E_\al^{(s)} (\ov\mu_\al^{(s)})}\to\int\,\,
\big|\frac {\pr v_\al^{(s)}}{\pr x}\big|^m\,dx\bigg\}^{\frac{m-1}m} \cdot
\bigg\{[\lm_s\rho_s]^m\,\underset\Om\to\int\,
\big|\frac{\pr z_s}{\pr x}\big|^m\,\,dx +{}
$$
\vskip-10pt
$$
{}+ \underset\Om\to\int |z_s (x)|^m\,dx\bigg\}^{\frac 1m} + C_{18}\,
\frac 1{\lm_s\rho_s}\,\bigg\{\underset{\al\in I^\pri_{1,s}}\to\sum\,\,
\underset{E_\al^{(s)} (\ov\mu_\al^{(s)})}\to\int\,\,
\big|\frac {\pr v_\al^{(s)}}{\pr x}\big|^2\,dx\bigg\}^{\frac 12} \cdot
\tag{3.32}
$$
\vskip-10pt
$$
\cdot \big\{[\lm_s\rho_s]^2\,\underset\Om\to\int\,
\big|\frac{\pr z_s}{\pr x}\big|^2\,dx + \underset\Om\to\int\,
|z_s (x)|^2\,dx\big\}^{\frac 12},
$$
where $v_\al^{(s)} = v_\al^{(s)} (x, q_\al^{(s)})$ and
$\ov\mu_\al^{(s)}$ is defined by (3.31). In the
right-hand side of (3.22) the factors containing $z_s (x)$ can be
estimated from above by $C_{19} \zeta_s$, where $\zeta_s$ is
defined by (3.23).
In order to check the equality
$$
\underset{s\to\infty}\to\lim\,\,J_s^{(3)} = 0,
\tag{3.33}
$$
it is sufficient to establish the estimate
$$
J_s^{(4)} := \underset{\al\in I^\pri_{1,s}}\to\sum\,\,
\underset{E_\al^{(s)} (\ov\mu_\al^{(s)})}\to\int\,\,
\bigg(\big|\frac {\pr v_\al^{(s)}}{\pr x}\big|^2 +
\big|\frac {\pr v_\al^{(s)}}{\pr x}\big|^m\bigg)\,dx \< C_{20}
[\lm_s\rho_s]^2 \cdot \zeta_s^{- \frac{m-2}{m-1}}.
\tag{3.34}
$$
This inequality follows from (2.2), (3.11), (3.14), (3.18), (3.31), and
condition B:
$$
\aligned
J_s^{(4)} & \< C_{21}\, [\lm_s\rho_s]^2 \,\,
\underset{\al\in I^\pri_{1,s}}\to\sum\,\,
|q_\al^{(s)}|^{2m-2}\,\, [\lm_s\rho_s]^n \<\\
& \< C_{22} \,[\lm_s\rho_s]^2 \, \zeta_s^{- \frac{m-2}{m-1}} \cdot
\underset\Om\to\int \,|q_s (x)|^m\,dx.
\endaligned
$$
This proves inequality (3.34) and concludes the proof of the
Convergence Theorem.
\head {4. Construction and properties of test functions}
\endhead
In this section we construct special functions which belong to the
space $\overset\circ\to W^1_m (\Om_s)$ and which will be used later
as test functions in the integral identity corresponding to the boundary
value problem (0.1), (0.2).
As in Section 3 we fix the sequences
$\rho_s$, $\mu_s$, $\lm_s$ introduced in (3.1), and the subdivision of
the domain $\Om$ defined by (1.13). For $s = 1,2,...\,$ and $\al\in
I_s$ we define $I_s (\al)$ as the set of all multi-indices
with integer coordinates such that
$K (2\rho_s\beta, \rho_s)\subset
K_s (\al)\setminus \overset\circ\to K_s^\pri (\al)$,
where $K_s (\al)$, $K_s^\pri (\al)$ are the cubes defined in (3.2) and
$\overset\circ\to K_s^\pri (\al)$ is the interior of the cube
$K_s^\pri (\al)$. For $\beta\in I_s (\al)$ we set
$x_{\al\beta}^{(s)}=2\rho_s\beta$ and
$K_s (\al,\beta) = K (x_{\al\beta}^{(s)}, \rho_s)$. Then we have the
following decomposition:
$$
K_s (\al)\setminus \overset\circ\to K_s^\pri (\al) =
\underset{\beta\in I_s(\al)}\to\bigcup\, K_s (\al,\beta).
\tag{4.1}
$$
Let $|I_s|$, $|I_s(\al)|$ be the numbers of multi-indices belonging to
the sets $I_s$ and $I_s (\al)$ respectively. It is easy to see that
$$
|I_s| \< C(\Om)\,[\lm_s \rho_s]^{-n},\quad
|I_s (\al)| \< 2 n \lm_s^{n-1},
\tag{4.2}
$$
where the constant $C(\Om)$ depends only on the measure of $\Om$.
Let $g(x)$ be an arbitrary function of class $C_0^\infty (\Om)$.
Let us consider the sequence
$$
q_s (x) = f_s (x) - u_0^{(s)} (x) - g (x),
\tag{4.3}
$$
where
$$
\aligned
f_s (x) = \frac 1{[\lm_s\rho_s]^n}\,\,&\underset{R^n}\to\int\,\,
K\bigg(\frac{|x-y|}{\lm_s\rho_s}\bigg)\, f(y)\,dy,\\
u_0^{(s)} (x) = \frac 1{[\lm_s\rho_s]^n}\,\,&\underset{R^n}\to\int\,\,
K\bigg(\frac{|x-y|}{\lm_s\rho_s}\bigg)\, u_0(y)\,dy,
\endaligned
$$
$f(x)$ is the boundary function from (0.2), $u_0 (x)$ is the weak
limit of the sequence $u_s (x)$, solutions of the boundary value
problem (0.1), (0.2), and the kernel $K (t)$ is the same as in (2.13).
We define new cut-off functions $\tilde\varphi_\al^{(s)} (x)$ by
$$
\tilde\varphi_\al^{(s)} (x) = \frac 2{\mu_\al^{(s)}}
\min\bigg\{\bigg[v_\al^{(s)}(x,1) - \frac{\mu_\al^{(s)}}2\bigg]_+,\,\,\,
\frac{\mu_\al^{(s)}}2\bigg\}\,,
\tag{4.4}
$$
where $v_\al^{(s)}(x,1)$ and $\mu_\al^{(s)}$ are the same as in (3.4)
and (3.7).
In accordance with [10], we can define two sequences of
nonnegative functions $\chi_{\al\beta}^{(s)} (x)$,
$\psi_{\al\beta}^{(s)} (x)$, for $\al\in I_s$,
$\beta\in I_s (\al)$, such that the following properties are satisfied:
1) there exists a number $s_2$ such that the inclusions
$$
\text {supp}\,\,\chi_{\al\beta}^{(s)} \sb K\big(x_{\al\beta}^{(s)},\,
\frac {3\rho_s}2\big) \quad\text{for}\quad
\al\in I_s,\,\,\beta\in I_s(\al)
\tag{4.5}
$$
holds for $s \> s_2$ where supp $\chi_{\al\beta}^{(s)}$ is the
support of the function $\chi_{\al\beta}^{(s)} (x)$;
2) for every point $x\in R^n$ in the sequence of numbers
$
\{\chi_{\al\beta}^{(s)} (x) : \al\in I_s,\,\,\beta\in I_s (\al)\}$
no more that $2^n$ numbers are non-zero and there exists a number $K_5$
depending only on $m, n, \nu_1, \nu_2, A$ such that the inequalities
$$
\chi_{\al\beta}^{(s)} (x) \< K_5,\quad \underset{R^n}\to\int\,\,
\bigg|\frac {\pr\chi_{\al\beta}^{(s)} (x)}{\pr x}\bigg|^m\,dx \<
K_5\,\mu_s^{1-m}\cdot \rho_s^n
\tag{4.6}
$$
holds for $s = 1,2,...,\,\,\al\in I_s,\,\,\beta\in I_s (\al)$;
3) the functions $\psi_{\al\beta}^{(s)} (x)$ are defined by the
equality
$$
\psi_{\al\beta}^{(s)} (x) = \chi_{\al\beta}^{(s)} (x)
\big\{1 - \underset{\g\in I_s}\to\sum\,\,\tilde\varphi_\g^{(s)} (x)\big\},
\quad x\in R^n;
\tag{4.7}
$$
4) the following equalities hold:
$$
\underset{\al\in I_s}\to\sum\,\, \underset{\beta\in I_s (\al)}\to\sum\,\,
\chi_{\al\beta}^{(s)} (x) = 1\quad\text{for}\quad
x\in \underset{\al\in I_s}\to\bigcup\,
\underset{\beta\in I_s (\al)}\to\bigcup\,\,
\{K_s (\al,\beta)\setminus \Om_s\}\,,
\tag{4.8}
$$
\vskip-20pt
$$
\underset{\al\in I_s}\to\sum\, \tilde \varphi_\al^{(s)} (x) +
\underset{\al\in I_s}\to\sum\,\, \underset{\beta\in I_s (\al)}\to\sum\,\,
\psi_{\al\beta}^{(s)} (x) = 1\quad\text{for}\quad
x\in \underset{\al\in I_s}\to\bigcup\,\{K_s (\al)\setminus \Om_s\}\,.
\tag{4.9}
$$
We shall assume later that
$$
s \> \max \{s_1, s_2\}.
\tag{4.10}
$$
Remark that from inclusions (3.8) and (4.5) we obtain that for every
$x\in R^n$, $\al, \g\in I_s$, $\beta\in I_s (\al)$ we have
$$
\chi_{\al\beta}^{(s)} (x)\,\, \tilde\varphi_\g^{(s)} (x) = 0\,,
\quad
\chi_{\al\beta}^{(s)} (x)\,\, \varphi_\g^{(s)} (x) = 0
\quad \text{if}\quad
\al\ne\g.
\tag{4.11}
$$
Let us introduce the sequence
$$
h_s (x) = f (x) - q_s (x) + r_s (x) +
\underset{i=1}\to{\overset 3\to\sum} r_s^{(i)} (x),
\tag{4.12}
$$
where
$$
\aligned
r_s^{(1)} (x) &= \underset{\al\in I_s}\to\sum\,\,
[q_s (x) - q_\al^{(s)}]\,\tilde\varphi_\al^{(s)} (x),\\
r_s^{(2)} (x) &= q_s (x) \,\underset{\al\in I_s}\to\sum\,\,
\underset{\beta\in I_s(\al)}\to\sum\,\,
\psi_{\al\beta}^{(s)} (x),\\
r_s^{(3)} (x) &= \underset{\al\in I_s}\to\sum\,\,
\underset{\beta\in I_s(\al)}\to\sum\,\,
[q_\al^{(s)}\,\tilde\varphi_\al^{(s)} (x)
- v_\al^{(s)} (x, q_\al^{(s)})\,\varphi_\al^{(s)} (x)]\,
\chi_{\al\beta}^{(s)} (x),
\endaligned
\tag{4.13}
$$
and the sequences
$r_s (x)$ and $q_s (x)$ are defined by
(1.14) and (4.3). The sequences
$q_\al^{(s)}$ and $\varphi_s^{(s)} (x)$ are the same as in Section 3, with
$q_s (x)$ defined by (4.3).
\medskip
{\bf Lemma 4.1.} {\it Assume that conditions A.1 and A.2 are
satisfied, and let $\ov g (x)$ be an arbitrary function in the space
$C_0^\infty (\Om)$. Then there exists a number $s_3 (\ov g)$,
depending on $\ov g (x)$, such that the inclusion
$$
\ov g (x) [u_s (x) - h_s (x)] \in \overset\circ\to W_m^1 (\Om_s)
\tag{4.14}
$$
holds for $ s \> \max \{s_1, s_2, s_3 (\ov g)\}$.}
\medskip
{\bf Proof.} By the definition of the functions
$v_\al^{(s)} (x, q_\al^{(s)}),\,\,\varphi_\al^{(s)} (x)$, Lemma 3.1,
and inclusion (4.8) we obtain that the function
% $$
% \aligned
% r_s^{(4)} (x)& := \underset{\al\in I_s}\to\sum\,\,
% [v_\al^{(s)} (x, q_\al^{(s)}) - q_\al^{(s)}]\,\varphi_\al^{(s)} (x) \cdot\\
% \cdot \big\{1 - &\,\underset{\g\in I_s}\to\sum\,\,
% \underset{\delta\in I_s(\g)}\to\sum\,\,\chi_{\g\delta}^{(s)} (x)\big\}
% \in \overset\circ\to W_m^1 (\Om_s^\pri),
% \endaligned
% \tag{4.15}
% $$
$$
r_s^{(4)} (x) := \underset{\al\in I_s}\to\sum\,\,
[v_\al^{(s)} (x, q_\al^{(s)})\,\varphi_\al^{(s)} (x)
- q_\al^{(s)}\,\tilde\varphi_\al^{(s)} (x)] \cdot
\big\{1 - \,\underset{\g\in I_s}\to\sum\,\,
\underset{\delta\in I_s(\g)}\to\sum\,\,\chi_{\g\delta}^{(s)} (x)\big\}
\tag{4.15}
$$
belongs to $\overset\circ\to W_m^1 (\Om_s^\pri)$,
where $\Om_s^\pri = \Om \setminus \{\underset{\al\in I_s}\to\bigcup\,\,
[K_s (\al)\setminus \Om_s]\}$.
{}From (4.9) we obtain the inclusion
$$
r_s^{(5)} (x) := q_s (x) \big\{ 1 - \underset{\al\in I_s}\to\sum\,
\tilde\varphi_\al^{(s)} (x) - \underset{\al\in I_s}\to\sum\,
\underset{\beta\in I_s(\al)}\to\sum\,\psi_{\al\beta}^{(s)} (x)\big\}
\in \overset\circ\to W_m^1 (\Om_s^\pri).
\tag{4.16}
$$
Taking (4.11) into account we obtain
$$
u_s (x) - h_s (x) = u_s (x) - f(x) - r_s^{(4)} (x) + r_s^{(5)} (x)
\in \overset\circ\to W_m^1 (\Om_s^\pri).
$$
Inclusion (4.14) follows now from the construction of the
subdivision (1.13) of the domain $\Om$ and from the choice of the
function $\ov g (x)$. The proof of lemma is complete.
\medskip
{\bf Lemma 4.2.} {\it Assume that conditions A.1, A.2, and B are
satisfied. Then the sequences $r_s^{(i)} (x)$, $i= 1,2,3$, defined
by (4.13), converge strongly to zero in the space $W_m^1 (\Om)$ as
$s\to\infty$.}
\medskip
{\bf Proof.} Assume that $s$ is large enough so that inequalities
(3.11) and (4.10) are satisfied. Using (2.2), (3.7), (4.4), and
condition B we have the estimate
$$
\left\Vert\frac{\pr \tilde\varphi_\al^{(s)} (x)}{\pr x}
\right\Vert^m_{L_m(\Om)}\,\<
2^m \,[\mu_\al^{(s)}]^{-m}\,\underset{\overline E_\al^{(s)}}\to\int\,
\left|\frac{\pr v_\al^{(s)} (x,1)}{\pr x}\right|^m\,dx \,\< C_{23}
\,\mu_s^{1-m} \,[\lm_s\rho_s]^n,
\tag{4.17}
$$
where $\overline E_\al^{(s)}=\big\{x\in \Om_0 :
\mu_\al^{(s)}/2 \<
v_\al^{(s)}(x,1) \< \mu_\al^{(s)}\big\}$.
Let us estimate the norm of the gradient of $r_s^{(1)} (x)$ in
$L_m (\Om)$:
$$
\aligned
&\bigg\Vert\frac\pr{\pr x}\,r_s^{(1)} (x)\bigg\Vert^m_{L_m (\Om)} \<
C_{24}\,\underset{\al\in I_s}\to\sum\,\underset{G_\al^{(s)}}\to\int\,
\bigg|\frac{\pr q_s (x)}{\pr x}\bigg|^m\,dx +{} \\
& {}+ C_{24}\,\underset{\al\in I_s}\to\sum\,\underset\Om\to\int\,
|g(x) - g_\al^{(s)}|^m \cdot
\bigg|\frac{\pr\tilde \varphi_\al^{(s)} (x)}{\pr x}\bigg|^m\,dx +{} \\
& {}+ C_{24}\,\underset{\al\in I_s}\to\sum\,\underset\Om\to\int\,
\{|f_s(x) - f_\al^{(s)}|^m + |u_0^{(s)}(x) - u_\al^{(s)}|^m\}
\bigg|\frac{\pr\tilde \varphi_\al^{(s)} (x)}{\pr x}\bigg|^m\,dx,
\endaligned
\tag{4.18}
$$
where $f_\al^{(s)}, u_\al^{(s)}, g_\al^{(s)}$ are the mean values of the
functions $f_s (x), u_0^{(s)} (x), g (x)$ in the cube
$K_s (\al)$. The first term in the right-hand side of (4.18) tends to
zero as $s\to\infty$ by Lemma 3.2, the strong convergence of the
sequence $q_s (x)$ in $W_m^1 (\Om)$, and the absolute continuity of
the integral. Since the function $g(x)$ is smooth,
the second term tends to zero by (4.17) and (3.1).
Using (4.17) and Lemma 2.4, the third term in the
right-hand side of (4.18) can be estimated from above by
$$
C_{25}\,\mu_s^{1-m} \,[\lm_s \rho_s]^m\,\underset\Om\to\int\,
\left[\left|\frac{\pr f(x)}{\pr x}\right|^m +
\left|\frac{\pr u_0(x)}{\pr x}\right|^m\right]\,dx\,,
$$
which vanishes as $s\to\infty$ by (3.1). This completes
the proof of the strong convergence of $r_s^{(1)} (x)$ to zero in
$W_m^1 (\Om)$.
Let $\Cal D_{\al\beta}^{(s)}$ be the support of the function
$\psi_{\al\beta}^{(s)} (x)$. Then from (4.2), (4.5) and (4.7) we have
$$
\underset{\al\in I_s}\to\sum\,\, \underset{\beta\in I_s(\al)}\to\sum\,\,
\text {\rm meas}\,\,\Cal D_{\al\beta}^{(s)} \< C_{26} \frac 1{\lm_s}.
\tag{4.19}
$$
We will use also the estimate
$$
\underset\Om\to\int\,\,\left|\frac{\pr v_\al^{(s)} (x, q)}{\pr x}
\right|^m\,\,[\chi_{\al\beta}^{(s)} (x)]^m\,dx \,\<
C_{27}\,[\mu_s^{1-m} |q|^m + 1]\,\rho_s^n,
\tag{4.20}
$$
which follows as in the proof of inequality (4.37) in [10].
{}From (4.7), (4.11) and from inequalities (4.6), (4.20) we obtain the estimate
$$
\underset\Om\to\int\,\
\left|\frac{\pr\psi_{\al\beta}^{(s)} (x)}{\pr x}\right|^m\,dx \,\<\,C_{28}
\,\mu_s^{1-2m}\,\rho_s^n.
\tag{4.21}
$$
Let us estimate the norm of $r_s^{(2)} (x)$ in $W_m^1 (\Om)$. We rewrite
$r_s^{(2)} (x)$ in the form
$$
r_s^{(2)} (x) = \underset{\al\in I_s}\to\sum\,\,
\underset{\beta\in I_s(\al)}\to\sum\,\,
\left\{ [q_s (x) - \ov q_\al^{(s)}] + \ov q_\al^{(s)} \right\}\,\,
\psi_{\al\beta}^{(s)} (x),
$$
where $\ov q_\al^{(s)}$ is the mean value of the function $q_s (x)$
in the cube $\tl K_s (\al) = K(x_\al^{(s)}, 2\lm_s\rho_s)$.
Using (4.6), (4.7), and (4.21) we obtain the inequality
$$
\aligned
\underset\Om\to\int\,
\left|\frac{\pr r_s^{(2)} (x)}{\pr x}\right|^m\,dx \,&\<\,C_{29}\,
\underset{\al\in I_s}\to\sum\,\,\underset{\beta\in I_s(\al)}\to\sum\,\,
\underset{\Cal D_{\al\beta}^{(s)}}\to\int\,
\left|\frac{\pr q_s (x)}{\pr x}\right|^m\,dx \,+{} \\
{}+ \,C_{29}\,\mu_s^{1-2m} & \underset{\al\in I_s}\to\sum\,\,
\underset{\beta\in I_s(\al)}\to\sum\,\,
\left|\ov q_\al^{(s)}\right|^m \cdot \rho_s^n +{} \\
{}+ C_{29} \,\underset{\al\in I_s}\to\sum\,
\underset{\tl K_s (\al)}\to\int &\,\left|q_s (x) - \ov q_\al^{(s)}\right|^m\,
\underset{\beta\in I_s(\al)}\to\sum\,\,
\left|\frac{\pr \psi_{\al\beta}^{(s)} (x)}{\pr x}\right|^m\,dx .
\endaligned
\tag{4.22}
$$
The first term in the right-hand side of (4.22) tends to zero as
$s\to\infty$ by (4.19) and the strong convergence of the sequence
$q_s (x)$ in $W_m^1 (\Om)$. Using (4.2) and (3.14), the
second term in the right-hand side of (4.22) is estimated from above by
$$
C_{30} \,\lm_s^{-1}\mu_s^{1-2m}\,\underset{\al\in I_s}\to\sum\,
\left|\ov q_\al^{s)}\right|^m\,[\lm_s \rho_s]^n\,\<
C_{31} \lm_s^{-1}\mu_s^{1-2m}\,\underset\Om\to\int\,
\left| q_s (x)\right|^m\,dx,
$$
which tends to zero by the choice of $\mu_s,\lm_s$.
Using Lemma 2.4 and inequalities (4.2), (4.21),
the third term in the right-hand side of (4.22) is estimated from above by
$$
\aligned
C_{32}\, \lm_s^{-1}\,\mu_s^{1-2m} & [\lm_s \rho_s]^m
\bigg\{\underset\Om\to\int\left[\left|\frac{\pr f(x)}{\pr x}\right|^m +
\left|\frac{\pr u_0(x)}{\pr x}\right|^m\right]\,dx +{} \\
{}+ & \underset {x\in\Om}\to\max\,\left|\frac{\pr g(x)}{\pr x}\right|^m \cdot
\,\text{\rm meas}\,\Om\bigg\},
\endaligned
$$
which converges to zero by (3.1). This concludes the proof of the
strong convergence to zero of $r_s^{(2)} (x)$ in $W_m^1 (\Om)$.
The same property for $r_s^{(3)} (x)$
follows from the inequality
$$
\aligned
\underset\Om\to\int\,\left|\frac{\pr r_s^{(3)} (x)}{\pr x}\right|^m\,& dx
\,\<\, C_{33}\,\underset{\al\in I_s}\to\sum\,
\underset{\beta\in I_s (\al)}\to\sum\,
\left\{|q_\al^{(s)}|^m \,\mu_s^{1-2m} + 1\right\}\,\rho_s^n \,\< \\
& \< C_{34} \lm_s^{-1} \underset\Om\to\int\,
\left\{\mu_s^{1-2m}\,|q_s (x)|^m + 1\right\}\,dx\,,
\endaligned
$$
which can be obtained by using (4.2), (4.6), (4.20), and (3.14). This completes the
proof of Lemma 4.2.
\medskip
Let $g(x)$ be the same function as before, and let $g_\al^{(s)}$ be its
mean value in the cube $K_s (\al)$. We introduce the sequence
$$
g_s (x) = g(x) + \rho_s (x) +
\underset{i=1}\to{\overset 3\to\sum}\,\rho_s^{(i)} (x),
\tag{4.23}
$$
where
$$
\aligned
&\rho_s (x) = - \underset{\al\in I_s}\to\sum\,\,\frac 1{\tl q_\al^{(s)}}
\,v_\al^{(s)} (x, \tl q_\al^{(s)})\,\varphi_\al^{(s)} (x) \,g_\al^{(s)},\\
&\rho_s^{(1)}(x) = \underset{\al\in I_s}\to\sum\,\,[g_\al^{(s)} - g(x)]\,
\tilde \varphi_\al^{(s)} (x),\\
&\rho_s^{(2)}(x) = - g(x)\underset{\al\in I_s}\to\sum\,
\underset{\beta\in I_s}\to\sum\,\psi_{\al\beta}^{(s)}(x),\\
&\rho_s^{(3)}(x) = - \underset{\al\in I_s}\to\sum\,
\underset{\beta\in I_s(\al)}\to\sum\,
\left[\tilde\varphi_\al^{(s)} (x) - \frac 1{\tl q_\al^{(s)}}
v_\al^{(s)} (x, \tl q_\al^{(s)})\, \varphi_\al^{(s)} (x) \right]\,
g_\al^{(s)}\,\chi_{\al\beta}^{(s)}(x).
\endaligned
\tag{4.24}
$$
Here $\varphi_\al^{(s)}(x)$, $\tilde\varphi_\al^{(s)} (x)$,
$\psi_{\al\beta}^{(s)}(x)$,
$\chi_{\al\beta}^{(s)}(x)$ are the same functions as in (4.13),
$$
\tl q_\al^{(s)} = q_\al^{(s)}\quad \text{for}\quad
\al\in I_s^\pri,\quad
\tl q_\al^{(s)} = 2\mu_s \quad \text{for}\quad
\al\in I_s^{\pri\pri},
$$
and $q_\al^{(s)}$ is the mean value in the cube
$K_s (\al)$ of the function $q_s (x)$ defined by (4.3).
\medskip
{\bf Lemma 4.3.} {\it Assume that conditions of Lemma 4.1 are satisfied.
Then there exists a number $s_4 (\ov g)$ depending on $\ov g (x)$
such that
$$
\ov g (x)\,g_s (x) \in \overset\circ\to W_m^1 (\Om_s)
\tag{4.25}
$$
for $s \> s_4 (\ov g)$.}
\medskip
The proof is analogous with the proof of Lemma 4.1.
\medskip
{\bf Lemma 4.4.} {\it Assume that conditions A.1, A.2, and B are satisfied.
Then the sequence $\rho_s (x)$ is bounded in $W_m^1 (\Om)$ and
converges to zero strongly in $W_p^1 (\Om)\,\,$ for $\,\,p