Let *G* be an open simply connected staircase polygon, let ${\gamma}_{j}$, $j=1,2,\dots ,N$, be its sides, including the ends, and let ${\alpha}_{j}\pi $, ${\alpha}_{j}\in \{\frac{1}{2},1,\frac{3}{2},2\}$, be the interior angle formed by the sides ${\gamma}_{j-1}$ and ${\gamma}_{j}$ (${\gamma}_{0}={\gamma}_{N}$). Furthermore, let *s* be the arc length measured along the boundary of *G* in the positive direction and ${s}_{j}$ be the value of *s* at the vertex ${A}_{j}={\gamma}_{j-1}\cap {\gamma}_{j}$, $({r}_{j},{\theta}_{j})$ be a polar system of coordinates with pole in ${A}_{j}$ and the angle ${\theta}_{j}$ taken counterclockwise from the side ${\gamma}_{j}$.

We consider the boundary value problem

$\mathrm{\Delta}u=0\phantom{\rule{1em}{0ex}}\text{on}G,\phantom{\rule{2em}{0ex}}u={\phi}_{j}\phantom{\rule{1em}{0ex}}\text{on}{\gamma}_{j},j=1,2,\dots ,N,$

(4.1)

where

${\phi}_{j}$ are given functions, and

${\phi}_{j}\in {C}^{6,\lambda}({\gamma}_{j}),\phantom{\rule{1em}{0ex}}0<\lambda <1,1\le j\le N.$

(4.2)

Moreover, at the vertices

${A}_{j}$ for

${\alpha}_{j}=\frac{1}{2}$ the conjugation conditions

${\phi}_{j}^{(2q)}({s}_{j})={(-1)}^{q}{\phi}_{j-1}^{(2q)}({s}_{j}),\phantom{\rule{1em}{0ex}}q=0,1,2,3,$

(4.3)

are satisfied. At the vertices ${A}_{j}$ for ${\alpha}_{j}\ne \frac{1}{2}$ no compatibility conditions for boundary functions are required; in particular the values of ${\phi}_{j-1}$ and ${\phi}_{j}$ at these vertices might be different. Additionally, it is required that when ${\alpha}_{j}\ne 1/2$, the boundary functions on ${\gamma}_{j-1}$ and ${\gamma}_{j}$ are given as algebraic polynomials of arclength *s* measured along *γ*.

Let

$E=\{j:{\alpha}_{j}\ne 1/2,j=1,2,\dots ,N\}$. We call the vertices

${A}_{j}$,

$j\in E$, the singular vertices of the polygon

*G*. We construct two fixed block sectors in the neighborhood of

${A}_{j}$,

$j\in E$, denoted by

${T}_{j}^{i}={T}_{j}({r}_{ji})\subset G$,

$i=1,2$, where

$0<{r}_{j2}<{r}_{j1}<min\{{s}_{j+1}-{s}_{j},{s}_{j}-{s}_{j-1}\}$,

${T}_{j}(r)=\{({r}_{j},{\theta}_{j}):0<{r}_{j}<r,0<{\theta}_{j}<{\alpha}_{j}\pi \}$. On the closed sector

${\overline{T}}_{j}^{1}$,

$j\in E$, we consider the function

${Q}_{j}({r}_{j},{\theta}_{j})$, which has the following properties:

- (i)
${Q}_{j}({r}_{j},{\theta}_{j})$ is harmonic and bounded on the open sector ${T}_{j}^{1}$;

- (ii)
continuous everywhere on ${\overline{T}}_{j}^{1}$ apart from the point ${A}_{j}$, $j\in E$ when ${\phi}_{j-1}\ne {\phi}_{j}$;

- (iii)
continuously differentiable on ${\overline{T}}_{j}^{1}\mathrm{\setminus}{A}_{j}$;

- (iv)
satisfies the given boundary conditions on ${\gamma}_{j-1}\cap {\overline{T}}_{j}^{1}$ and ${\gamma}_{j}\cap {\overline{T}}_{j}^{1}$, $j\in E$.

The function ${Q}_{j}({r}_{j},{\theta}_{j})$ with properties (i)-(iv) is given in [17].

Let

${R}_{j}({r}_{j},{\theta}_{j},\eta )=\frac{1}{{\alpha}_{j}}\sum _{k=0}^{1}{(-1)}^{k}R({\left(\frac{r}{{r}_{j2}}\right)}^{1/{\alpha}_{j}},\frac{\theta}{{\alpha}_{j}},{(-1)}^{k}\frac{\eta}{{\alpha}_{j}}),\phantom{\rule{1em}{0ex}}j\in E,$

(4.4)

where

$R(r,\theta ,\eta )=\frac{1-{r}^{2}}{2\pi (1-2rcos(\theta -\eta )+{r}^{2})}$

(4.5)

is the kernel of the Poisson integral for a unit circle.

The approximation of the integral representation given in the following lemma is used to construct an approximate solution of problem (4.1) around the singular vertices ${A}_{j}$, $j\in E$.

**Lemma 4.1** *The solution* *u* *of problem* (2.1), (2.2)

*can be represented on* ${\overline{T}}_{j}^{2}\mathrm{\setminus}{V}_{j}$,

$j\in E$,

*in the form* $u({r}_{j},{\theta}_{j})={Q}_{j}({r}_{j},{\theta}_{j})+{\int}_{0}^{{\alpha}_{j}\pi}(u({r}_{j2},\eta )-{Q}_{j}({r}_{j2},\eta )){R}_{j}({r}_{j},{\theta}_{j},\eta )\phantom{\rule{0.2em}{0ex}}d\eta ,$

(4.6)

*where* ${V}_{j}$ *is the curvilinear part of the boundary of the sector* ${T}_{j}^{2}$.

*Proof* The proof follows from Theorems 3.1 and 5.1 in [17]. □

We define the approximate solution in the whole polygon *G* by applying a version of the Block-Grid method introduced in [8] (see also [9]).

Let us consider, in addition to the sectors

${T}_{j}^{1}$,

${T}_{j}^{2}$, the sectors

${T}_{j}^{3}$ and

${T}_{j}^{4}$, which are also in the neighborhood of each vertex

${A}_{j}$,

$j\in E$, of the polygon

*G*, with

$0<{r}_{j4}<{r}_{j3}<{r}_{j2}$,

${r}_{j3}=({r}_{j2}+{r}_{j4})/2$ and

${T}_{k}^{3}\cap {T}_{l}^{3}=\mathrm{\varnothing}$,

$k\ne l$, where

$k,l\in E$. Furthermore, let

${G}_{T}=G\mathrm{\setminus}({\bigcup}_{j\in E}{T}_{j}^{4})$. We give the description of the Block-Grid method on a hexagonal grid:

- (i)
All singular corners ${A}_{j}$, $j\in E$, are separated by the double sectors ${T}_{j}^{i}={T}_{j}({r}_{ji})$, $i=2,3$, with ${r}_{j3}<{r}_{j2}$, ${T}_{k}^{2}\cap {T}_{l}^{3}=\mathrm{\varnothing}$, $k\ne l$ and $k,l\in E$. The polygon is covered by overlapping rectangles ${\mathrm{\Pi}}_{k}$, $k=1,2,\dots ,M$, and sectors ${T}_{j}^{3}$, $j\in E$, such that the distance from ${\mathrm{\Pi}}_{k}$ to a singular point ${A}_{j}$ is greater than ${r}_{j4}$ for all $k=1,2,\dots ,M$ and $j\in E$.

- (ii)
On each rectangle ${\mathrm{\Pi}}_{k}$, the seven point difference scheme for the approximation of Laplace’s equation on a hexagonal grid is used, with step size ${h}_{k}\le h$, *h* is a parameter, and as an approximate solution on ${\overline{T}}_{j}^{3}$, $j\in E$, the harmonic function (4.6) is used.

- (iii)
We use the matching operator ${S}^{4}$ constructed in Section 3 to connect the subsystems.

For obtaining the numerical solution of the algebraic system of equations (2.1), (2.2), we outline the procedure: Let ${\mathrm{\Pi}}_{k}\subset {G}_{T}$, $k=1,2,\dots ,M$, be certain fixed open rectangles with sides ${a}_{1k}$ and ${a}_{2k}$ parallel to the sides of *G*, and $G\subset ({\bigcup}_{k=1}^{M}{\mathrm{\Pi}}_{k})\cup ({\bigcup}_{j\in E}{T}_{j}^{3})\subset G$. We use ${\eta}_{k}$ to denote the boundary of the rectangle ${\mathrm{\Pi}}_{k}$, ${V}_{j}$ is the curvilinear part of the boundary of the sector ${T}_{j}^{2}$ and ${t}_{j}=({\bigcup}_{k=1}^{M}{\eta}_{k})\cap {\overline{T}}_{j}^{3}$.

The overlapping condition is imposed on the arrangement of the rectangles ${\mathrm{\Pi}}_{k}$, $k=1,2,\dots ,M$: any point *P* lying on ${\eta}_{k}\cap {G}_{T}$, $1\le k\le M$, or located on ${V}_{j}\cap G$, $j\in E$, falls inside at least one of the rectangles ${\mathrm{\Pi}}_{k(P),}1\le k(P)\le M$, where the distance from *P* to ${G}_{T}\cap {\eta}_{k(P)}$ is not less than some constant ${\varkappa}_{0}$ independent of *P*. The quantity ${\varkappa}_{0}$ is called the gluing depth of the rectangles ${\mathrm{\Pi}}_{k}$, $k=1,2,\dots ,M$.

We introduce the parameter

$h\in (0,{\varkappa}_{0}/4]$ and consider a hexagonal grid on

${\mathrm{\Pi}}_{k}$,

$k=1,2,\dots ,M$, with maximal possible step

${h}_{k}\le min\{h,min\{{a}_{1k},{a}_{2k}\}/4\}$. Let

${\mathrm{\Pi}}_{k}^{h}$ be the set of nodes on

${\mathrm{\Pi}}_{k}$, let

${\eta}_{k}^{h}$ be the set of nodes on

${\eta}_{k}$, and let

${\overline{\mathrm{\Pi}}}_{k}^{h}={\mathrm{\Pi}}_{k}^{h}\cap {\eta}_{k}^{h}$. We denote the set of nodes on the closure of

${\eta}_{k}\cap {G}_{T}$ by

${\eta}_{k0}^{h}$, and the set of nodes on

${\mathrm{\Pi}}_{k}^{h}$ whose distance from the boundary

${\eta}_{k}\cap {G}_{T}$ of

${\mathrm{\Pi}}_{k}$ is

$\frac{h}{2}$ by

${\eta}_{k0}^{\ast h}$. We also have

${\mathrm{\Pi}}_{k}^{\ast h}$ denoting the set of nodes whose distance from the boundary

${\eta}_{k1}$ of

${\mathrm{\Pi}}_{k}$ is

$\frac{h}{2}$ and

${\mathrm{\Pi}}_{k}^{0h}={\mathrm{\Pi}}_{k}^{h}\mathrm{\setminus}({\mathrm{\Pi}}_{k}^{\ast h}\cup {\eta}_{k0}^{\ast h})$. Let

${t}_{j}^{h}$ be the set of nodes on

${t}_{j}$, and let

${\eta}_{k1}^{h}$ be the set of remaining nodes on

${\eta}_{k}$. We also specify a natural number

$n\ge [{ln}^{1+\varkappa}{h}^{-1}]+1$, where

$\varkappa >0$ is a fixed number and the quantities

$n(j)=max\{4,[{\alpha}_{j}n]\}$,

${\beta}_{j}={\alpha}_{j}\pi /n(j)$, and

${\theta}_{j}^{m}=(m-1/2){\beta}_{j}$,

$j\in E$,

$1\le m\le n(j)$. On the arc

${V}_{j}$ we choose the points

$({r}_{j2},{\theta}_{j}^{m})$,

$1\le m\le n(j)$, and denote the set of these points by

${V}_{j}^{n}$. Finally, let

${\omega}^{h,n}=\left(\bigcup _{k=1}^{M}{\eta}_{k0}^{h}\right)\cup \left(\bigcup _{k=1}^{M}{\eta}_{k0}^{\ast h}\right)\cup \left(\bigcup _{j\in E}{V}_{j}^{n}\right),\phantom{\rule{2em}{0ex}}{\overline{G}}_{\ast}^{h,n}={\omega}^{h,n}\cup \left(\bigcup _{k=1}^{M}{\overline{\mathrm{\Pi}}}_{k}^{h}\right).$

Consider the system of equations

${u}_{h}=S{u}_{h}\phantom{\rule{1em}{0ex}}\text{on}{\mathrm{\Pi}}_{k}^{0h},$

(4.7)

${u}_{h}={S}_{m}^{\ast}{u}_{h}+{E}_{mh}^{\ast}({\phi}_{m})\phantom{\rule{1em}{0ex}}\text{on}{\mathrm{\Pi}}_{k}^{\ast h},{\eta}_{k1}^{h}\cap {\gamma}_{m}\ne \mathrm{\varnothing},$

(4.8)

${u}_{h}={\phi}_{m}\phantom{\rule{1em}{0ex}}\text{on}{\eta}_{k1}^{h}\cap {\gamma}_{m},$

(4.9)

$\begin{array}{c}{u}_{h}({r}_{j},{\theta}_{j})={Q}_{j}({r}_{j},{\theta}_{j})\hfill \\ \phantom{{u}_{h}({r}_{j},{\theta}_{j})=}+{\beta}_{j}\sum _{k=1}^{n(j)}{R}_{j}({r}_{j},{\theta}_{j},{\theta}_{j}^{k})({u}_{h}({r}_{j2},{\theta}_{j}^{k})-{Q}_{j}({r}_{j2},{\theta}_{j}^{k}))\phantom{\rule{1em}{0ex}}\text{on}{t}_{j}^{h},\hfill \end{array}$

(4.10)

${u}_{h}={S}^{4}({u}_{h},\phi )\phantom{\rule{1em}{0ex}}\text{on}{\omega}^{h,n},$

(4.11)

where $1\le k,m\le M$, $j\in E$, $\phi ={\{{\phi}_{j}\}}_{j=1}^{N}$; $S{u}_{h}$, ${S}_{m}^{\ast}{u}_{h}$ and ${E}_{mh}^{\ast}({\phi}_{m})$ are defined as equations (2.9), (2.10), and (2.11) in Section 2, respectively.

The solution of the system of equations (4.7)-(4.11) is a numerical solution of problem (2.1), (2.2) on ${\overline{G}}_{T}$ (‘nonsingular’ part of the polygon *G*).

**Theorem 4.2** *There is a natural number* ${n}_{0}$ *such that for all* $n\ge {n}_{0}$ *and* $h\in (0,\frac{{\varkappa}_{0}}{4}]$, *where* ${\varkappa}_{0}$ *is the gluing depth*, *the system of equations* (4.7)-(4.11) *has a unique solution*.

*Proof* Let

${v}_{h}$ be a solution of the system of equations

$\begin{array}{c}{u}_{h}=S{u}_{h}\phantom{\rule{1em}{0ex}}\text{on}{\mathrm{\Pi}}_{k}^{0h},\hfill \\ {u}_{h}={S}_{m}^{\ast}{u}_{h}\phantom{\rule{1em}{0ex}}\text{on}{\mathrm{\Pi}}_{k}^{\ast h},{\eta}_{k1}^{h}\cap {\gamma}_{m}\ne \mathrm{\varnothing},\hfill \\ {u}_{h}=0\phantom{\rule{1em}{0ex}}\text{on}{\eta}_{k1}^{h}\cap {\gamma}_{m},\hfill \end{array}$

(4.12)

$\begin{array}{c}{u}_{h}({r}_{j},{\theta}_{j})={\beta}_{j}\sum _{k=1}^{n(j)}{R}_{j}({r}_{j},{\theta}_{j},{\theta}_{j}^{k}){u}_{h}({r}_{j2},{\theta}_{j}^{k})\phantom{\rule{1em}{0ex}}\text{on}{t}_{j}^{h},\hfill \\ {u}_{h}={S}^{4}{u}_{h}\phantom{\rule{1em}{0ex}}\text{on}{\omega}^{h,n},\hfill \end{array}$

(4.13)

where

$1\le k,m\le M$,

$j\in E$. To prove the given theorem, we show that

${max}_{{\overline{G}}_{\ast}^{h,n}}|{v}_{h}|=0$. On the basis of the structure of operators

*S* and

${S}_{j}^{\ast}$, and the forms (3.5), (3.6), (3.7), (3.8), (3.9), and (3.10)-(3.11) of the matching operator

${S}^{4}$ and by the maximum principle (see Ch. 4, [

11]) it follows that the nonzero maximum value of the function

${v}_{h}$ can be at the points on

${\bigcup}_{j\in E}{t}_{j}^{h}$. From estimation (2.29) in [

18] the existence of the positive constants

${n}_{0}$ and

$\sigma >0$ such that for

$n\ge {n}_{0}$ $\underset{({r}_{j},{\theta}_{j})\in {\overline{{T}_{j}}}^{3}}{max}{\beta}_{j}\sum _{q=1}^{n(j)}{R}_{j}({r}_{j},{\theta}_{j},{\theta}_{j}^{q})\le \sigma <1$

(4.14)

follows. However, taking (4.14) into account in (4.13) we have that the nonzero maximum value can not be at the points on ${\bigcup}_{j\in E}{t}_{j}^{h}$ either. Since the set ${\overline{G}}_{\ast}^{h,n}$ is connected, from equation (4.12) it follows that ${max}_{{\overline{G}}_{\ast}^{h,n}}|{v}_{h}|=0$. □

Let

${u}_{h}$ be the solution of the system of equations (4.7)-(4.11). The function

${U}_{h}({r}_{j},{\theta}_{j})={Q}_{j}({r}_{j},{\theta}_{j})+{\beta}_{j}\sum _{q=1}^{n(j)}{R}_{j}({r}_{j},{\theta}_{j},{\theta}_{j}^{q})({u}_{h}({r}_{j2},{\theta}_{j}^{q})-{Q}_{j}({r}_{j2},{\theta}_{j}^{q}))$

(4.15)

is the approximation of the integral representation (4.6) with the use of the composite mid-point rule. We use the function ${U}_{h}({r}_{j},{\theta}_{j})$ as an approximate solution of problem (2.1), (2.2) on the closed block ${\overline{T}}_{j}^{3}$, $j\in E$ (‘singular’ parts of the polygon *G*).

Let

${\u03f5}_{h}={u}_{h}-u,$

(4.16)

where

${u}_{h}$ is the solution of system (4.7)-(4.11) and

*u* is the trace of the solution of (2.1), (2.2) on

${\overline{G}}_{\ast}^{h,n}$. On the basis of (2.1), (2.2), (4.7)-(4.11), and (4.16),

${\u03f5}_{h}$ satisfies the following difference equations:

${\u03f5}_{h}=S{\u03f5}_{h}+{r}_{h}^{1}\phantom{\rule{1em}{0ex}}\text{on}{\mathrm{\Pi}}_{k}^{0h},$

(4.17)

${\u03f5}_{h}={S}_{m}^{\ast}{\u03f5}_{h}+{r}_{h}^{2}\phantom{\rule{1em}{0ex}}\text{on}{\mathrm{\Pi}}_{k}^{\ast h},{\eta}_{k1}^{h}\cap {\gamma}_{m}\ne \mathrm{\varnothing},\phantom{\rule{2em}{0ex}}{\u03f5}_{h}=0\phantom{\rule{1em}{0ex}}\text{on}{\eta}_{k1}^{h}\cap {\gamma}_{m},$

(4.18)

${\u03f5}_{h}({r}_{j},{\theta}_{j})={\beta}_{j}\sum _{k=1}^{n(j)}{R}_{j}({r}_{j},{\theta}_{j},{\theta}_{j}^{k}){\u03f5}_{h}({r}_{j2},{\theta}_{j}^{k})+{r}_{jh}^{3},\phantom{\rule{1em}{0ex}}({r}_{j},{\theta}_{j})\in {t}_{kj}^{h},$

(4.19)

${\u03f5}_{h}={S}^{4}{\u03f5}_{h}+{r}_{h}^{4}\phantom{\rule{1em}{0ex}}\text{on}{\omega}^{h,n},$

(4.20)

where

$1\le k,m\le M$,

$j\in E$ and

${r}_{h}^{1}=Su-u\phantom{\rule{1em}{0ex}}\text{on}\bigcup _{k=1}^{M}{\mathrm{\Pi}}_{k}^{0h},\phantom{\rule{2em}{0ex}}{r}_{h}^{2}={S}_{m}^{\ast}u+{E}_{mh}^{\ast}({\phi}_{m})-u\phantom{\rule{1em}{0ex}}\text{on}\bigcup _{1\le k\le M}{\mathrm{\Pi}}_{k}^{\ast h},$

(4.21)

$\begin{array}{c}\begin{array}{rl}{r}_{jh}^{3}=& {\beta}_{j}\sum _{k=1}^{n(j)}{R}_{j}({r}_{j},{\theta}_{j},{\theta}_{j}^{k})(u({r}_{j2},{\theta}_{j}^{k})-{Q}_{j}({r}_{j2},{\theta}_{j}^{k}))\\ -(u({r}_{j},{\theta}_{j})-{Q}_{j}({r}_{j},{\theta}_{j}))\phantom{\rule{1em}{0ex}}\text{on}\bigcup _{j\in E}{t}_{j}^{h},\end{array}\hfill \\ {r}_{h}^{4}={S}^{4}(u,\phi )-u\phantom{\rule{1em}{0ex}}\text{on}{\omega}^{h,n}.\hfill \end{array}$

(4.22)

Since all the rectangles

${\mathrm{\Pi}}_{k}$,

$k=1,2,\dots ,M$ are located away from the singular vertices

${A}_{j}$,

$j\in E$ of the polygon

*G*, at a distance greater than

${r}_{j4}>0$ independent of

*h*, by virtue of the conditions (2.3) and (2.4), up to sixth order derivatives of the solution of problem (2.1), (2.2) are bounded on

${\bigcup}_{k=1}^{M}{\mathrm{\Pi}}_{k}$. Then, by Taylor’s formula, from (4.21) we obtain

$\underset{{\bigcup}_{k=1}^{M}{\mathrm{\Pi}}_{k}^{0h}}{max}\left|{r}_{h}^{1}\right|\le {c}_{1}{h}^{6},\phantom{\rule{2em}{0ex}}\underset{{\bigcup}_{k=1}^{M}{\mathrm{\Pi}}_{k}^{\ast h}}{max}\left|{r}_{h}^{2}\right|\le {c}_{2}{h}^{4}.$

(4.23)

Furthermore, as

${\omega}^{h,n}\subset {\bigcup}_{k=1}^{M}{\mathrm{\Pi}}_{k}$ from (4.22) and Theorem 3.3, we have

$\underset{{\omega}^{h,n}}{max}\left|{r}_{h}^{4}\right|\le {c}_{3}{h}^{4}.$

(4.24)

By analogy to the proof of Lemma 6.2 in [

9], it is shown that there exists a natural number

${n}_{0}$, such that for all

$n\ge max\{{n}_{0},[{ln}^{1+\varkappa}{h}^{-1}]+1\}$,

$\varkappa >0$ being a fixed number,

$\underset{j\in E}{max}\left|{r}_{jh}^{3}\right|\le {c}_{4}{h}^{4}.$

(4.25)

**Theorem 4.3** *There exists a natural number* ${n}_{0}$ *such that for all* $n\ge max\{{n}_{0},[{ln}^{1+\varkappa}{h}^{-1}]\}$,

$\varkappa >0$ *being a fixed number*,

$\underset{{\overline{G}}_{\ast}^{h,n}}{max}|{u}_{h}-u|\le c{h}^{4}.$

(4.26)

*Proof* The proof follows from estimations (4.23)-(4.25) and the principle of maximum by analogy to the proof of Theorem 6.1 in [9]. □

**Theorem 4.4** *Let* ${u}_{h}$ *be the solution of the system of equations* (4.7)-(4.11),

*and let an approximate solution of problem* (2.1), (2.2)

*be found on blocks* ${\overline{T}}_{j}^{3}$,

$j\in E$,

*by* (4.15).

*There is a natural number* ${n}_{0}$ *such that for all* $n\ge max\{{n}_{0},[{ln}^{1+\varkappa}{h}^{-1}]\}$,

$\varkappa >0$ *being a fixed number*,

*the following estimations hold*:

$|{U}_{h}({r}_{j},{\theta}_{j})-u({r}_{j},{\theta}_{j})|\le {c}_{0}{h}^{4}\phantom{\rule{1em}{0ex}}\mathit{\text{on}}{\overline{T}}_{j}^{3},j\in E,$

(4.27)

$\left|\frac{{\partial}^{p}}{\partial {x}^{p-q}\phantom{\rule{0.2em}{0ex}}\partial {y}^{q}}({U}_{h}({r}_{j},{\theta}_{j})-u({r}_{j},{\theta}_{j}))\right|\le {c}_{p}{h}^{4}/{r}_{j}^{p-1/{\alpha}_{j}}\phantom{\rule{1em}{0ex}}\mathit{\text{on}}{\overline{T}}_{j}^{3}\mathrm{\setminus}{A}_{j},j\in E,$

(4.28)

*where* $0\le q\le p$, $p=0,1,\dots $ .

*Proof* Estimation (4.27) is obtained from the integral representation (4.6) and formula (4.15) by using estimations (4.25) and (4.26). Estimation (4.28) for $p=0,1,\dots $ is obtained by using inequality (4.27) and Lemma 6.12 in [17]. □