On Estimates of Functions in Norms of Weighted Spaces in the Neighborhoods of Singularity Points

Abstract

A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence and uniqueness of the $R_{\nu }$-generalized solution, its coercive and differential properties of biharmonic boundary value problems with a corner singularity. This paper establishes estimates of a function in the neighborhood of a singularity point in the norms of weighted Lebesgue spaces through its norms in weighted Sobolev spaces over the entire domain, with a minimum weight exponent. In addition, we obtain an estimate of the function norm in a boundary strip for the degeneration of a function on the entire boundary of the domain. These estimates will be useful not only for studying differential problems with singularity, but also in estimating the convergence rate of an approximate solution to an exact one in the weighted finite element method.

1. Introduction

The singularity of a solution to a boundary value problem for partial differential equations can be caused by features of the initial data, the presence of re-entrant corners on the boundary of the domain, or a change in the type of boundary conditions. In this case, the solution to the problem for second-order equations does not belong to the Sobolev space $W_2^2$, and for fourth-order equations, it does not belong to the space $W_2^3$. The lack of the necessary smoothness of the solution to a differential problem leads to a loss of accuracy in finding an approximate solution using difference schemes or the classical finite element method (FEM). This follows from the principle of consistent estimates (for more details, see [1,2,3]).

We have proposed to define an $R_{\nu }$-generalized solution in weighted Sobolev spaces for boundary value problems with singularity [4]. This solution coincides with a weak solution if a weak solution exists. A weighted FEM [5] and a weighted vector FEM [6] were created to find an approximate $R_{\nu }$-generalized solution. The presence of a weight function in the definition of a solution suppresses the influence of a singularity and allows one to find an approximate solution to a boundary value problem with a singularity without loss of accuracy. For example, an approximate solution using the weighted FEM with a given accuracy of ${10}^{-3}$ is calculated millions of times faster than using the classical FEM.

For problems with corner singularity, FEMs are well known for finding an approximate solution with a convergence rate equal to the convergence rate of an FEM for problems with a regular solution. These methods include extended FEM (XFEM) [7,8,9,10,11], field-enriched FEM [12,13], and smoothed FEM [14,15,16,17,18]. In addition, meshless/meshfree methods have been developed to find an approximate solution to problems with a corner singularity [19,20,21,22,23]. The weighted FEM was developed and investigated for elasticity problems with corner singularity in [24,25,26,27] and hydrodynamics problems in [28,29,30]. Preliminarily, the existence, uniqueness, coercivity and differential properties of the $R_{\nu }$-generalized solution were investigated for these problems [31,32,33,34].

The study of differential properties of the $R_{\nu }$-generalized solution is carried out in weighted Sobolev spaces. The peculiarity of functions from these spaces lies in their behavior in the neighborhoods of singularity points. In this paper, we establish estimates of functions in norms of weighted spaces in neighborhoods of singularity points. The root cause of these singularities is either individual points on the boundary (Section 2) or the entire boundary of the domain (Section 3).

2. Theorems on Estimating the Norm of a Function in the Neighborhood of a Singularity Point

Let $\Omega$ be a two-dimensional convex bounded domain with piecewise smooth boundary $\partial \Omega$, and let $\overline{\Omega }$ – the closure of $\Omega$, i.e., $\overline{\Omega }=\Omega \cup \partial \Omega$. By $\partial {\Omega }^{\circ }={\cup }_{i=1}^n\partial {\Omega }^{\left(i\right)}$, we denote the set of points of the boundary, including the points of intersection of its smooth pieces. The diameter of the circle inscribed in $\Omega$ will be denoted by $l_{\Omega }$.

Let $\rho \left(x\right)$ be a weight function that is infinitely differentiable and positive everywhere, except for the set of points $\partial {\Omega }^{\circ }$, coinciding in some neighborhood ${\Omega }_i$ of each point $\partial {\Omega }^{\left(i\right)}$ $(i=1,\dots ,n)$ with the distance to it, i.e.,

$$\rho \left(x_1,x_2\right)={\left({\left(x_1-x_1^{\left(i\right)}\right)}^2+{\left(x_2-x_2^{\left(i\right)}\right)}^2\right)}^{{\alpha }_2},$$

where ${\alpha }_2\ge {\displaystyle \frac{1}{2}}$, $\left(x_1^{\left(i\right)},x_2^{\left(i\right)}\right)=\partial {\Omega }^{\left(i\right)}$.

We will assume that ${\Omega }_i\cap {\Omega }_j=\varnothing$ $(i\ne j)$. Let $\chi$ be the radius of the largest of the neighborhoods ${\Omega }_i$. We denote by ${\Omega }_{\chi }$ the union of the domains ${\Omega }_i$ $(i=1,\dots ,n)$, ${\Omega }_{\chi }={\cup }_{i=1}^n{\Omega }_i$.

Furthermore, let the derivatives of the function $\rho \left(x\right)$ satisfy the inequality

$$\left|{\displaystyle \frac{\partial \rho }{\partial x_s}}\right|\le \delta ,\hspace{0.33em}s=1,2.$$

Now we introduce the weighed spaces $H_{2,\alpha }^k\left(\Omega \right)$ and $W_{2,\alpha }^k\left(\Omega \right)$ with norms

$${\parallel u\parallel }_{H_{2,\alpha }^k\left(\Omega \right)}={\left(\sum _{\left|\lambda \right|\le k}{\int }_{\Omega }{\rho }^{2(\alpha +|\lambda |-k)}{\left|D^{\lambda }u\right|}^{2}dx\right)}^{1/2},$$

$${\parallel u\parallel }_{W_{2,\alpha }^k\left(\Omega \right)}={\left(\sum _{\left|\lambda \right|\le k}{\int }_{\Omega }{\rho }^{2\alpha }{\left|D^{\lambda }u\right|}^{2}dx\right)}^{1/2}.$$

(1)

Here, $D^{\lambda }={\displaystyle \frac{{\partial }^{\left|\lambda \right|}}{\partial x_1^{{\lambda }_1}\partial x_2^{{\lambda }_2}}}$, $\lambda =\left({\lambda }_1,{\lambda }_2\right)$ and $\left|\lambda \right|={\lambda }_1+{\lambda }_2$; ${\lambda }_1$, ${\lambda }_2$ are integer nonnegative numbers, $\alpha$ is some real number, and k is an integer nonnegative number. For $k=0$ we use the notation $H_{2,\alpha }^0\left(\Omega \right)=W_{2,\alpha }^0\left(\Omega \right)=L_{2,\alpha }\left(\Omega \right)$. For $\alpha =0$, we have

$${\parallel u\parallel }_{W_{2,0}^k\left(\Omega \right)}={\parallel u\parallel }_{W_2^k\left(\Omega \right)}.$$

By ${\left|u\right|}_{H_{2,\alpha }^k\left(\Omega \right)}$ and ${\left|u\right|}_{W_{2,\alpha }^k\left(\Omega \right)}$ we denote the seminorms of the function u.

Note. 

The type of the weight function $\rho \left(x\right)$ in the definition of weighted Sobolev spaces is determined by and coincides with the weight function of the singular component of the solution of boundary value problems with a corner singularity (see, for example, [25,27]).

Let us formulate and prove the main lemmas and theorems that are used to estimate the norm of a function in the neighborhood of a singularity point.

Lemma 1 

([35]). $\left(A\right)$ Let $u\in H_{2,\alpha }^1\left(\Omega \right)$. Then ${\rho }^{\alpha }u\in W_{2,0}^1\left(\Omega \right)$, ${\rho }^{\alpha -1}u\in L_{2,0}\left(\Omega \right)$ and

$${\left|{\rho }^{\alpha }u\right|}_{W_{2,0}^1\left(\Omega \right)}+{∥{\rho }^{\alpha -1}u∥}_{L_{2,0}\left(\Omega \right)}\le C_1{∥u∥}_{H_{2,\alpha }^1\left(\Omega \right)},$$

where $C_1$ is a positive constant independent of u.

$\left(B\right)$ Let ${\rho }^{\alpha }u\in W_{2,0}^1\left(\Omega \right)$ and ${\rho }^{\alpha -1}u\in L_{2,0}\left(\Omega \right)$. Then, $u\in H_{2,\alpha }^1\left(\Omega \right)$ and there exist positive constants $C_2$ and $C_3$ independent of u such that the inequality

$$C_2{\left|{\rho }^{\alpha }u\right|}_{W_{2,0}^1\left(\Omega \right)}+C_3{∥{\rho }^{\alpha -1}u∥}_{L_{2,0}\left(\Omega \right)}\ge {∥u∥}_{H_{2,\alpha }^1\left(\Omega \right)}$$

holds.

Theorem 1 

([35]). $\left(A\right)$ If $u\in H_{2,\alpha }^k\left(\Omega \right)$, then ${\rho }^{\alpha -\left(k-s\right)}u\in W_{2,0}^s\left(\Omega \right)$ $\left(s=0,\dots ,k\right)$ and

$${\left|{\rho }^{\alpha }u\right|}_{W_{2,0}^k\left(\Omega \right)}+{\left|{\rho }^{\alpha -1}u\right|}_{W_{2,0}^{k-1}\left(\Omega \right)}+\cdots +{\left|{\rho }^{\alpha -k}u\right|}_{L_{2,0}\left(\Omega \right)}\le C_4{∥u∥}_{H_{2,\alpha }^k\left(\Omega \right)},$$

where $C_4$ is a positive constant independent of u.

$\left(B\right)$ If ${\rho }^{\alpha -(k-s)}u\in W_{2,0}^s\left(\Omega \right)$ $\left(s=0,\dots ,k\right)$, then $u\in H_{2,\alpha }^k\left(\Omega \right)$ and there exist positive constants $C_0^{*},\dots ,C_k^{*}$ independent of u such that the inequality

$$C_k^{*}{\left|{\rho }^{\alpha }u\right|}_{W_{2,0}^k\left(\Omega \right)}+C_{k-1}^{*}{\left|{\rho }^{\alpha -1}u\right|}_{W_{2,0}^{k-1}\left(\Omega \right)}+\cdots +C_0^{*}{\left|{\rho }^{\alpha -k}u\right|}_{L_{2,0}\left(\Omega \right)}\ge {∥u∥}_{H_{2,\alpha }^k\left(\Omega \right)}.$$

is valid.

Further, we will assume that the function u has a singularity at one point $\partial {\Omega }^{\left(1\right)}$ of the boundary $\partial \Omega$. Let ${\Omega }_{\theta }$ be the $\theta$-neighborhood of the point $\partial {\Omega }^{\left(1\right)}$, where $\theta$ $\left(\theta \le \chi \right)$ is a real number.

Theorem 2. 

Let $u\in W_{2,\alpha }^1\left(\Omega \right)$ and $\alpha >1$, then for any $\epsilon \in \left(0,1\right]$, the inequality

$${∥u∥}_{L_{2,\alpha }\left({\Omega }_{\theta }\right)}\le {\theta }^{\epsilon }\left(C_5{\left|u\right|}_{W_{2,\alpha +1-\epsilon }^1\left({\Omega }_{\theta }\right)}+C_6{∥u∥}_{W_{2,\alpha +\frac{1}{2}-\epsilon }^1\left(\Omega \setminus {\Omega }_{\theta }\right)}\right)$$

(2)

holds, where $C_5$, $C_6$ are positive constants independent of u and mes ${\Omega }_{\theta }.$

Proof. 

Let us introduce a coordinate system $O{x}_1{x}_2$ on the plane so that the origin of coordinates coincides with the point $\partial {\Omega }^{\left(1\right)}$ and the entire domain $\Omega$ is located in the upper half-plane. This is always possible since $\Omega$ is a convex domain.

Let $\overline{u}\left(x\right)$ be a function with support containing the domain $\Omega$, coinciding with $u\left(x\right)$ on this domain. Let us introduce the notation: ${\alpha }_1={\displaystyle \underset{x\in {\Omega }_{\theta }}{inf}}{x}_1$, $b_1={\displaystyle \underset{x\in {\Omega }_{\theta }}{sup}}{x}_1$ and define the rectangles ${\prod }_1=\left[-b,b\right]\times \left[0,c\right]$, ${\prod }_1^{+}=\left[0,b\right]\times \left[0,c\right]$, where $b>{\displaystyle \underset{x\in {\Omega }_{\theta }}{sup}}\left|x_1\right|$, $c>\theta$.

We fix some ${\overline{x}}_2$ from the segment $[0,\theta ]$ and denote by $x_1^{\left(1\right)}$ and $x_1^{\left(2\right)}$ the abscissas of the points $x^{\left(1\right)}=\left(x_1^{\left(1\right)},{\overline{x}}_2\right)$ and $x^{\left(2\right)}=\left(x_1^{\left(2\right)},{\overline{x}}_2\right)$, which belong to $\partial {\Omega }_{\theta }$ (see Figure 1).

Let $x_1$ be any point from the segment $\left[0,x_1^{\left(2\right)}\right]$, and let $y_1$ be any point from the segment $\left[x_1^{\left(2\right)},b\right]$. The equalities

$$\overline{u}\left(x_1,{\overline{x}}_2\right)=\overline{u}\left(y_1,{\overline{x}}_2\right)-{\int }_{x_1}^{y_1}{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}dt$$

and

$$\left|\overline{u}\left(x_1,{\overline{x}}_2\right)\right|=\left|\overline{u}\left(y_1,{\overline{x}}_2\right)-{\int }_{x_1}^{x_1^{\left(2\right)}}{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}dt-{\int }_{x_1^{\left(2\right)}}^{y_1}{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}dt\right|$$

hold. From the last equality, we have

$$\left|\overline{u}\left(x_1,{\overline{x}}_2\right)\right|\le \left|\overline{u}\left(y_1,{\overline{x}}_2\right)\right|+{\int }_{x_1}^{x_1^{\left(2\right)}}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt+{\int }_{x_1^{\left(2\right)}}^{y_1}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt.$$

(3)

We multiply both sides of (3) by $x_1^{\alpha }$ and square the resulting inequality. Applying the $\epsilon$-inequality, we will have

$$x_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x_1,{\overline{x}}_2\right)\le C_7{x}_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(y_1,{\overline{x}}_2\right)+C_8{\left(x_1^{\alpha }{\int }_{x_1}^{x_1^{\left(2\right)}}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^2+C_9{x}_1^{2\alpha }{\left({\int }_{x_1^{\left(2\right)}}^{y_1}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^2,$$

(4)

where $C_7$, $C_8$, $C_9$ are positive constants independent of u and mes ${\Omega }_{\theta }$.

We integrate inequality (4) over $x_1$ on the segment $\left[0,x_1^{\left(2\right)}\right]$. As a result we obtain

$${\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x_1,{\overline{x}}_2\right)d{x}_1\le C_7{\overline{u}}^2\left(y_1,{\overline{x}}_2\right){\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha }d{x}_1+C_8{\int }_0^{x_1^{\left(2\right)}}{\left(x_1^{\alpha }{\int }_{x_1}^{x_1^{\left(2\right)}}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^{2}d{x}_1$$

$$+C_9{\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha }d{x}_1{\left({\int }_{x_1^{\left(2\right)}}^{y_1}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^2.$$

(5)

We estimate the second term in the right-hand side of (5) using the Hardy inequality (see, for example, [36]) for $\alpha >1$ and $0<\epsilon \le 1$

$${\int }_0^{x_1^{\left(2\right)}}{\left(x_1^{\alpha }{\int }_{x_1}^{x_1^{\left(2\right)}}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^{2}d{x}_1\le C_{10}{\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha +2}{\left({\displaystyle \frac{\partial \overline{u}\left(x_1,{\overline{x}}_2\right)}{\partial x_1}}\right)}^{2}d{x}_1,$$

$$\le C_{11}{\theta }^{2\epsilon }{\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha +2-2\epsilon }{\left({\displaystyle \frac{\partial \overline{u}\left(x_1,{\overline{x}}_2\right)}{\partial x_1}}\right)}^{2}d{x}_1.$$

Taking into account the definition of the weight function $\rho \left(x\right)$, we obtain

$${\int }_0^{x_1^{\left(2\right)}}{\left(x_1^{\alpha }{\int }_{x_1}^{x_1^{\left(2\right)}}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^{2}d{x}_1\le C_{11}{\theta }^{2\epsilon }{\int }_0^{x_1^{\left(2\right)}}{\rho }^{2\alpha +2-2\epsilon }\left(x_1,{\overline{x}}_2\right){\left({\displaystyle \frac{\partial \overline{u}\left(x_1,{\overline{x}}_2\right)}{\partial x_1}}\right)}^{2}d{x}_1.$$

Substituting the obtained estimate into (5), we will have

$${\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x_1,{\overline{x}}_2\right)d{x}_1\le C_{12}{x}_1^{{\left(2\right)}^{2\alpha +1}}{\overline{u}}^2\left(x_1,{\overline{x}}_2\right)$$

$$+C_{13}{\theta }^{2\epsilon }{\int }_0^{x_1^{\left(2\right)}}{\rho }^{2\alpha +2-2\epsilon }\left(x_1,{\overline{x}}_2\right){\left({\displaystyle \frac{\partial \overline{u}\left(x_1,{\overline{x}}_2\right)}{\partial x_1}}\right)}^{2}d{x}_1$$

$$+C_{14}{x}_1^{{\left(2\right)}^{2\alpha +1}}{\left({\int }_{x_1^{\left(2\right)}}^{y_1}\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^2.$$

(6)

Due to the definition of weight function $\rho \left(x\right)$ and conditions $\alpha >1$ and $\epsilon \in \left(0,1\right]$ of the theorem, the inequalities

$$x_1^{{\left(2\right)}^{2\alpha +1}}=x_1^{{\left(2\right)}^{2\epsilon }}{x}_1^{{\left(2\right)}^{2\alpha +1-2\epsilon }}\le {\theta }^{2\epsilon }{y}_1^{2\alpha +1-2\epsilon }\le {\theta }^{2\epsilon }{\rho }^{2\alpha +1-2\epsilon }\left(y_1,{\overline{x}}_2\right)$$

(7)

are valid.

Taking into account (7), we estimate the first and third terms in the right-hand side of the inequality (6), and then integrate the obtained inequality over $y_1$ on the segment $\left[x_1^{\left(2\right)},b\right]$. As a result, we obtain

$${\int }_{x_1^{\left(2\right)}}^{b}d{y}_1{\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x_1,{\overline{x}}_2\right)d{x}_1\le C_{12}{\theta }^{2\epsilon }{\int }_{x_1^{\left(2\right)}}^b{\rho }^{2\alpha +1-2\epsilon }\left(y_1,{\overline{x}}_2\right)\hspace{0.33em}{\overline{u}}^2\left(y_1,{\overline{x}}_2\right)d{y}_1,$$

$$+C_{13}{\theta }^{2\epsilon }{\int }_{x_1^{\left(2\right)}}^{b}d{y}_1{\int }_0^{x_1^{\left(2\right)}}{\rho }^{2\alpha +2-2\epsilon }\left(x_1,{\overline{x}}_2\right){\left({\displaystyle \frac{\partial \overline{u}\left(x_1,{\overline{x}}_2\right)}{\partial x_1}}\right)}^{2}d{x}_1,$$

$$+C_{14}{\theta }^{2\epsilon }{\int }_{x_1^{\left(2\right)}}^b{\left({\int }_{x_1^{\left(2\right)}}^{y_1}{t}^{\alpha +\frac{1}{2}-\epsilon }\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^{2}d{y}_1.$$

(8)

Considering the definition of the weight function $\rho \left(x\right)$, we estimate the third term in the right-hand side of (8) using the Cauchy–Schwarz inequality:

$${\int }_{x_1^{\left(2\right)}}^b{\left({\int }_{x_1^{\left(2\right)}}^{y_1}{t}^{\alpha +\frac{1}{2}-\epsilon }\left|{\displaystyle \frac{\partial \overline{u}\left(t,{\overline{x}}_2\right)}{\partial t}}\right|dt\right)}^{2}d{y}_1\le C_{15}{\int }_{x_1^{\left(2\right)}}^b{\rho }^{2\alpha +1-2\epsilon }\left(y_1,{\overline{x}}_2\right){\left({\displaystyle \frac{\partial \overline{u}\left(y_1,{\overline{x}}_2\right)}{\partial y_1}}\right)}^{2}d{y}_1.$$

(9)

Since the value ${\displaystyle \frac{1}{\left(b-x_1^{\left(2\right)}\right)}}$ does not depend on u and mes ${\Omega }_{\theta }$, then from inequalities (8) and (9), we have

$${\int }_0^{x_1^{\left(2\right)}}{x}_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x_1,{\overline{x}}_2\right)d{x}_1\le C_{16}{\theta }^{2\epsilon }{\int }_{x_1^{\left(2\right)}}^b{\rho }^{2\alpha +1-2\epsilon }\left(y_1,{\overline{x}}_2\right){\overline{u}}^2\left(y_1,{\overline{x}}_2\right)d{y}_1,$$

$$+C_{17}{\theta }^{2\epsilon }{\int }_{x_1^{\left(2\right)}}^b{\rho }^{2\alpha +1-2\epsilon }\left(y_1,{\overline{x}}_2\right){\left({\displaystyle \frac{\partial \overline{u}\left(y_1,{\overline{x}}_2\right)}{\partial y_1}}\right)}^{2}d{y}_1,$$

$$+C_{18}{\theta }^{2\epsilon }{\int }_0^{x_1^{\left(2\right)}}{\rho }^{2\alpha +2-2\epsilon }\left(x_1,{\overline{x}}_2\right){\left({\displaystyle \frac{\partial \overline{u}\left(x_1,{\overline{x}}_2\right)}{\partial x_1}}\right)}^{2}d{x}_1.$$

Integrating the last inequality over $x_2$ on the segment $\left[0,\theta \right]$, we obtain

$${\int }_{{\Omega }_{\theta }^{+}}{x}_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x\right)dx\le C_{16}{\theta }^{2\epsilon }{∥\overline{u}∥}_{L_{2,\alpha +1/2-\epsilon }\left({\prod }_1^{+}\setminus {\Omega }_{\theta }^{+}\right)}^2$$

$$+C_{17}{\theta }^{2\epsilon }{∥{\displaystyle \frac{\partial \overline{u}}{\partial x_1}}∥}_{L_{2,\alpha +1/2-\epsilon }\left({\prod }_1^{+}\setminus {\Omega }_{\theta }^{+}\right)}^2+C_{18}{\theta }^{2\epsilon }{∥{\displaystyle \frac{\partial \overline{u}}{\partial x_1}}∥}_{L_{2,\alpha +1-\epsilon }\left({\Omega }_{\theta }^{+}\right)}^2.$$

(10)

Here, ${\Omega }_{\theta }^{+}$ is a subdomain of ${\Omega }_{\theta }$ located in the first quadrant.

We note that in the domain ${\Omega }_{\theta }^{-}={\Omega }_{\theta }\setminus {\Omega }_{\theta }^{+}$ an estimate similar to (10) is valid. Therefore, the inequality

$${\int }_{{\Omega }_{\theta }}{x}_1^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x\right)dx\le C_{19}{\theta }^{2\epsilon }{∥\overline{u}∥}_{L_{2,\alpha +1/2-\epsilon }\left({\prod }_1\setminus {\Omega }_{\theta }\right)}^2$$

$$+C_{20}{\theta }^{2\epsilon }{∥{\displaystyle \frac{\partial \overline{u}}{\partial x_1}}∥}_{L_{2,\alpha +1/2-\epsilon }\left({\prod }_1\setminus {\Omega }_{\theta }\right)}^2+C_{21}{\theta }^{2\epsilon }{∥{\displaystyle \frac{\partial \overline{u}}{\partial x_1}}∥}_{L_{2,\alpha +1-\epsilon }\left({\Omega }_{\theta }\right)}^2$$

(11)

holds.

We fix some ${\overline{x}}_1$ from the segment $\left[a_1,b_1\right]$ (see Figure 2). Assuming ${\prod }_2=\left[a_1,b_1\right]\times \left[0,c\right]$ and, reasoning by analogy with the above, we establish the validity of the inequality

$${\int }_{{\Omega }_{\theta }}{x}_2^{2\alpha }\hspace{0.33em}{\overline{u}}^2\left(x\right)dx\le C_{22}{\theta }^{2\epsilon }{∥\overline{u}∥}_{L_{2,\alpha +1/2-\epsilon }\left({\prod }_2\setminus {\Omega }_{\theta }\right)}^2$$

$$+C_{23}{\theta }^{2\epsilon }{∥{\displaystyle \frac{\partial \overline{u}}{\partial x_2}}∥}_{L_{2,\alpha +1/2-\epsilon }\left({\prod }_2\setminus {\Omega }_{\theta }\right)}^2+C_{24}{\theta }^{2\epsilon }{∥{\displaystyle \frac{\partial \overline{u}}{\partial x_2}}∥}_{L_{2,\alpha +1-\epsilon }\left({\Omega }_{\theta }\right)}^2.$$

(12)

Let us add inequalities (11) and (12) to obtain

$${\int }_{{\Omega }_{\theta }}\left(x_1^{2\alpha }+x_2^{2\alpha }\right)\hspace{0.33em}{\overline{u}}^2\left(x\right)dx\le {\theta }^{2\epsilon }\left(C_{25}{\left|\overline{u}\right|}_{W_{2,\alpha +1-\epsilon }^1\left({\Omega }_{\theta }\right)}^2+C_{26}{∥\overline{u}∥}_{H_{2,\alpha +1/2-\epsilon }^1\left({\prod }_1\setminus {\Omega }_{\theta }\right)}^2\right).$$

(13)

Using the Clarkson inequalities (see, for example, [37]) for $\alpha >1$ and taking into account the definition of the weight function $\rho \left(x\right)$, we estimate the left-hand side of (13) from below:

$${\int }_{{\Omega }_{\theta }}\left(x_1^{2\alpha }+x_2^{2\alpha }\right){\overline{u}}^2\left(x\right)dx\ge C_{27}{∥\overline{u}∥}_{L_{2,\alpha }\left({\Omega }_{\theta }\right)}^2.$$

(14)

Taking into account the definition of the function $\overline{u}\left(x\right)$, from (13) and (14), the validity of estimate (2) follows. Theorem 2 is proved. □

Theorem 3. 

Let $u\in H_{2,\alpha }^1\left(\Omega \right)$ and parameters α and ${\alpha }_1$ satisfy the conditions ${\alpha }_1\le 1$ and $\alpha >{\alpha }_1-{\displaystyle \frac{1}{2}}$, then

$${∥u∥}_{L_{2,\alpha -{\alpha }_1}\left({\Omega }_{\theta }\right)}\le {\theta }^{1-{\alpha }_1}\left(C_{28}{\left|u\right|}_{W_{2,\alpha }^1\left({\Omega }_{\theta }\right)}+C_{29}{∥u∥}_{W_{2,\alpha }^1\left(\Omega \setminus {\Omega }_{\theta }\right)}\right),$$

where $C_{28}$, $C_{29}$ are positive constants independent of u and mes ${\Omega }_{\theta }$.

This theorem was proved in the general case in [38].

Corollary 1. 

If $u\in W_{2,\alpha }^1\left(\Omega \right)$ and $\alpha >-{\displaystyle \frac{1}{2}}$, then the estimate

$${∥u∥}_{L_{2,\alpha }\left({\Omega }_{\theta }\right)}\le C_{30}\theta {∥u∥}_{W_{2,\alpha }^1\left(\Omega \right)}$$

is true, where $C_{30}$ is a positive constant independent of u and mes ${\Omega }_{\theta }$.

Corollary 2. 

If $u\in W_{2,\alpha }^2\left(\Omega \right)$ and $\alpha >-{\displaystyle \frac{1}{2}}$, then the estimate

$${∥u∥}_{W_{2,\alpha }^1\left({\Omega }_{\theta }\right)}\le C_{31}\theta {∥u∥}_{W_{2,\alpha }^2\left(\Omega \right)}$$

is true, where $C_{31}$ is a positive constant independent of u and mes ${\Omega }_{\theta }$.

The proofs of these statements follow from Theorem 3.

3. On the Estimate of the Norm of a Function in the Boundary Strip

For boundary value problems with uncoordinated degeneration of the initial data with a singularity on the entire boundary of the domain $\Omega$, an estimate of the norm of the function in the boundary strip is necessary (see, for example, [39]).

Let $\Omega$ be a convex domain and let the function $\overline{\rho }\left(x\right)$ be the distance from the point x to the boundary $\partial \Omega$. In this section, the norms (1) for the spaces $H_{2,\alpha }^k\left(\Omega \right)$ and $W_{2,\alpha }^k\left(\Omega \right)$ are defined with the weight function $\overline{\rho }\left(x\right)$.

Let us set a real number $\tau <l_{\Omega }/2$, where $l_{\Omega }$ is the diameter of the circle inscribed in $\Omega$. We denote by ${\Omega }_{\tau }$ the boundary strip of width $\tau$ (the boundary strip is the set of points from $\Omega$ that are at a distance of no more than $\tau$ from the boundary $\partial \Omega$).

Let u be a function from the space $H_{2,\alpha }^1\left(\Omega \right)$. We will establish an estimate of the norm of a function in the space $L_{2,\alpha }\left({\Omega }_{\tau }\right)$.

Lemma 2 

([40]). If u belongs to the space $W_2^1\left(\Omega \right)$, then the estimate

$${∥u∥}_{L_2\left({\Omega }_{\tau }\right)}\le C_{32}{\tau }^{\frac{1}{2}}{∥u∥}_{W_2^1\left(\Omega \right)}$$

is valid, and the positive constant $C_{32}$ does not depend on the function u or the width of the boundary strip τ.

Theorem 4. 

If u is some function from the space $H_{2,\alpha }^1\left(\Omega \right)$ and α is a nonnegative real number, then the estimate

$${∥u∥}_{L_{2,\alpha }\left({\Omega }_{\tau }\right)}\le C_{33}{\tau }^{\frac{1}{2}}{∥u∥}_{H_{2,\alpha }^1\left(\Omega \right)}$$

(15)

holds, where $C_{33}$ is a positive constant independent of u and τ.

Proof. 

By the condition of the theorem, the function $u\in H_{2,\alpha }^1\left(\Omega \right)$. Then, based on Lemma 2, the function $\overline{u}={\overline{\rho }}^{\alpha }u$ belongs to the space $W_{2,0}^1\left(\Omega \right)$ and the estimate

$${∥\overline{u}∥}_{W_2^1\left(\Omega \right)}^2\le C_{34}{∥u∥}_{H_{2,\alpha }^1\left(\Omega \right)}^2$$

is true.

From the statement of Lemma 2 and the last inequality we obtain

$${∥u∥}_{L_{2,\alpha }\left({\Omega }_{\tau }\right)}^2={∥\overline{u}∥}_{L_2\left({\Omega }_{\tau }\right)}^2\le C_{35}\tau {∥\overline{u}∥}_{W_2^1\left(\Omega \right)}^2\le C_{36}\tau {∥u∥}_{H_{2,\alpha }^1\left(\Omega \right)}^2.$$

(16)

From inequality (16) the estimate (15) follows. □

Corollary 3. 

If $u\in H_{2,\alpha }^2\left(\Omega \right)$ and α is a nonnegative real number, then the inequality

$${∥u∥}_{H_{2,\alpha }^1\left({\Omega }_{\tau }\right)}\le C_{37}{\tau }^{\frac{1}{2}}{∥u∥}_{H_{2,\alpha }^2\left({\Omega }_{\tau }\right)}$$

holds, where $C_{37}$ is a positive constant independent of u and τ.

4. Conclusions

The estimates of functions in the neighborhoods of singularity points established here will be used to study the existence, uniqueness, coercivity, and differential properties of the $R_{\nu }$-generalized solution of a biharmonic boundary value problem with a singularity. In addition, the presented theorems will be useful in proving estimates of the rate of convergence of approximate solutions by the weighted finite element method to the exact $R_{\nu }$-generalized solution of the boundary value problem for a biharmonic equation with a singularity.