Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 512109, 20 pages
http://dx.doi.org/10.5402/2011/512109
Research Article

On Removable Sets of the First Boundary-Value Problem for Degenerated Elliptic Equations

1Department of Nonlinear Analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Azerbaijan
2Department of Mathematics, Azad University, Fouman Branch, Iran

Received 26 June 2011; Accepted 10 August 2011

Academic Editor: V. Kravchenko

Copyright © 2011 Tair S. Gadjiev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the paper, the necessary and sufficient condition of compact removability is obtained.

1. Introduction

The questions of compact removability for Laplace equation is studied by Carleson [1]. The uniform elliptic equation of the seconds order of divergent structure is studied by Moiseev [2]. The compact removability for elliptic and parabolic equations of nondivergent structure is considered by Landis [3]. Gadjiev, Mamedova [4]. The removability condition of compact in the space of continuous functions are constructed in the papers Harvey and Polking [5], Kilpeläinen and Zhong [6]. The different questions of qualitative properties of solutions of uniformly degenerated elliptic equations is studied by Chanillo and Wreeden [7]. In paper [8] the second order uniform divergent elliptic operator is considered.

Let 𝐸𝑛 be 𝑛 dimensional Euclidean space of the points 𝑥=(𝑥1,,𝑥𝑛). Denote the ball {𝑥|𝑥𝑥0|<𝑅} by 𝐵𝑅(𝑥0𝑅) for 𝑅>0 and the cylinder 𝐵𝑅(𝑥0)(0,𝑇) by 𝑄𝑅𝑇(𝑥0𝑅). Further, let for 𝑥0𝐸𝑛,𝑅>0 and 𝑘>0𝜀𝑟,𝑘(𝑥0) be an ellipsoid {𝑥𝑛𝑖=1((𝑥𝑖𝑥0𝑖)2/𝑅𝛼𝑖)<(𝑘𝑅)2}. Let 𝐷 be an bounded domain 𝐸𝑛 with the domain 𝜕𝐷,0𝐷. 𝜀 is a such king of ellipsoid that 𝐷𝜀,𝔅(𝜀) is a set of all functions, satisfying in 𝜀 the uniform Lipschitz condition and having zero near the 𝜕𝜀.

Denote by 𝛼 and (𝛼1,,𝛼𝑛) the vector 𝛼=𝛼1,,𝛼𝑛.

Denote by 𝑊12,𝛼(𝐷) the Banach space of the functions 𝑢(𝑥) given on 𝐷 with the finite norm𝑢𝑊12,𝛼(𝐷)=𝐷𝑢2+𝑛𝑖=1𝜆𝑖(𝑥)𝑢2𝑖𝑑𝑥1/2,(1.1) where𝑢𝑖=𝜕𝑢𝜕𝑥𝑖𝜆,𝑖=1,,𝑛.𝑖(𝑥)=|𝑥|𝜆𝛼𝑖,|𝑥|𝛼=𝑛𝑖=1||𝑥𝑖||22+𝛼𝑖,0𝛼𝑖<2.𝑛1(1.2) Further, let 𝑊12,𝛼(𝐷) be a degenerated set of all functions from 𝐶0(𝐷) by the norm of the space 𝑊12,𝛼(𝐷). Denote by (𝐷) the set of all bounded in 𝐷 functions.

Let 𝐸𝐷 be some compact. Denote by 𝐴𝐸(𝐷) the totality of all functions 𝑢(𝑥)𝐶(𝐷), each of which there exists some neighbourhood of the compact 𝐸 in which 𝑢(𝑥)=0.

The compact 𝐸 is called the removable relative to the first boundary value problem for the operator 𝐿 in the space (𝐷) if all generalized solution of the equation 𝑢=0 in 𝜕/𝐸 formed in zero on 𝜕𝐷 and belonging to the space (𝐷) identically equal to zero. We will say that the function 𝑢(𝑥)𝑊12,𝛼(𝜀) is nonnegative on the set 𝐻𝜀, in the sense 𝑊12,𝛼(𝜀) if there exists the sequence of the functions {𝑢(𝑚)(𝑥)}, 𝑚=1,2, such that 𝑢𝑚(𝑥)𝔅(𝜀), 𝑢𝑚(𝑥)0 for 𝑥𝐻 and lim𝑚𝑢(𝑚)𝑢𝑊12,𝛼(𝜀)=0.

The function 𝑢(𝑥)𝑊12,𝛼(𝐷) is nonnegative and 𝜕D in sense 𝑊12,𝛼(𝐷) if there exists the sequence of the functions {𝑢𝑚(𝑥)}, 𝑚=1,2, such that 𝑢(𝑚)(𝑥)𝐶1(𝐷), 𝑢𝑚(𝑥)0 for 𝑥𝜕𝐷 and lim𝑚𝑢(𝑚)𝑢𝑊12,𝛼(𝜀)=0. It is easy to determine the inequalities 𝑢(𝑥)const, 𝑢(𝑥)𝑣(𝑥), 𝑢(𝑥)0, and also equality 𝑢(𝑥)=1 on the set 𝐻 in the sense 𝑊12,𝛼(𝜀) if at the same time 𝑢(𝑥)1 and 𝑢(𝑥)1 on 𝐻, in the sense 𝑊12,𝛼(𝜀).

Let 𝜔(𝑥) be measurable function in 𝐷, finite and positive for a.e. 𝑥𝐷. Denote by 𝑝,𝜔(𝐷) the Banach space of the functions given on 𝐷, with the norm𝑢𝑝,𝜔(𝐷)=𝐷(𝜔(𝑥))𝑝/2|𝑢|𝑝𝑑𝑥1/𝑝,1<𝑝<.(1.3)

Let 𝑊1𝑝,𝛼(𝐷) be a Banach space of the functions given on 𝑢(𝑥), with the finite norm 𝐷𝑢𝑊1𝑝,𝛼(𝐷)=𝐷|𝑢|𝑝+𝑛𝑖=1𝜆𝑖(𝑥)𝑝/2||𝑢𝑖||𝑝𝑑𝑥1/𝑝,1<𝑝<.(1.4)

Analogously to 𝑊12,𝛼(𝐷), it is introduced the subspace 𝑊1𝑝,𝛼(𝐷) for 1<𝑝<. The space conjugated to 𝑊1𝑝,𝛼(𝐷) we will denote by 𝑊1𝑝,𝛼(𝐷).

We will consider the elliptic operator in the bounded domain 𝐷𝐸𝑛=𝑛𝑖,𝑗=1𝜕𝜕𝑥𝑖𝑎𝑖𝑗𝜕(𝑥)𝜕𝑥𝑗.(1.5) In assumption that 𝑎𝑖𝑗(𝑥) is a real symmetric matrix with measurable in 𝐷 elements, moreover, for all 𝜉𝐸𝑛 and a.e. 𝑥𝐷, the condition𝛾𝑛𝑖=1𝜆𝑖(𝑥)𝜉2𝑖𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝜉𝑖𝜉𝑗𝛾𝑛1𝑖=1𝜆𝑖(𝑥)𝜉2𝑖.(1.6) Here, 𝛾(0,1] is a constant.

The function 𝑢(𝑥)𝑊12,𝛼(𝐷) is called the generalized solution of the equation 𝑢=𝑓(𝑥) in 𝐷, if for any function 𝜂(𝑥)𝑊12,𝛼(𝐷) the integral identity 𝐷𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑥𝑖𝜂𝑥𝑗𝑑𝑥=𝐷𝑓𝜂𝑑𝑥(1.7) be fulfilled.

Here, 𝑓(𝑥) is a given function from 2(𝐷).

Let 𝐸𝐷 be some compact. The function 𝑢(𝑥)𝑊12,𝛼(𝐷𝐸) is called generalized solution of the equation 𝑢=𝑓(𝑥) in 𝐷𝐸 vanishing on 𝜕𝐷 if integral identity (1.7) is fulfilled for any function 𝜂(𝑥)𝐴𝐸(𝐷).

We will assume that the coefficients of the operator continued in 𝐸𝑛𝐷 with saving condition (1.2), (1.6). For this, it is sufficient, for example, to assume that 𝑎𝑖𝑗(𝑥)=𝛿𝑖𝑗𝜆𝑖(𝑥) for 𝑥𝐸𝑛𝐷, 𝑖,𝑗=1,,𝑛, where 𝛿𝑖𝑗 is a Croneker symbol.

Let (𝑥)𝑊12,𝛼(𝐷), 𝑓0(𝑥)2(𝐷), 𝑓𝑖(𝑥)2,𝜆1(𝐷), 𝑖=1,2,,𝑛, be a given functions. Let us consider the first boundary value problem𝑢=𝑓0(𝑥)+𝑛𝑖=1𝜕𝑓𝑖(𝑥)𝜕𝑥𝑖,𝑥𝐷,(𝑢(𝑥)(𝑥))𝑊12,𝛼(𝐷).(1.8) The function 𝑢(𝑥)𝑊12,𝛼(𝐷) we will call generalized solution of problem (1.8) if for any function 𝜂(𝑥)𝑊12,𝛼(𝐷), the integral identity𝐷𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑥𝑖𝜂𝑥𝑗𝑑𝑥=𝐷𝑓0𝜂+𝑛𝑖=1𝑓𝑖𝜂𝑥𝑖𝑑𝑥(1.9) is fulfilled.

Our aim to get the necessary and sufficient condition of compact removability 𝐸 in the class of bounded functions.

2. Preliminaries Statements

At first, we introduce some auxiliary statements.

Lemma 2.1. If relative to the coefficients of the operator condition (1.2), (1.6) are fulfilled, then the first boundary value problem (1.8) has a unique generalized solution 𝑢(𝑥) at any (𝑥)𝑊12,𝛼(𝐷), 𝑓0(𝑥)2(𝐷), 𝑓𝑖(𝑥)𝐿2,𝜆𝑖1(𝐷), 𝑖=1,2,,𝑛. At this, there exists 𝑃0(𝛼,𝑛) such that if 𝑝>𝑝0, (𝑥)𝑊1𝑝,𝛼(𝐷), 𝑓0(𝑥)𝑝(𝐷), 𝑓𝑖(𝑥)𝐿2,𝜆𝑖1(𝐷), 𝑖=1,2,,𝑛, 𝜕𝐷𝐶1, then solution 𝑢(𝑥) is continuous in 𝐷.

Lemma 2.2. Let relative to the coefficients of the operator conditions (1.2), (1.6) be fulfilled. Then, any generalized solution of the equation 𝑢=0 in 𝐷 is continuous by Holder at each strictly internal domain 𝜕.

Lemma 2.3. Let relative to the coefficients of the operator , conditions (1.2), (1.6) be fulfilled and 𝜀𝑅,1<𝐷. Then, for any positive generalized solution 𝑢(𝑥), the equation 𝑢=0 in 𝐷 the Harnack inequality is true sup𝜀𝑅,1(0)𝑢𝐶1(𝛾,𝛼,𝑛)inf𝜀𝑅,1(0)𝑢.(2.1) If at this 𝑦𝜕𝜀𝑅,2(0) and 𝜀𝑅,1(0)𝐷, then the inequality of form (2.1) is true in ellipsoid 𝜀𝑅,1(𝑦).

Lemma 2.4. Let relative to the coefficients of the operator conditions (1.2), (1.6) be fulfilled and 𝑢(𝑥) generalized solution of the first boundary-value problem (1.8) at 𝑓𝑖(𝑥)0, 𝑖=0,,𝑛. Then, if (𝑥) is bounded on 𝜕𝐷 in the sense 𝑊12,𝛼(𝐷), then for solution 𝑢(𝑥) the following maximum principle is true: inf𝜕𝐷inf𝐷𝑢sup𝐷sup𝜕𝐷,(2.2) where inf𝜕𝐷(sup𝜕𝐷) is an exact lower (upper) bound those numbers 𝑎, for which (𝑥)𝑎((𝑥)a)  on 𝜕𝐷 in the sense 𝑊12,𝛼(𝐷).
These lemmas are proved analogously to paper [7]. Therefore, the proof of these lemmas is not given.
Let 𝐻𝜀 be some compact and 𝑉𝐻 a set of all functions 𝜑(𝑥)𝑊12,𝛼(𝜀) such that 𝜑(𝑥)1 on 𝐻, in the sense 𝑊12,𝛼(𝜀). Let one considers the functional 𝐽𝜃(𝜑)=𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝜑𝑖𝜑𝑗𝑑𝑥,𝜑(𝑥)𝑉𝐻,(2.3) is a 𝐻 compact capacity relative to ellipsoid 𝜀 and is called the value inf𝜑𝑉𝐻𝐽𝜃(𝑢) and denoted by cap(𝜀)(𝐻). In case 𝜀=𝐸𝑛, the corresponding value is called capacity of the compact 𝐻 and denoted by cap(𝐻).

Lemma 2.5. There exists the unique function 𝑢(𝑥)𝑊12,𝛼(𝜀) such that 𝑢(𝑥)1 on 𝐻 in the sense 𝑊12,𝛼(𝜀) and cap(𝜀)(𝐻)=𝐽(𝑢).

Proof. It is easy to see that 𝑉𝐻 is convex closed set in 𝑊12,𝛼(𝜀). From the fact that 𝑊12,𝛼(𝜀) is a Hilbert space, it follows the existence of unique function 𝑢(𝑥)𝑉𝐻, which achieved an exact lower bound of the functional 𝐽(𝜑). Next, {𝑢(𝑥)}1=𝑢(𝑥)if𝑢(𝑥)1,1if𝑢(𝑥)>1.(2.4)
It is clear that {𝑢(𝑥)}1𝑊12,𝛼(𝜀). Moreover, {𝑢(𝑥)}1𝑉𝐻. Denote by 𝐴+={𝑥𝑥𝜀,𝑢(𝑥)>1}. We have 𝐽𝑢(𝑥)1=𝐴++𝜀𝐴+𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥){𝑢}1𝑖{𝑢}1𝑗𝑑𝑥=𝜀𝐴+𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝑢𝑗𝑑𝑥.(2.5) On the other side, according to (1.2), 𝐴+𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝑢𝑗𝑑𝑥0.(2.6) From (2.5) and (2.6), we conclude 𝐽𝑢(𝑥)1𝐽(𝑢)=inf𝜑𝑉𝐻𝐽(𝜑),(2.7) that is, 𝐽{𝑢(𝑥)1}=𝐽(𝑢). From uniqueness extreme function, it follows that {𝑢(𝑥)}1=𝑢(𝑥), and lemma is proved.
The function 𝑢(𝑥), which achieved an exact lower bound of the functional 𝐽(𝜑) on the set 𝑉𝐻 is called capacity of the compact potential 𝐻 relative to the ellipsoid 𝜀.

Lemma 2.6. Let be a capacity potential 𝑢(𝑥) of the compact 𝐻 relative to 𝜀 which is a generalized solution of the equation 𝑢=0 in 𝜀𝐻, vanishing on 0 and 𝜕𝜀 in 1 on 𝜕𝐻 in the sense 𝑊12,𝛼(𝜀).

Proof. It is sufficient to show the truth of the first part of assertion of lemma. Let 𝜂(𝑥)𝑊12,𝛼(𝜀) and 𝜂(𝑥)0 on 𝐻 in the sense 𝑊12,𝛼(𝜀). Then, for any 𝜀>0(𝑢(𝑥)+𝜀𝜂(𝑥))𝑉𝐻. Therefore, 𝐽(𝑢+𝜀𝜂)𝐽(𝑢).(2.8) Thus, 𝐽(𝑢)+𝜀2𝐽(𝜂)+2𝜀𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜂𝑗𝑑𝑥𝐽(𝑢),(2.9) that is, 𝐽(𝑢)+2𝜀𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜂𝑗𝑑𝑥0.(2.10) Tending 𝜀 to zero, we conclude 𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜂𝑗𝑑𝑥0.(2.11) It is easy to see as 𝜂(𝑥) in (2.11), we can take any function from 𝐶1(𝜀) with compact support in 𝜀𝐻. Then, 𝑛𝜀𝐻𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜂𝑗𝑑𝑥0.(2.12) Substituting 𝜂(𝑥) on 𝜂(𝑥), we arrive to 𝑛𝜀𝐻𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜂𝑗𝑑𝑥=0.(2.13)
Lemma is proved.

Let 𝜇 be a charge of bounded variation, given on 𝜀. We will say that the function 𝑢(𝑥)𝐿1(𝜀) is a weak solution of the equation 𝑢=𝜇, equaling to zero on 𝜕𝜀 if for any function 𝜑(𝑥)𝑊12,𝛼(𝜀)𝐶(𝜀), the integral identity𝜀𝑢𝜑𝑑𝑥=𝜀𝜑𝑑𝜇(2.14) is fulfilled.

According to Lemma 2.1 (at =0), there exists the continuous linear operator 𝐻 from 𝑊12,𝛼(𝜀) in 𝑊12,𝛼(𝜀) such that for any functional 𝑇𝑊12,𝛼(𝜀), the function 𝑢=𝐻(𝑇) is unique in 𝑊12,𝛼(𝜀) generalized solution of the equation 𝑢=𝑇.

The operator 𝐻 is called Green operator.

By Lemma 2.1, this operator at 𝑝>𝑝0 we transform 𝑊12,𝛼(𝜀) to 𝐶(𝜀). It is easy to see that the function 𝑢(𝑥) is weak solution of the equation 𝑢=𝜇, equaling to zero on 𝜕𝜀 if and only if for any function 𝜓(𝑥)𝐶(𝜀) the integral identity𝜀𝑢𝜓𝑑𝑥=𝜀𝐻(𝜓)𝑑𝜇(2.15) is fulfilled.

By analogy with [8], we can show that for each measure 𝜇 on 𝜀, there exists the unique weak solution of the equation 𝑢=𝜇 equaling to zero on 𝜕𝜀.

Let us say that the charge 𝜇𝑊12,𝛼(𝜀) if there exists the vector 𝑓(𝑥)=(𝑓(𝑥),𝑓1(𝑥),,𝑓𝑛(𝑥))𝑓0(𝑥)2(𝜀), 𝑓𝑖(𝑥)𝐿2,𝜆𝑖(𝜀), 𝑖=1,2,,𝑛, for any function 𝜑(𝑥)𝑊12,𝛼(𝜀)𝐶(𝜀), the integral identity𝜇(𝜑)=𝜀𝜑𝑑𝜇=𝜀𝑓𝜑𝑛𝑖=1𝑓𝑖𝜑𝑖𝑑𝑥(2.16) is true.

At this, it is evident that||||||𝜀||||||𝜑𝑑𝜇𝐶2𝑓𝜑𝑊12,𝛼(𝜀).(2.17)

Lemma 2.7. The weak solution 𝑢(𝑥) of the equation 𝑢=𝜇, equaling to zero on 𝜕𝜀, belongs to 𝑊12,𝛼(𝜀), if and only if 𝜇𝑊12,𝛼(𝜀).

Proof. At first, we will show that if the function 𝜑(𝑥)𝑊12,𝛼(𝜀) satisfies the integral identity 𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜑𝑗𝑑𝑥=𝜀𝜑𝑑𝜇,(2.18) for any function 𝜑(𝑥)𝑊12,𝛼(𝜀)𝐶(𝜀)zero on 𝜕𝜀, then it is weak solution of the equation 𝑢=𝜇, equaling to zero on 𝜕𝜀. Really, assuming that 𝜑=𝐻(𝜓), 𝜓(𝑥)𝐶(𝜀), we obtain 𝜀𝐻(𝜓)𝑑𝜇=𝜀𝜑𝑑𝜇=𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜑𝑗𝑑𝑥=𝜀𝑢𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝜑𝑗𝑖=𝑑𝑥𝜀𝑢𝜑𝑑𝑥=𝜀𝑢𝜓𝑑𝑥,(2.19) and now, it is sufficient to use the identity (2.15). We will show that 𝜇𝑊12,𝛼(𝜀). For this, it is sufficient to prove that if 𝑓𝑖(𝑥)=𝑛𝑖=1𝑎𝑖𝑗(𝑥)𝑢𝑖(𝑥), then 𝑓𝑖(𝑥)𝐿2,𝜆𝑖1(𝜀), 𝑖=1,2,,𝑛,. Assume in condition (2.18) that 𝜉1==𝜉𝑖1=𝜉𝑖+1==𝜉𝑛=0, 𝜉𝑖=1/𝜆𝑖(𝑥).
We will obtain 𝑎𝛾𝑖𝑖(𝑥)𝜆𝑖(𝑥)𝛾1;𝑖=1,,𝑛.(2.20)
Let 𝑖𝑗. Assuming that 𝜉𝑘=0 at 𝑘𝑗 and 𝑘𝑖, 𝜉𝑖=1/𝜆𝑖(𝑥), 𝜉𝑗=1/𝜆𝑗(𝑥), we will obtain 𝑎2𝛾𝑖𝑖(𝑥)𝜆𝑖+𝑎(𝑥)𝑗𝑗(𝑥)𝜆𝑗+(𝑥)2𝑎𝑖𝑗(𝑥)𝜆𝑖(𝑥)𝜆𝑗(𝑥)2𝛾1.(2.21) Using (2.20), we conclude ||𝑎𝑖𝑗||(𝑥)𝜆𝑖(𝑥)𝜆𝑗(𝑥)𝛾1𝛾;𝑖,𝑗=1,,𝑛;𝑖𝑗.(2.22) From (2.20) and (2.22), it follows that ||𝑎𝑖𝑗||(𝑥)𝜆𝑖(𝑥)𝜆𝑗(𝑥)𝛾1;𝑖,𝑗=1,,𝑛.(2.23) Thus, from (2.23), take out for 𝑗=1,,𝑛𝜀1𝜆𝑗𝑓(𝑥)𝑗2𝑑𝑥=𝜀1𝜆𝑗(𝑥)𝑛𝑖=1𝑎𝑖𝑗(𝑥)𝑢𝑖2𝑑𝑥𝛾2𝑛𝑛𝑖=1𝜀𝜆𝑖(𝑥)𝑢2𝑖𝑑𝑥<𝛼.(2.24) So, 𝜇𝑊12,𝛼(𝜀). Inversely, if 𝑢(𝑥) is a weak solution of the equation 𝑢=𝜇, vanishing on 𝜕𝜀, then there exists 𝜇𝑊12,𝛼(𝜀) such that 𝑓𝜑𝑛𝑖=1𝑓𝑖𝜑𝑖𝑑𝑥=𝜀𝜑𝑑𝜇=𝜀𝑢𝜑𝑑𝑥=𝜀𝑢𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝜑𝑗𝑖𝑑𝑥=𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜑𝑗𝑑𝑥,(2.25) for any function 𝜑(𝑥)𝑊12,𝛼(𝜀)𝐶(𝜀), 𝜑(𝑥)𝐶(𝜀).
Then, from Lemma 2.1, we obtain that 𝑢(𝑥)𝑊12,𝛼(𝜀). The lemma is proved.

Let now 𝛿(𝑥) be Dirac measure, concentrated at the point 0, and 𝑦 an arbitrary fixed point 𝜀.

The weak solution 𝑔(𝑥,𝑦) of the equation 𝑦=𝛿(𝑥𝑦), vanishing on 𝜕𝜀, is called Green function of the operator in 𝜀.

In case 𝜀=𝐸𝑛 the corresponding function is called the fundamental solution of the operator and denoted by 𝐺(𝑥,𝑦).

According to the above proved, if 𝜓(𝑥) is an arbitrary function from 𝐶(𝜀), then the generalized solution 𝜑(𝑥)𝑊12,𝛼(𝜀) of the equation 𝜑=𝜓 can be introduced in the following form:𝜑(𝑦)=𝜀𝑔(𝑥,𝑦)𝜓(𝑥)𝑑𝑥.(2.26) We can show that 𝑔(𝑥,𝑦) is nonnegative in 𝜀×𝜀 moreover; 𝑔(𝑥,𝑦)=𝑔(𝑦,𝑥).

Lemma 2.8. For any charge of bounded variation on 𝜀, the integral 𝑢(𝑥)=𝜀𝑔(𝑥,𝑦)𝑑𝜇(𝑦)(2.27) exists, finite a.e. in 𝜀 and is weak solution of the equation 𝑢=𝜇, equaling to zero on 𝜕𝜀.

Proof. Without losing generality, we will assume that the charge 𝜇 is the measure in 𝜀. Let 𝜑(𝑥)𝐶(𝜀), 𝜓(𝑥)0 in 𝜀. Denote by 𝜑(𝑥)𝑊12,𝛼(𝜀) the generalized solution of the equation 𝜑=𝜓(𝑥). Then, 𝜑(𝑥)𝐶(𝜀) according to Lemma 2.1 and 𝜓(𝑥)0 according to Lemma 2.4. At this, 𝜑(𝑦)=𝜀𝑔(𝑥,𝑦)𝜓(𝑥)𝑑𝑥.(2.28) Then, by the Fubini theorem, we conclude that the integral 𝜀𝑔(𝑥,𝑦)𝑑𝜇(𝑦) there exists for almost all 𝑥𝜀; moreover, 𝜀𝐻(𝜓)𝑑𝜇(𝑦)=𝜀𝜑(𝑦)𝑑𝜇(𝑦)=𝜀×𝜀𝑔(𝑥,𝑦)𝜓(𝑥)𝑑𝑥𝑑𝜇(𝑦)=𝜀𝜓(𝑥)𝑢(𝑥)𝑑𝑥.(2.29)
Let us note that (2.29) is fulfilled for weak nonnegative and continuous in 𝜀 function 𝜓(𝑥). Now, it is sufficient to remember the identity (2.15) and lemma is proved.
Let us consider now -capacity of the potential 𝑢(𝑥) of the compact 𝐻 relative to the ellipsoid 𝜀. Before, it was proved that 𝑢(𝑥) satisfies (2.11) at any nonnegative on 𝐻 the function 𝜂(𝑥)𝐶0(𝜀). By the Schwartz theorem [9, 10], there exists the measure 𝜇 on 𝐻 such that 𝜀𝑛𝑖,𝑗=1𝑎𝑖𝑗(𝑥)𝑢𝑖𝜂𝑗𝑑𝑥=𝜀𝜂𝑑𝜇.(2.30) Further, since 𝑢=1 on 𝐻 in the sense 𝑊12,𝛼(𝜀), then the carrier of the measure 𝜇 is situated on 𝜕𝐻. The measure 𝜇 is called -capacity of the compact 𝐻.
According to Lemma 2.8,  -capacity potential 𝑢(𝑥) is weak solution of the equation 𝑢=𝜇, equaling to zero on 𝜕𝜀 and can be represented in the following form: 𝑢(𝑥)=𝜀𝑔(𝑥,𝑧)𝑑𝜇(𝑧).(2.31) On the other side, there exists the sequence of the functions {𝜂(𝑚)(𝑥)}; 𝑚=1,2, such that 𝜂(𝑚)(𝑥)𝔅(𝜀), 𝜂(𝑚)(𝑥)=1 for 𝑥𝐻 and lim𝑚𝜂(𝑚)𝑢𝑊12,𝛼(𝜀)=0. Assuming in (2.15) 𝜂(𝑚)(𝑥) instead of 𝜂(𝑚), we conclude that it first fart is equal to 𝜇(𝐻) at any natural 𝑚, while the left part tends to cap(𝜀)(𝐻) as 𝑚. Thus, cap(𝜀)(𝐻)=𝜇(𝐻).(2.32)

Lemma 2.9. Let relative to coefficients of the operator conditions (1.2)–(1.6), 𝑦𝜕𝜀𝑅,2(0), 𝜀𝑅,1(0)𝐷, 𝑥𝜕𝜀𝑅,1(𝑦) be fulfilled. Then, for the Green function 𝑔(𝑥,𝑦) the following estimations are true: 𝐶3(𝛾,𝛼,𝑛)cap(𝜀)𝜀𝑅,1(𝑦)1𝑔(𝑥,𝑦)𝐶4(𝛾,𝛼,𝑛)cap(𝜀)𝜀𝑅,1(𝑦)1.(2.33) If 𝜀𝑅,1(0)𝐷, 𝑥𝜕𝜀𝑅,1(0) then, 𝐶3cap(𝜀)𝜀𝑅,1(0)1𝑔(𝑥,0)𝐶4cap(𝜀)𝜀𝑅,1(0)1.(2.34)

Proof. Without loss of generality, we can assume that the coefficients of the operator are continuously differentiable in 𝜀. The general case is obtained by means of limit passage. Then, at 𝑥𝑦, the function 𝑔(𝑥,𝑦) is continuous by 𝑥 and 𝑦 moreover, lim𝑥𝑦𝑔(𝑥,𝑦)=.(2.35)
Let 𝑎 be a positive number, which will be chosen later, 𝐾𝑎={𝑥𝑔(𝑥,𝑦)𝑎}, where 𝑦 is an arbitrary fixed point on 𝜕𝜀𝑅,2(0). From (2.35), it follows that 𝑦 is internal point 𝑦 of the compact 𝐾𝑎. Then, is capacity potential 𝐾𝑎, represented in the form (2.31), so it means that it equal to zero in it. Thus, 1=𝜀𝑦(𝑦,𝑧)𝑑𝜇𝑎(𝑧),(2.36) where 𝜇 is a -capacity distribution of the compact 𝐾𝑎. Allowing for the carrier of the measure 𝜇𝑎 is situated on 𝜕𝐾𝑎, where 𝑔(𝑦,𝑧) = 𝑎and using (2.32), we obtain 𝜇𝑎𝐾𝑎=cap(𝜀)𝐾𝑎=1𝑎.(2.37) Let us assume now that 𝑎=inf𝑥𝜕𝜀𝑅,1(𝑦)𝑔(𝑥,𝑦). According to maximum principle, 𝜀𝑅,1(𝑦)𝐾𝑎. Therefore, from (2.37), we conclude cap(𝜀)𝜀𝑅,1(𝑦)cap(𝜀)𝐾𝑎=1inf𝑥𝜕𝜀𝑅,1(𝑦)𝑔.(𝑥,𝑦)(2.38) If we will assume 𝑏=sup𝑥𝜕𝜀𝑅,1(𝑦)𝑔(𝑥,𝑦), then 𝜀𝑅,1(𝑦)𝐾𝑎; that is, cap(𝜀)𝜀𝑅,1(𝑦)cap(𝜀)𝐾𝑏=1sup𝑥𝜕𝜀𝑅,1(𝑦)𝑔.(𝑥,𝑦)(2.39) From (2.38) and (2.39), follows that inf𝑥𝜕𝜀𝑅,1(𝑦)𝑔(𝑥,𝑦)cap(𝜀)𝜀𝑅,1(𝑦)1sup𝑥𝜕𝜀𝑅,1(𝑦)𝑔(𝑥,𝑦).(2.40) On the other side, according to Lemma 2.3, sup𝑥𝜕𝜀𝑅,1(𝑦)𝑔(𝑥,𝑦)𝐶5(𝛾,𝛼,𝑛)inf𝑥𝜕𝜀𝑅,1(𝑦)𝑔(𝑥,𝑦).(2.41) Now, the required estimations (2.33) follows from (2.40) and (2.41). Absolutely analogously the truth of (2.34) is proved.

Corollary 2.10. Let the conditions of the lemma and 𝑦𝜕𝜀𝑅,2(0) be fulfilled, 𝜀𝑅,1(0)𝐷, 𝑥𝜕𝜀𝑅,1(0) or 𝑦=0, 𝜀𝑅,1(0)𝐷, 𝑥𝜕𝜀𝑅,1(0). Then, for fundamental solution 𝐺(𝑥,𝑦), the estimations 𝐶3cap(𝜀)𝜀𝑅,1(0)1𝐺(𝑥,𝑦)𝐶4cap(𝜀)𝜀𝑅,1(0)1(2.42) are true.

3. Removability Criterion of the Compact in the Space 𝑀(𝐷)

Theorem 3.1. Let relative to the coefficients of the operator , conditions (1.2)–(1.6) be fulfilled. Then, for removability of the compact 𝐸𝐷 relative to the first boundary value problem for the operator in the space (𝐷), it is necessary and sufficient that cap(𝐸)=0.(3.1)

Proof. Let the ellipsoid 𝜀 have the same sense as above. It is easy to see that if condition (3.1) is fulfilled, then cap(𝜀)(𝐸)=0.(3.2) Not losing generality, we can limit the case, when the coefficients of the operator is continuously differentiable in 𝜀. Let us fix an arbitrary 𝜀>0 and 𝑥0𝐷𝐸. By virtue of (3.1), there exists the neighbourhood 𝐻 of the compact 𝐸 such that cap(𝜀)𝐻<𝜀.(3.3) At this, we can assume that 𝜀 is such small that 𝑥dist0,𝐻12𝑥dist0.,𝐸(3.4) Denote by 𝑉𝐻(𝑥) and 𝜇𝐻 the -capacity potential of the compact 𝐻 relative to the ellipsoid 𝜀 and -capacity of the distribution 𝐻, respectively. According to above proved, 𝑉𝐻(𝑥)=𝜀𝑔(𝑥,𝑦)𝑑𝜇𝐻(𝑦),(3.5) moreover, the function 𝑉𝐻(𝑥) is the generalized solution of the equation 𝑉𝐻=0 in 𝜀𝐻, vanishing on 0 and in 𝜕𝜀 on 1 in 𝜕𝐻 in the sense 𝑊12,𝛼(𝜀). Let now 𝑢(𝑥)(𝐷) be an arbitrary solution of the equation 𝑢=0 in 𝐷𝐸, vanishing on 𝜕𝐷, 𝑀=sup𝐷|𝑢|. It is easy to see that the function 𝑉𝐻(𝑥) is nonnegative on 𝜕𝐷, in the sense 𝑊12,𝛼(𝐷). Hence, it follows, that the function 𝑢(𝑥)𝑀𝑉𝐻(𝑥) is generalized solution of the equation 𝑢=0 in 𝐷𝐻, is non-positive on 𝜕(𝐷𝐻). According to Lemma 2.4,  𝑢(𝑥)𝑀𝑉𝐻(𝑥)0 and 𝐷𝐻 in particular 𝑢𝑥0𝑀𝑉𝐻𝑥0𝑀sup𝑦𝜕𝐻𝑔𝑥0𝜇,𝑦𝐻𝐻=𝑀sup𝑦𝜕𝐻𝑔𝑥0,𝑦cap(𝜀)𝐻.(3.6) By virtue of continuity of the function 𝑔(𝑥,𝑦) at 𝑥𝑦 and (3.4), we obtain sup𝑦𝜕𝐻𝑔𝑥0,𝑦𝐶6𝛾,𝛼,𝑛,𝑥0.,𝐸(3.7) Thus, from (3.3) and (3.6), we conclude 𝑢𝑥0𝑀𝐶6𝜀.(3.8) Using an arbitrariness 𝜀, we have 𝑢𝑥00.(3.9) Making analogous considerations with the function 𝑢(𝑥)+𝑀𝑉𝐻(𝑥), we obtain 𝑢𝑥00.(3.10) From (3.8)-(3.9) and an arbitrariness of the point 𝑥0 it follows that 𝑢(𝑥)0 in 𝐷𝐸. Thereby, the sufficiency of condition (3.1) is proved. Let us prove its necessity. Let us assume that cap(𝐸)>0. Denote by 𝜀 the ellipsoid such that 𝜀𝛿, 𝐸𝜀. Assume 𝐷=𝜀  potential of the compact 𝐸 relative to the ellipsoid 𝜀 and -capacity distribution 𝐸, respectively. Following to [11], we can give the equivalent definition of Vallee-Poussin type of -capacity of the compact 𝐸, relative to the ellipsoid 𝜀. Let 𝑔(𝑥,𝑦) be a Green function of the operator in 𝜀. Let us call the measure 𝜇 on 𝐸, -admissible if 𝜇𝐸 and 𝑉𝐸𝜇(𝑥)=𝜀𝑔(𝑥,𝑦)𝑑𝜇(𝑦)1for𝑥sup𝑝𝜇.(3.11) The value sup𝜇(𝐸)=cap(𝜀)(𝐸), where an exact upper boundary is taken on all -admissible measures and is called -capacity of the compact 𝐸 relative to the ellipsoid 𝜀.
Analogously, the -capacity cap(𝐸) is determined. At this, by the standard method, we show that there exists the unique measure on which an exact upper boundary of the functional 𝜇(𝐸) is reached by the set of all -admissible measures 𝜇. This measure is -capacity distribution of the compact 𝐸.
According to the above proved, the function 𝑢𝐸(𝑥) is generalized solution of the equation 𝑢𝐸=0 in 𝜀𝐸, equaling to zero on 𝜕𝜀. Besides, from (3.10) and maximum principle, it follows that 𝑢𝐸(𝑥)𝑀(𝜀). On the other side, 𝑢𝐸(𝑥)0, as 𝑉𝐻(𝐸)>0. Theorem is proved.

Lemma 3.2. Let relative to the coefficients of the operator condition (1.2) be fulfilled. Then, if 𝑦𝜕𝜀𝑅,2(0), then 𝐶7(𝛾,𝛼,𝑛)𝑅𝑛+(𝛼/2)2cap𝜀𝑅,1(𝑦)𝐶8(𝛾,𝛼,𝑛)𝑅𝑛+(𝛼/2)2.(3.12)

Proof. Let 0=𝑛𝑖=1(𝜕/𝜕𝑥𝑖)(𝜆𝑖(𝑥)(𝜕/𝜕𝑥𝑖)). Then, according to (1.2), 𝛾cap0𝜀𝑅,1(𝑦)cap𝜀𝑅,1(𝑦)𝛾1cap0𝜀𝑅,1.(𝑦)(3.13) Let 𝑢(𝑥)𝐶0(𝜀𝑅,(3/2)(𝑦)), 𝑢(𝑥)=1 for 𝜀𝑅,1(𝑦); moreover, ||𝑢𝑖(||𝐶𝑥)9(𝜆,𝑛)𝑅1+(𝛼𝑖/2);𝑖=1,,𝑛.(3.14) Then, cap0𝜀𝑅,1(𝑦)𝜀𝑅,(3/2)𝑛(𝑦)𝑖=1𝜆𝑖(𝑥)𝑢2𝑖𝑑𝑥.(3.15) On the other side, as 𝑦𝜕𝜀𝑅,2(0), then 𝑛𝑖=1(𝑦2𝑖/𝑅𝛼𝑖)=4𝑅2, and thereby ||𝑦𝑖||2𝑅1+(𝛼𝑖/2);𝑖=1,,𝑛.(3.16) Besides, as 𝑥𝜀𝑅,(3/2)(𝑦), then ||𝑥𝑖𝑦𝑖||32𝑅1+(𝛼𝑖/2);𝑖=1,,𝑛.(3.17) Thus, ||𝑥𝑖||||𝑦𝑖||+||𝑥𝑖𝑦𝑖||72𝑅1+(𝛼𝑖/2);𝑖=1,,𝑛.(3.18) Hence, it follows that |𝑥|𝛼𝑅𝑛𝑖=1𝑧22/(2+𝜆𝑖).(3.19) Therefore, 𝜆𝑖(𝑥)𝐶𝛼𝑖10𝑅𝛼𝑖𝐶𝛼+10𝑅𝛼𝑖;𝑖=1,,𝑛.(3.20) where 𝛼+=max{𝛼1,,𝛼𝑛}.
Allowing for (3.14) and (3.20) in (3.15), we obtain cap0𝜀𝑅,1(𝑦)𝐶10(𝛼,𝑛)𝑅2𝜀mes𝑅,(3/2)(𝑦)=𝐶11(𝛼,𝑛)𝑅𝑛+(𝛼/2)2,(3.21) and by virtue of (3.13), the estimation from upper in (3.11) is proved.
For showing the truth of the estimations from lower in (3.11), we note that cap0𝜀𝑅,1(𝑦)cap0𝜀𝑅,(1/2𝑛)(𝑦).(3.22) Besides, considering the same as in [8], we conclude cap0𝜀𝑅,(1/2𝑛)(𝑦)𝐶12(𝛼,𝑛)cap(𝜀0)0𝜀𝑅,(1/2𝑛)(𝑦),(3.23) where 𝜀0=𝜀𝑅,(1/𝑛)(𝑦).
Let 𝑊={𝑢(𝑥)𝑢(𝑥)𝐶0(𝜀0),𝑢(𝑥)=1for𝑥𝜀𝑅,(1/2𝑛)(𝑦)}. Then, cap(𝜀0)0𝜀𝑅,(1/2𝑛)(𝑦)=inf𝑢𝑊𝜀0𝑛𝑖=1𝜆𝑖(𝑥)𝑢2𝑖𝑑𝑥.(3.24) On the other side, if 𝑦𝜕𝜀𝑅,2(0), then we can find 𝑖0, 1𝑖0𝑛 such that 𝑦2𝑖04𝑅2+𝛼𝑖0/𝑛, that is; ||𝑦𝑖0||4𝑅1+(𝛼𝑖0/2)𝑛.(3.25) Besides, as 𝑥𝜀0, then ||𝑥𝑖0𝑦𝑖0||𝑅1+(𝛼𝑖0/2)𝑛.(3.26) Therefore, ||𝑥𝑖0||||𝑦𝑖0||||𝑥𝑖0𝑦𝑖0||𝑅1+(𝛼𝑖0/2)𝑛.(3.27) Thereby, 𝜆𝑖(𝑥)𝑛1/(2+𝛼𝑖0)𝑅;𝑖=1,,𝑛,(3.28) where 𝛼=min{𝛼1,,𝛼𝑛}.
Allowing for (3.28) in (3.24), we obtain cap(𝜀0)0𝜀𝑅,(1/2𝑛)(𝑦)=𝐶13(𝛼,𝑛)inf𝑢𝑊𝜀0𝑛𝑖=1𝑅𝛼𝑖𝑢2𝑖𝑑𝑥.(3.29) Denote by 𝐵𝑅(𝑧) the ball {𝑥|𝑥𝑧|<𝑅}. Let us make in (3.30) the substitution of the variables 𝑣𝑖=𝑥𝑖/𝑅1+(𝛼𝑖/2); 𝑖=1,,𝑛, and let ̃𝑦 be an image of the point 𝑦, where 𝑊={̃𝑢(𝑣)̃𝑢(𝜏)𝐶0(𝐵0),̃𝑢(𝜏)=1for𝑣𝐵(1/2𝑛)(̃𝑦)}. Then, from (3.30), we deduce 𝐵0=𝐵(1/2𝑛)(̃𝑦) where by (3.30), cap(𝜀0)0𝜀𝑅,(1/2𝑛)(𝑦)𝐶13𝑅𝑛+(𝛼/2)2inf̃𝑢𝑊𝐵0𝑛𝑖=1𝜕̃𝑢𝜕𝑣𝑖2𝑑𝜏=𝐶13𝑅𝑛+(𝛼/2)2cap(𝐵0)𝐵(1/2𝑛)(,̃𝑦)(3.30) we will denote by cap(𝐵0)(𝐵(1/2𝑛)(̃𝑦)) Wiener capacity of the compact 𝐵(1/2𝑛)(̃𝑦), relative to the ball 𝐵0. Now, it is sufficient to note that cap(𝐵0)(𝐵(1/2𝑛)(̃𝑦))=𝐶14(𝑛), and required estimation follows from (3.22), (3.23), and (3.31). Lemma is proved.

Lemma 3.3. Let relative to the coefficients of the operator condition (1.2) be fulfilled. Then, 𝐶15(𝛾,𝛼,𝑛)𝑅𝑛+(𝛼/2)2cap𝜀𝑅,1(𝑦)𝐶16(𝛾,𝛼,𝑛)𝑅𝑛+(𝛼/2)2.(3.31)
Upper estimation in (3.32) is proved analogously to the estimation in (3.11). For the proofing of the lower estimation, it is sufficient to note that 𝜀𝑅,(1/4)(𝑦)𝜀𝑅,1(0), that is, cap𝜀𝑅,(1/4)𝑦<cap𝜀𝑅,1,(0)(3.32) where 𝑦=((1/2)𝑅1+(𝛼/2),0,,0) and repeat the consideration of the proofing of the previous lemma.

Corollary 3.4. If conditions (1.2)–(1.6) 𝑦𝜕𝜀𝑅,2(0) be fulfilled, then for any 𝜌(0,𝑅], the estimation cap𝜀𝜌,1𝑦𝐶17(𝛾,𝛼,𝑛)𝜌𝑛+(𝛼/2)21+𝑛𝑖=1𝑅𝜌𝛼𝑖(3.33) is true.
Then, 𝑣(𝑥)𝐶0(𝜀𝜌,(3/2)(𝑦)), 𝑣(𝑥)=1 for 𝑥𝜀𝜌,1(𝑦)||𝑣𝑖(||𝐶𝑥)18(𝛼,𝑛)𝜌1+(𝛼𝑖/2);𝑖=1,,𝑛,cap0𝜀𝜌,1𝑦=𝛾1𝐶218𝜌2𝜀𝜌,(3/2)𝑛(𝑦)𝑖=1𝜆𝑖(𝑥)𝜌𝛼𝑖𝑑𝑥.(3.34) On the other side, arguing the same, as well as in the proof of Lemma 3.2, we have 𝜆𝑖(𝑥)<𝐶19(𝛼,𝑛)(𝑅+𝜌)𝛼𝑖,𝑥𝜀𝜌,(3/2)(𝑦);𝑖=1,,𝑛.(3.35) Now, it is sufficient to take into account that 𝑛𝑖=1𝑅1+𝜌𝛼𝑖𝑛𝑖=1𝑅1+𝜌𝛼𝑖𝑛1+𝑛𝑖=1𝑅𝜌𝛼i,(3.36) and from (3.34)-(3.35) follows the required estimation (3.33).

Corollary 3.5. If conditions (1.2)– (1.6) 𝑦0 are fulfilled, then at 𝑥𝜀𝑑|𝑦|𝑑,1(𝑦), 𝑥𝑦 for the fundamental solution 𝐺(𝑥,𝑦), the estimation 𝐺(𝑥,𝑦)𝐶20(||||𝛾,𝛼,𝑛)𝑥𝑦𝛼2𝑛(𝛼/2)1+𝑛𝑖=1||𝑦||𝛼/||||𝑥𝑦𝛼𝛼𝑖(3.37) is true.
If 𝑦=0, then estimation (3.37) is true for all 𝑥0. Here, 𝑑=1/𝑛22/(2+𝛼).
For proving, at first, let us show that if 𝑦0, then 𝑦𝜀𝑑|𝑦|𝑑,2(0). Really, as ||𝑦||𝛼=𝑛𝑖=1||𝑦𝑖||2/(2+𝛼𝑖),(3.38) then there exists 𝑖0, 1𝑖0𝑛 such that ||𝑦0||2/(2+𝛼𝑖0)||𝑦||𝛼𝑛.(3.39) Thus, ||𝑦2𝑖0||||𝑦||𝛼𝛼𝑖0||𝑦||𝛼2𝑛2+𝛼𝑖.(3.40) Thereby, 𝑛𝑖=1𝑦2𝑖𝑑||𝑦||𝛼𝛼𝑖𝑦2𝑖0𝑑||𝑦||𝛼𝛼𝑖0𝑑||𝑦||𝛼2(𝑑𝑛)2+𝛼𝑖0=4𝑑||𝑦||𝛼222/(2+𝛼𝑖0)𝑑𝑛2+𝛼𝑖0.(3.41) Now, it is sufficient to note that 22/(2+𝛼𝑖0)𝑑𝑛22/(2+𝛼)𝑑𝑛=1, and the required assertion is proved. On the other side from (3.38), it follows that for all 𝑖, 1𝑖𝑛||𝑦𝑖||2/(2+𝛼𝑖)||𝑦||𝛼,(3.42) that is, 𝑛𝑖=1𝑦2𝑖||𝑦||𝛼𝛼𝑖||𝑦||𝑛𝛼2.(3.43) So, one will show that 𝜀|𝑦|𝛼,𝑛(0) if only 𝑦0.
Let now for 𝑦0, 𝑥𝜀𝑑|𝑦|𝑑,1(𝑦) and 𝑥𝑦. Denote by |𝑥𝑦|𝛼 the 𝜌. It is easy to see that there exists 𝑖1, 1𝑖1𝑛 such that ||𝑥𝑖1𝑦𝑖1||2/(2+𝛼𝑖1)𝜌𝑛.(3.44) Hence, it follows that 𝑛𝑖=1𝑥𝑖𝑦𝑖2𝜌𝛼𝑖𝑥𝑖1𝑦𝑖12𝜌𝛼1𝜌2𝑛2+𝛼𝑖1𝜌2𝑛2+𝛼.(3.45) Thus, 𝑥𝜀𝜌;𝑑1(𝑦), where 𝑑1=1/𝑛1+(𝛼/2). Analogously, it is proved that 𝑥𝜀𝜌,𝑛(𝑦). Now, the required estimation (3.37) at 𝑦0 follows from (2.42) and Corollary 3.4 from Lemma 3.2. If 𝑦=0, then (3.37) immediately follows from (2.42) and Lemma 2.7.
Let 𝐹(𝑥,𝑦) be a positive function, determined in 𝐸𝑛×𝐸𝑛, continuous at 𝑥𝑦, moreover lim𝑥𝑦𝐹(𝑥,𝑦)= (condition (A)).
Further, let 𝐸𝐸𝑛 be some compact. Let us call the measure 𝜇 on 𝐸[𝐹] admissible if sup𝑝𝜇𝐸 and 𝑉𝐸𝜇(𝑥)=𝐸𝐹(𝑥,𝑦)𝑑𝜇(𝑦)1, for 𝑥sup𝑝𝜇.
The value sup𝜇(𝐸)=cap[𝐹](𝐸), where an exact upper boundary is taken by all [𝐹] admissible measures, is called [𝐹]-capacity of the compact 𝐸.

Theorem 3.6. Let relative to the coefficients of the operator conditions (1.2)–(1.6) be fulfilled. Then, for removability of the compact 𝐸𝐷 relative to the first boundary-value problem for the operator in the space (𝐷) it is sufficient that cap[𝐹1](𝐸)=0,(3.46) where 𝐹1(𝑥,𝑦)=[1+𝑛𝑖=1(|𝑦|𝛼/|𝑥𝑦|𝛼)𝛼𝑖]1(|𝑥𝑦|𝛼)2𝑛(𝛼/2).

Proof. We will use the following assertion, which is proved in [11]. Let the function 𝐹(𝑥,𝑦) be satisfied condition (A), the compact 𝐸 has zero [𝐹]-capacity, 𝜇 zero measure concentrated on 𝐸. Then, there exists the point 𝑥sup𝑝𝜇, such that 𝑉𝐸𝜇(𝑥)=. At this the potential of the measure sup𝑝𝜇 cannot be bounded on any portion 𝐵, that is, for any open set 𝐵 at 𝐸sup𝑝𝜇𝐵; the potential 𝑉𝐸𝜇(𝑥) is not bound 𝐵. In particular, if 𝐵 is an arbitrary neighborhood of the point 𝑥 that 𝑉𝐸𝜇(𝑥)=.
Let the condition (3.46) be fulfilled, 𝜇 an arbitrary measure, concentrated on 𝐸, 𝑥sup𝑝𝜇 is a point, corresponding to the above-stated assertion at 𝐹=𝐹1. Let us assume at first that 𝑥0. Then, |𝑥|𝛼=𝑣>0. Further, let 𝐵 be such small neighborhood of the point 𝑥 that if 𝐸sup𝑝𝜇𝐵, then sup𝑦𝐸||𝑦||𝛼(1+𝜀)𝑟,inf𝑦𝐸||𝑦||𝛼(1+𝜀)𝑟,(3.47) where the number 𝜀>0 will be chosen later. Let us consider the ellipsoids 𝜀𝑑|𝑦|𝑑,1(𝑦) at 𝑦𝐸. Let us choose 𝜀 such small than 𝑥0𝜀𝑑|𝑦|𝑑,1(𝑦) for all 𝑦𝐸. Then, according to Corollary 3.5 from Lemma 2.7, we obtain 𝑉𝐸𝜇𝑥0=𝐸𝐺𝑥0,𝑦𝑑𝜇(𝑦)𝐸𝐺𝑥0,𝑦𝑑𝜇(𝑦)𝐶20𝐸𝐹1𝑥0,𝑦𝑑𝜇(𝑦)=𝐶20𝑉𝐸𝜇𝑥0=.(3.48) Hence, it follows that any zero measure 𝜇, concentrated on 𝐸 cannot be admissible. Thus, cap(𝐸)=0, and the required assertion follows from Theorem 3.1.
Let now 𝑥=0. Then, using the equality 𝐺(𝑥,𝑦)=𝐺(𝑦,𝑥) and Corollary 3.5 from Lemma 2.7, we conclude 𝑉𝐸𝜇(0)=𝐸𝐺(0,𝑦)𝑑𝜇(𝑦)=𝐸𝐺(𝑦,0)𝑑𝜇(𝑦)𝐶20𝐸𝐹1(𝑦,0)𝑑𝜇(𝑦)=𝐶20𝐸𝐹1(0,𝑦)𝑑𝜇(𝑦)=𝐶20𝑉𝐸𝜇(0)=.(3.49) Theorem is proved.

Remark 3.7. Let conditions of the real theorem be fulfilled and the compact 𝐸𝐷 removable relative to the first boundary-value problem for the operator in the space (𝐷). Then, mes(𝐸)=0.
At first, let us note for proofing that the discussion are the same, as at the conclusion of estimation (3.37), we can show that at 𝑥𝜀𝑑|𝑦|𝑑,1(𝑦), 𝑥𝑦(𝑦0), and at 𝑥𝑦(𝑦=0) the estimations 𝐺(𝑥,𝑦)𝐶21||||(𝛾,𝛼,𝑛)𝑥𝑦𝑑2𝑛(𝛼/2)(3.50) are true.
Further, analogously to Theorem 3.1, it is shown that if the compact 𝐸 is removable, then according to cap[𝐹2](𝐸)=0, where 𝐹2(𝑥,𝑦)=(|𝑥𝑦|𝑑)2𝑛(𝛼/2).
Hence, it follows that if mes (𝐸)>0, then there exists the point 𝑥2𝐸, such that 𝑉𝐸(𝑥1)=, where 𝑉𝐸(𝑥)=𝐸𝐹2(𝑥,𝑦)𝑑𝑦.(3.51) Moreover, if 𝐵 is an arbitrary neighborhood of the point 𝐸=𝐵𝐸, then the potential 𝑉𝐸(𝑥) is not bounded on 𝐸. Let us consider the case 𝑥10. Choose small neighborhood 𝐵 of the point 𝑥1 that at all 𝑥𝐸, 𝑦𝐸 the inequality |𝑥𝑖𝑦𝑖|1; 𝑖=1,,𝑛 are fulfilled. For 𝑥𝐸, we have 𝑉𝐸(𝑥)=𝐸𝑛𝑖=1||𝑥𝑖𝑦𝑖||(2/2+𝛼𝑖)2𝑛(𝛼/2)𝑑𝑦𝐸𝑛𝑖=1||𝑥𝑖𝑦𝑖||2𝑛(𝛼/2)𝑑𝑦𝐸||||𝑥𝑦2𝑛(𝛼/2)𝑑𝑦𝐵|𝑧|2𝑛(𝛼/2)𝑑𝑦,(3.52) where 𝐵 is a ball of the radius 𝑛 with the center origin of the coordinate. Now, it is sufficient to note that according to condition (1.6) 𝛼/2𝑛/(𝑛1)3/2, and the assertion the corollary is proved.

Acknowledgment

This paper was performed under the financial support of Science Foundation under the President of Azerbaijan.

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