International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 512109 |

Tair S. Gadjiev, Asghar I. Shariffar, Rafig A. Rasulov, "On Removable Sets of the First Boundary-Value Problem for Degenerated Elliptic Equations", International Scholarly Research Notices, vol. 2011, Article ID 512109, 20 pages, 2011.

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

Academic Editor: V. Kravchenko
Received26 Jun 2011
Accepted10 Aug 2011
Published17 Oct 2011


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], KilpelΓ€inen 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 π‘₯=(π‘₯1,…,π‘₯𝑛). Denote the ball {π‘₯∢|π‘₯βˆ’π‘₯0|<𝑅} by 𝐡𝑅(π‘₯0𝑅) for 𝑅>0 and the cylinder 𝐡𝑅(π‘₯0)βˆͺ(0,𝑇) by 𝑄𝑅𝑇(π‘₯0𝑅). Further, let for π‘₯0βˆˆπΈπ‘›,𝑅>0 and π‘˜>0πœ€π‘Ÿ,π‘˜(π‘₯0) be an ellipsoid βˆ‘{π‘₯βˆΆπ‘›π‘–=1((π‘₯π‘–βˆ’π‘₯0𝑖)2/𝑅𝛼𝑖)<(π‘˜π‘…)2}. Let 𝐷 be an bounded domain 𝐸𝑛 with the domain πœ•π·,0∈𝐷. πœ€ is a such king of ellipsoid that π·βŠ‚πœ€,𝔅(πœ€) is a set of all functions, satisfying in πœ€ the uniform Lipschitz condition and having zero near the πœ•πœ€.

Denote by 𝛼 and (𝛼1,…,𝛼𝑛) the vector βŸ¨π›ΌβŸ©=𝛼1,…,𝛼𝑛.

Denote by π‘Š12,𝛼(𝐷) the Banach space of the functions 𝑒(π‘₯) given on 𝐷 with the finite normβ€–π‘’β€–π‘Š12,𝛼(𝐷)=ξƒ©ξ€œπ·ξƒ©π‘’2+𝑛𝑖=1πœ†π‘–(π‘₯)𝑒2𝑖ξƒͺξƒͺ𝑑π‘₯1/2,(1.1) where𝑒𝑖=πœ•π‘’πœ•π‘₯π‘–πœ†,𝑖=1,…,𝑛.𝑖(π‘₯)=|π‘₯|πœ†ξ€Έπ›Όπ‘–,|π‘₯|𝛼=𝑛𝑖=1||π‘₯𝑖||22+𝛼𝑖,0≀𝛼𝑖<2.π‘›βˆ’1(1.2) Further, let βˆ˜π‘Š12,𝛼(𝐷) be a degenerated set of all functions from 𝐢∞0(𝐷) by the norm of the space π‘Š12,𝛼(𝐷). Denote by β„³(𝐷) the set of all bounded in 𝐷 functions.

Let πΈβŠ‚π· be some compact. Denote by 𝐴𝐸(𝐷) the totality of all functions 𝑒(π‘₯)∈𝐢∞(𝐷), each of which there exists some neighbourhood of the compact 𝐸 in which 𝑒(π‘₯)=0.

The compact 𝐸 is called the removable relative to the first boundary value problem for the operator 𝐿 in the space β„³(𝐷) if all generalized solution of the equation ℒ𝑒=0 in πœ•/𝐸 formed in zero on πœ•π· and belonging to the space β„³(𝐷) identically equal to zero. We will say that the function 𝑒(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€) is nonnegative on the set π»βŠ‚πœ€, in the sense βˆ˜π‘Š12,𝛼(πœ€) if there exists the sequence of the functions {𝑒(π‘š)(π‘₯)}, π‘š=1,2,… such that π‘’π‘š(π‘₯)βˆˆπ”…(πœ€), π‘’π‘š(π‘₯)β‰₯0 for π‘₯∈𝐻 and limπ‘šβ†’βˆžβ€–π‘’(π‘š)βˆ’π‘’β€–π‘Š12,𝛼(πœ€)=0.

The function 𝑒(π‘₯)βˆˆπ‘Š12,𝛼(𝐷) is nonnegative and πœ•D in sense π‘Š12,𝛼(𝐷) if there exists the sequence of the functions {π‘’π‘š(π‘₯)}, π‘š=1,2,… such that 𝑒(π‘š)(π‘₯)∈𝐢1(𝐷), π‘’π‘š(π‘₯)β‰₯0 for π‘₯βˆˆπœ•π· and limπ‘šβ†’βˆžβ€–π‘’(π‘š)βˆ’π‘’β€–π‘Š12,𝛼(πœ€)=0. It is easy to determine the inequalities 𝑒(π‘₯)β‰₯const, 𝑒(π‘₯)β‰₯𝑣(π‘₯), 𝑒(π‘₯)≀0, and also equality 𝑒(π‘₯)=1 on the set 𝐻 in the sense βˆ˜π‘Š12,𝛼(πœ€) if at the same time 𝑒(π‘₯)β‰₯1 and 𝑒(π‘₯)≀1 on 𝐻, in the sense βˆ˜π‘Š12,𝛼(πœ€).

Let πœ”(π‘₯) be measurable function in 𝐷, finite and positive for a.e. π‘₯∈𝐷. Denote by ℒ𝑝,πœ”(𝐷) the Banach space of the functions given on 𝐷, with the norm‖𝑒‖ℒ𝑝,πœ”(𝐷)=ξ‚΅ξ€œπ·(πœ”(π‘₯))𝑝/2|𝑒|𝑝𝑑π‘₯1/𝑝,1<𝑝<∞.(1.3)

Let π‘Š1𝑝,𝛼(𝐷) be a Banach space of the functions given on 𝑒(π‘₯), with the finite norm π·β€–π‘’β€–π‘Š1𝑝,𝛼(𝐷)=ξƒ©ξ€œπ·ξƒ©|𝑒|𝑝+𝑛𝑖=1ξ€·πœ†π‘–ξ€Έ(π‘₯)𝑝/2||𝑒𝑖||𝑝ξƒͺξƒͺ𝑑π‘₯1/𝑝,1<𝑝<∞.(1.4)

Analogously to βˆ˜π‘Š12,𝛼(𝐷), it is introduced the subspace βˆ˜π‘Š1𝑝,𝛼(𝐷) for 1<𝑝<∞. The space conjugated to βˆ˜π‘Š1𝑝,𝛼(𝐷) we will denote by βˆ—π‘Š1𝑝,𝛼(𝐷).

We will consider the elliptic operator in the bounded domain π·βŠ‚πΈπ‘›β„’=𝑛𝑖,𝑗=1πœ•πœ•π‘₯π‘–ξ‚΅π‘Žπ‘–π‘—πœ•(π‘₯)πœ•π‘₯𝑗.(1.5) In assumption that β€–π‘Žπ‘–π‘—(π‘₯)β€– is a real symmetric matrix with measurable in 𝐷 elements, moreover, for all πœ‰βˆˆπΈπ‘› and a.e. π‘₯∈𝐷, the condition𝛾𝑛𝑖=1πœ†π‘–(π‘₯)πœ‰2𝑖≀𝑛𝑖,𝑗=1π‘Žπ‘–π‘—(π‘₯)πœ‰π‘–πœ‰π‘—β‰€π›Ύπ‘›βˆ’1𝑖=1πœ†π‘–(π‘₯)πœ‰2𝑖.(1.6) Here, π›Ύβˆˆ(0,1] is a constant.

The function 𝑒(π‘₯)βˆˆπ‘Š12,𝛼(𝐷) is called the generalized solution of the equation ℒ𝑒=𝑓(π‘₯) in 𝐷, if for any function πœ‚(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(𝐷) the integral identity ξ€œπ·π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)𝑒π‘₯π‘–πœ‚π‘₯π‘—ξ€œπ‘‘π‘₯=π·π‘“πœ‚π‘‘π‘₯(1.7) be fulfilled.

Here, 𝑓(π‘₯) is a given function from β„’2(𝐷).

Let πΈβŠ‚π· be some compact. The function 𝑒(π‘₯)βˆˆπ‘Š12,𝛼(𝐷⧡𝐸) is called generalized solution of the equation ℒ𝑒=𝑓(π‘₯) in 𝐷⧡𝐸 vanishing on πœ•π· if integral identity (1.7) is fulfilled for any function πœ‚(π‘₯)∈𝐴𝐸(𝐷).

We will assume that the coefficients of the operator β„’ continued in 𝐸𝑛⧡𝐷 with saving condition (1.2), (1.6). For this, it is sufficient, for example, to assume that π‘Žπ‘–π‘—(π‘₯)=π›Ώπ‘–π‘—πœ†π‘–(π‘₯) for π‘₯βˆˆπΈπ‘›β§΅π·, 𝑖,𝑗=1,…,𝑛, where 𝛿𝑖𝑗 is a Croneker symbol.

Let β„Ž(π‘₯)βˆˆπ‘Š12,𝛼(𝐷), 𝑓0(π‘₯)βˆˆβ„Ž2(𝐷), 𝑓𝑖(π‘₯)βˆˆβ„’2,πœ†βˆ’1(𝐷), 𝑖=1,2,…,𝑛, be a given functions. Let us consider the first boundary value problemℒ𝑒=𝑓0(π‘₯)+𝑛𝑖=1πœ•π‘“π‘–(π‘₯)πœ•π‘₯𝑖,π‘₯∈𝐷,(𝑒(π‘₯)βˆ’β„Ž(π‘₯))βˆˆβˆ˜π‘Š12,𝛼(𝐷).(1.8) The function 𝑒(π‘₯)βˆˆπ‘Š12,𝛼(𝐷) we will call generalized solution of problem (1.8) if for any function πœ‚(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(𝐷), the integral identityξ€œπ·π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)𝑒π‘₯π‘–πœ‚π‘₯π‘—ξ€œπ‘‘π‘₯=π·ξƒ©βˆ’π‘“0πœ‚+𝑛𝑖=1π‘“π‘–πœ‚π‘₯𝑖ξƒͺ𝑑π‘₯(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 β„’ condition (1.2), (1.6) are fulfilled, then the first boundary value problem (1.8) has a unique generalized solution 𝑒(π‘₯) at any β„Ž(π‘₯)βˆˆπ‘Š12,𝛼(𝐷), 𝑓0(π‘₯)βˆˆβ„Ž2(𝐷), 𝑓𝑖(π‘₯)∈𝐿2,πœ†π‘–βˆ’1(𝐷), 𝑖=1,2,…,𝑛. At this, there exists 𝑃0(𝛼,𝑛) such that if 𝑝>𝑝0, β„Ž(π‘₯)βˆˆπ‘Š1𝑝,𝛼(𝐷), 𝑓0(π‘₯)βˆˆβ„Žπ‘(𝐷), 𝑓𝑖(π‘₯)∈𝐿2,πœ†π‘–βˆ’1(𝐷), 𝑖=1,2,…,𝑛, πœ•π·βˆˆπΆ1, then solution 𝑒(π‘₯) is continuous in 𝐷.

Lemma 2.2. Let relative to the coefficients of the operator β„’ conditions (1.2), (1.6) be fulfilled. Then, any generalized solution of the equation ℒ𝑒=0 in 𝐷 is continuous by Holder at each strictly internal domain πœ•.

Lemma 2.3. Let relative to the coefficients of the operator β„’, conditions (1.2), (1.6) be fulfilled and πœ€π‘…,1<𝐷. Then, for any positive generalized solution 𝑒(π‘₯), the equation ℒ𝑒=0 in 𝐷 the Harnack inequality is true supπœ€π‘…,1(0)𝑒≀𝐢1(𝛾,𝛼,𝑛)infπœ€π‘…,1(0)𝑒.(2.1) If at this π‘¦βˆˆπœ•πœ€π‘…,2(0) and πœ€π‘…,1(0)βŠ‚π·, then the inequality of form (2.1) is true in ellipsoid πœ€π‘…,1(𝑦).

Lemma 2.4. Let relative to the coefficients of the operator β„’ conditions (1.2), (1.6) be fulfilled and 𝑒(π‘₯) generalized solution of the first boundary-value problem (1.8) at 𝑓𝑖(π‘₯)≑0, 𝑖=0,…,𝑛. Then, if β„Ž(π‘₯) is bounded on πœ•π· in the sense π‘Š12,𝛼(𝐷), then for solution 𝑒(π‘₯) the following maximum principle is true: infπœ•π·β„Žβ‰€inf𝐷𝑒≀sup𝐷≀supπœ•π·β„Ž,(2.2) where infπœ•π·β„Ž(supπœ•π·β„Ž) is an exact lower (upper) bound those numbers π‘Ž, for which β„Ž(π‘₯)β‰₯π‘Ž(β„Ž(π‘₯)≀a)  on πœ•π· in the sense π‘Š12,𝛼(𝐷).
These lemmas are proved analogously to paper [7]. Therefore, the proof of these lemmas is not given.
Let π»βŠ‚πœ€ be some compact and 𝑉𝐻 a set of all functions πœ‘(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€) such that πœ‘(π‘₯)β‰₯1 on 𝐻, in the sense βˆ˜π‘Š12,𝛼(πœ€). Let one considers the functional π½πœƒξ€œ(πœ‘)=πœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)πœ‘π‘–πœ‘π‘—π‘‘π‘₯,πœ‘(π‘₯)βˆˆπ‘‰π»,(2.3)β„’ is a 𝐻 compact capacity relative to ellipsoid πœ€ and is called the value infπœ‘βˆˆπ‘‰π»π½πœƒ(𝑒) and denoted by capβ„’(πœ€)(𝐻). In case πœ€=𝐸𝑛, the corresponding value is called β„’ capacity of the compact 𝐻 and denoted by capβ„’(𝐻).

Lemma 2.5. There exists the unique function 𝑒(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€) such that 𝑒(π‘₯)β‰₯1 on 𝐻 in the sense βˆ˜π‘Š12,𝛼(πœ€) and capβ„’(πœ€)(𝐻)=𝐽ℒ(𝑒).

Proof. It is easy to see that 𝑉𝐻 is convex closed set in βˆ˜π‘Š12,𝛼(πœ€). From the fact that βˆ˜π‘Š12,𝛼(πœ€) is a Hilbert space, it follows the existence of unique function 𝑒(π‘₯)βˆˆπ‘‰π», which achieved an exact lower bound of the functional 𝐽ℒ(πœ‘). Next, {𝑒(π‘₯)}1=𝑒(π‘₯)if𝑒(π‘₯)≀1,1if𝑒(π‘₯)>1.(2.4)
It is clear that {𝑒(π‘₯)}1βˆˆβˆ˜π‘Š12,𝛼(πœ€). Moreover, {𝑒(π‘₯)}1βˆˆπ‘‰π». Denote by 𝐴+={π‘₯∢π‘₯βˆˆπœ€,𝑒(π‘₯)>1}. We have 𝐽ℒ𝑒(π‘₯)1ξ€Ύ=ξ‚΅ξ€œπ΄++ξ€œπœ€β§΅π΄+𝑛𝑖,𝑗=1π‘Žπ‘–π‘—(π‘₯){𝑒}1𝑖{𝑒}1π‘—ξ€œπ‘‘π‘₯=πœ€β§΅π΄+𝑛𝑖,𝑗=1π‘Žπ‘–π‘—(π‘₯)𝑒𝑖𝑒𝑗𝑑π‘₯.(2.5) On the other side, according to (1.2), ξ€œπ΄+𝑛𝑖,𝑗=1π‘Žπ‘–π‘—(π‘₯)𝑒𝑖𝑒𝑗𝑑π‘₯β‰₯0.(2.6) From (2.5) and (2.6), we conclude 𝐽ℒ𝑒(π‘₯)1≀𝐽ℒ(𝑒)=infπœ‘βˆˆπ‘‰π»π½β„’(πœ‘),(2.7) that is, 𝐽ℒ{𝑒(π‘₯)1}=𝐽ℒ(𝑒). From uniqueness extreme function, it follows that {𝑒(π‘₯)}1=𝑒(π‘₯), and lemma is proved.
The function 𝑒(π‘₯), which achieved an exact lower bound of the functional 𝐽ℒ(πœ‘) on the set 𝑉𝐻 is called β„’ capacity of the compact potential 𝐻 relative to the ellipsoid πœ€.

Lemma 2.6. Let β„’ be a capacity potential 𝑒(π‘₯) of the compact 𝐻 relative to πœ€ which is a generalized solution of the equation ℒ𝑒=0 in πœ€β§΅π», vanishing on 0 and πœ•πœ€ in 1 on πœ•π» in the sense π‘Š12,𝛼(πœ€).

Proof. It is sufficient to show the truth of the first part of assertion of lemma. Let πœ‚(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€) and πœ‚(π‘₯)β‰₯0 on 𝐻 in the sense βˆ˜π‘Š12,𝛼(πœ€). Then, for any πœ€>0(𝑒(π‘₯)+πœ€πœ‚(π‘₯))βˆˆπ‘‰π». Therefore, 𝐽ℒ(𝑒+πœ€πœ‚)β‰₯𝐽ℒ(𝑒).(2.8) Thus, 𝐽ℒ(𝑒)+πœ€2π½β„’ξ€œ(πœ‚)+2πœ€πœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‚π‘—π‘‘π‘₯β‰₯𝐽ℒ(𝑒),(2.9) that is, π½β„’ξ€œ(𝑒)+2πœ€πœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‚π‘—π‘‘π‘₯β‰₯0.(2.10) Tending πœ€ to zero, we conclude ξ€œπœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‚π‘—π‘‘π‘₯β‰₯0.(2.11) It is easy to see as πœ‚(π‘₯) in (2.11), we can take any function from 𝐢1(πœ€) with compact support in πœ€β§΅π». Then, ξ€œπ‘›πœ€β§΅π»ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‚π‘—π‘‘π‘₯β‰₯0.(2.12) Substituting πœ‚(π‘₯) on βˆ’πœ‚(π‘₯), we arrive to ξ€œπ‘›πœ€β§΅π»ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‚π‘—π‘‘π‘₯=0.(2.13)
Lemma is proved.

Let πœ‡ be a charge of bounded variation, given on πœ€. We will say that the function 𝑒(π‘₯)∈𝐿1(πœ€) is a weak solution of the equation ℒ𝑒=βˆ’πœ‡, equaling to zero on πœ•πœ€ if for any function πœ‘(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€)∩𝐢(πœ€), the integral identityξ€œπœ€ξ€œπ‘’β„’πœ‘π‘‘π‘₯=πœ€πœ‘π‘‘πœ‡(2.14) is fulfilled.

According to Lemma 2.1 (at β„Ž=0), there exists the continuous linear operator 𝐻 from βˆ—π‘Š12,𝛼(πœ€) in βˆ˜π‘Š12,𝛼(πœ€) such that for any functional π‘‡βˆˆβˆ—π‘Š12,𝛼(πœ€), the function 𝑒=𝐻(𝑇) is unique in βˆ˜π‘Š12,𝛼(πœ€) generalized solution of the equation ℒ𝑒=𝑇.

The operator 𝐻 is called Green operator.

By Lemma 2.1, this operator at 𝑝>𝑝0 we transform βˆ—π‘Š12,𝛼(πœ€) to 𝐢(πœ€). It is easy to see that the function 𝑒(π‘₯) is weak solution of the equation ℒ𝑒=βˆ’πœ‡, equaling to zero on πœ•πœ€ if and only if for any function πœ“(π‘₯)∈𝐢(πœ€) the integral identityξ€œπœ€ξ€œπ‘’πœ“π‘‘π‘₯=πœ€π»(πœ“)π‘‘πœ‡(2.15) is fulfilled.

By analogy with [8], we can show that for each measure πœ‡ on πœ€, there exists the unique weak solution of the equation ℒ𝑒=βˆ’πœ‡ equaling to zero on πœ•πœ€.

Let us say that the charge πœ‡βˆˆβˆ—π‘Š12,𝛼(πœ€) if there exists the vector 𝑓(π‘₯)=(π‘“βˆ˜(π‘₯),𝑓1(π‘₯),…,𝑓𝑛(π‘₯))𝑓0(π‘₯)βˆˆβ„Ž2(πœ€), 𝑓𝑖(π‘₯)∈𝐿2,πœ†π‘–(πœ€), 𝑖=1,2,…,𝑛, for any function πœ‘(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€)∩𝐢(πœ€), the integral identityξ€œπœ‡(πœ‘)=πœ€ξ€œπœ‘π‘‘πœ‡=πœ€ξƒ©π‘“βˆ˜πœ‘βˆ’π‘›ξ“π‘–=1π‘“π‘–πœ‘π‘–ξƒͺ𝑑π‘₯(2.16) is true.

At this, it is evident that||||||ξ€œπœ€||||||πœ‘π‘‘πœ‡β‰€πΆ2ξ‚€π‘“ξ‚β€–πœ‘β€–π‘Š12,𝛼(πœ€).(2.17)

Lemma 2.7. The weak solution 𝑒(π‘₯) of the equation ℒ𝑒=βˆ’πœ‡, equaling to zero on πœ•πœ€, belongs to βˆ˜π‘Š12,𝛼(πœ€), if and only if πœ‡βˆˆβˆ—π‘Š12,𝛼(πœ€).

Proof. At first, we will show that if the function πœ‘(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€) satisfies the integral identity ξ€œπœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‘π‘—ξ€œπ‘‘π‘₯=βˆ’πœ€πœ‘π‘‘πœ‡,(2.18) for any function πœ‘(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€)∩𝐢(πœ€)zero on πœ•πœ€, then it is weak solution of the equation ℒ𝑒=βˆ’πœ‡, equaling to zero on πœ•πœ€. Really, assuming that πœ‘=𝐻(πœ“), πœ“(π‘₯)∈𝐢(πœ€), we obtain ξ€œπœ€ξ€œπ»(πœ“)π‘‘πœ‡=πœ€ξ€œπœ‘π‘‘πœ‡=βˆ’πœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‘π‘—ξ€œπ‘‘π‘₯=πœ€π‘’π‘›ξ“π‘–,𝑗=1ξ€·π‘Žπ‘–π‘—(π‘₯)πœ‘π‘—ξ€Έπ‘–=ξ€œπ‘‘π‘₯πœ€ξ€œπ‘’β„’πœ‘π‘‘π‘₯=πœ€π‘’πœ“π‘‘π‘₯,(2.19) and now, it is sufficient to use the identity (2.15). We will show that πœ‡βˆˆβˆ—π‘Š12,𝛼(πœ€). For this, it is sufficient to prove that if π‘“π‘–βˆ‘(π‘₯)=𝑛𝑖=1π‘Žπ‘–π‘—(π‘₯)𝑒𝑖(π‘₯), then 𝑓𝑖(π‘₯)∈𝐿2,πœ†π‘–βˆ’1(πœ€), 𝑖=1,2,…,𝑛,. Assume in condition (2.18) that πœ‰1=β‹―=πœ‰π‘–βˆ’1=πœ‰π‘–+1=β‹―=πœ‰π‘›=0, πœ‰π‘–βˆš=1/πœ†π‘–(π‘₯).
We will obtain π‘Žπ›Ύβ‰€π‘–π‘–(π‘₯)πœ†π‘–(π‘₯)β‰€π›Ύβˆ’1;𝑖=1,…,𝑛.(2.20)
Let 𝑖≠𝑗. Assuming that πœ‰π‘˜=0 at π‘˜β‰ π‘— and π‘˜β‰ π‘–, πœ‰π‘–βˆš=1/πœ†π‘–(π‘₯), πœ‰π‘—βˆš=1/πœ†π‘—(π‘₯), we will obtain π‘Ž2𝛾≀𝑖𝑖(π‘₯)πœ†π‘–+π‘Ž(π‘₯)𝑗𝑗(π‘₯)πœ†π‘—+(π‘₯)2π‘Žπ‘–π‘—(π‘₯)βˆšπœ†π‘–(π‘₯)πœ†π‘—(π‘₯)≀2π›Ύβˆ’1.(2.21) Using (2.20), we conclude ||π‘Žπ‘–π‘—||(π‘₯)βˆšπœ†π‘–(π‘₯)πœ†π‘—(π‘₯)β‰€π›Ύβˆ’1βˆ’π›Ύ;𝑖,𝑗=1,…,𝑛;𝑖≠𝑗.(2.22) From (2.20) and (2.22), it follows that ||π‘Žπ‘–π‘—||(π‘₯)βˆšπœ†π‘–(π‘₯)πœ†π‘—(π‘₯)β‰€π›Ύβˆ’1;𝑖,𝑗=1,…,𝑛.(2.23) Thus, from (2.23), take out for 𝑗=1,…,π‘›ξ€œπœ€1πœ†π‘—ξ€·π‘“(π‘₯)𝑗2ξ€œπ‘‘π‘₯=πœ€1πœ†π‘—ξƒ©(π‘₯)𝑛𝑖=1π‘Žπ‘–π‘—(π‘₯)𝑒𝑖ξƒͺ2𝑑π‘₯β‰€π›Ύβˆ’2𝑛𝑛𝑖=1ξ€œπœ€πœ†π‘–(π‘₯)𝑒2𝑖𝑑π‘₯<𝛼.(2.24) So, πœ‡βˆˆβˆ—π‘Š12,𝛼(πœ€). Inversely, if 𝑒(π‘₯) is a weak solution of the equation ℒ𝑒=βˆ’πœ‡, vanishing on πœ•πœ€, then there exists πœ‡βˆˆβˆ—π‘Š12,𝛼(πœ€) such that ξƒ©π‘“βˆ˜πœ‘βˆ’π‘›ξ“π‘–=1π‘“π‘–πœ‘π‘–ξƒͺξ€œπ‘‘π‘₯=πœ€ξ€œπœ‘π‘‘πœ‡=πœ€ξ€œπ‘’β„’πœ‘π‘‘π‘₯=πœ€π‘’π‘›ξ“π‘–,𝑗=1ξ€·π‘Žπ‘–π‘—(π‘₯)πœ‘π‘—ξ€Έπ‘–ξ€œπ‘‘π‘₯=βˆ’πœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‘π‘—π‘‘π‘₯,(2.25) for any function πœ‘(π‘₯)βˆˆβˆ—π‘Š12,𝛼(πœ€)∩𝐢(πœ€), β„’πœ‘(π‘₯)∈𝐢(πœ€).
Then, from Lemma 2.1, we obtain that 𝑒(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€). The lemma is proved.

Let now 𝛿(π‘₯) be Dirac measure, concentrated at the point 0, and 𝑦 an arbitrary fixed point πœ€.

The weak solution 𝑔(π‘₯,𝑦) of the equation ℒ𝑦=βˆ’π›Ώ(π‘₯βˆ’π‘¦), vanishing on πœ•πœ€, is called Green function of the operator β„’ in πœ€.

In case πœ€=𝐸𝑛 the corresponding function is called the fundamental solution of the operator β„’ and denoted by 𝐺(π‘₯,𝑦).

According to the above proved, if πœ“(π‘₯) is an arbitrary function from 𝐢(πœ€), then the generalized solution πœ‘(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€) of the equation β„’πœ‘=βˆ’πœ“ can be introduced in the following form:ξ€œπœ‘(𝑦)=πœ€π‘”(π‘₯,𝑦)πœ“(π‘₯)𝑑π‘₯.(2.26) We can show that 𝑔(π‘₯,𝑦) is nonnegative in πœ€Γ—πœ€ moreover; 𝑔(π‘₯,𝑦)=𝑔(𝑦,π‘₯).

Lemma 2.8. For any charge of bounded variation on πœ€, the integral ξ€œπ‘’(π‘₯)=πœ€π‘”(π‘₯,𝑦)π‘‘πœ‡(𝑦)(2.27) exists, finite a.e. in πœ€ and is weak solution of the equation ℒ𝑒=βˆ’πœ‡, equaling to zero on πœ•πœ€.

Proof. Without losing generality, we will assume that the charge πœ‡ is the measure in πœ€. Let πœ‘(π‘₯)∈𝐢(πœ€), πœ“(π‘₯)β‰₯0 in πœ€. Denote by πœ‘(π‘₯)βˆˆβˆ˜π‘Š12,𝛼(πœ€) the generalized solution of the equation β„’πœ‘=βˆ’πœ“(π‘₯). Then, πœ‘(π‘₯)∈𝐢(πœ€) according to Lemma 2.1 and πœ“(π‘₯)β‰₯0 according to Lemma 2.4. At this, ξ€œπœ‘(𝑦)=πœ€π‘”(π‘₯,𝑦)πœ“(π‘₯)𝑑π‘₯.(2.28) Then, by the Fubini theorem, we conclude that the integral βˆ«πœ€π‘”(π‘₯,𝑦)π‘‘πœ‡(𝑦) there exists for almost all π‘₯βˆˆπœ€; moreover, ξ€œπœ€ξ€œπ»(πœ“)π‘‘πœ‡(𝑦)=πœ€ξ€πœ‘(𝑦)π‘‘πœ‡(𝑦)=πœ€Γ—πœ€ξ€œπ‘”(π‘₯,𝑦)πœ“(π‘₯)𝑑π‘₯π‘‘πœ‡(𝑦)=πœ€πœ“(π‘₯)𝑒(π‘₯)𝑑π‘₯.(2.29)
Let us note that (2.29) is fulfilled for weak nonnegative and continuous in πœ€ function πœ“(π‘₯). Now, it is sufficient to remember the identity (2.15) and lemma is proved.
Let us consider now β„’-capacity of the potential 𝑒(π‘₯) of the compact 𝐻 relative to the ellipsoid πœ€. Before, it was proved that 𝑒(π‘₯) satisfies (2.11) at any nonnegative on 𝐻 the function πœ‚(π‘₯)∈𝐢∞0(πœ€). By the Schwartz theorem [9, 10], there exists the measure πœ‡ on 𝐻 such that ξ€œπœ€π‘›ξ“π‘–,𝑗=1π‘Žπ‘–π‘—(π‘₯)π‘’π‘–πœ‚π‘—ξ€œπ‘‘π‘₯=πœ€πœ‚π‘‘πœ‡.(2.30) Further, since 𝑒=1 on 𝐻 in the sense βˆ˜π‘Š12,𝛼(πœ€), then the carrier of the measure πœ‡ is situated on πœ•π». The measure πœ‡ is called β„’-capacity of the compact 𝐻.
According to Lemma 2.8,  ℒ-capacity potential 𝑒(π‘₯) is weak solution of the equation ℒ𝑒=βˆ’πœ‡, equaling to zero on πœ•πœ€ and can be represented in the following form: ξ€œπ‘’(π‘₯)=πœ€π‘”(π‘₯,𝑧)π‘‘πœ‡(𝑧).(2.31) On the other side, there exists the sequence of the functions {πœ‚(π‘š)(π‘₯)}; π‘š=1,2,… such that πœ‚(π‘š)(π‘₯)βˆˆπ”…(πœ€), πœ‚(π‘š)(π‘₯)=1 for π‘₯∈𝐻 and limπ‘šβ†’βˆžβ€–πœ‚(π‘š)βˆ’π‘’β€–π‘Š12,𝛼(πœ€)=0. Assuming in (2.15) πœ‚(π‘š)(π‘₯) instead of πœ‚(π‘š), we conclude that it first fart is equal to πœ‡(𝐻) at any natural π‘š, while the left part tends to capβ„’(πœ€)(𝐻) as π‘šβ†’βˆž. Thus, capβ„’(πœ€)(𝐻)=πœ‡(𝐻).(2.32)

Lemma 2.9. Let relative to coefficients of the operator β„’ conditions (1.2)–(1.6), π‘¦βˆˆπœ•πœ€π‘…,2(0), πœ€π‘…,1(0)βŠ‚π·, π‘₯βˆˆπœ•πœ€π‘…,1(𝑦) be fulfilled. Then, for the Green function 𝑔(π‘₯,𝑦) the following estimations are true: 𝐢3(𝛾,𝛼,𝑛)capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έξ‚„(𝑦)βˆ’1≀𝑔(π‘₯,𝑦)≀𝐢4(𝛾,𝛼,𝑛)capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έξ‚„(𝑦)βˆ’1.(2.33) If πœ€π‘…,1(0)βŠ‚π·, π‘₯βˆˆπœ•πœ€π‘…,1(0) then, 𝐢3capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έξ‚„(0)βˆ’1≀𝑔(π‘₯,0)≀𝐢4capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έξ‚„(0)βˆ’1.(2.34)

Proof. Without loss of generality, we can assume that the coefficients of the operator β„’ are continuously differentiable in πœ€. The general case is obtained by means of limit passage. Then, at π‘₯≠𝑦, the function 𝑔(π‘₯,𝑦) is continuous by π‘₯ and 𝑦 moreover, limπ‘₯→𝑦𝑔(π‘₯,𝑦)=∞.(2.35)
Let π‘Ž be a positive number, which will be chosen later, πΎπ‘Ž={π‘₯βˆΆπ‘”(π‘₯,𝑦)β‰₯π‘Ž}, where 𝑦 is an arbitrary fixed point on πœ•πœ€π‘…,2(0). From (2.35), it follows that 𝑦 is internal point 𝑦 of the compact πΎπ‘Ž. Then, β„’ is capacity potential πΎπ‘Ž, represented in the form (2.31), so it means that it equal to zero in it. Thus, ξ€œ1=πœ€π‘¦(𝑦,𝑧)π‘‘πœ‡π‘Ž(𝑧),(2.36) where πœ‡ is a β„’-capacity distribution of the compact πΎπ‘Ž. Allowing for the carrier of the measure πœ‡π‘Ž is situated on πœ•πΎπ‘Ž, where 𝑔(𝑦,𝑧) = π‘Žand using (2.32), we obtain πœ‡π‘Žξ€·πΎπ‘Žξ€Έ=capβ„’(πœ€)ξ€·πΎπ‘Žξ€Έ=1π‘Ž.(2.37) Let us assume now that π‘Ž=infπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔(π‘₯,𝑦). According to maximum principle, πœ€π‘…,1(𝑦)βŠ‚πΎπ‘Ž. Therefore, from (2.37), we conclude capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έ(𝑦)≀capβ„’(πœ€)ξ€·πΎπ‘Žξ€Έ=1infπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔.(π‘₯,𝑦)(2.38) If we will assume 𝑏=supπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔(π‘₯,𝑦), then πœ€π‘…,1(𝑦)βŠ‚πΎπ‘Ž; that is, capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έ(𝑦)≀capβ„’(πœ€)𝐾𝑏=1supπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔.(π‘₯,𝑦)(2.39) From (2.38) and (2.39), follows that infπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔(π‘₯,𝑦)≀capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έξ‚„(𝑦)βˆ’1≀supπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔(π‘₯,𝑦).(2.40) On the other side, according to Lemma 2.3, supπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔(π‘₯,𝑦)≀𝐢5(𝛾,𝛼,𝑛)infπ‘₯βˆˆπœ•πœ€π‘…,1(𝑦)𝑔(π‘₯,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 π‘¦βˆˆπœ•πœ€π‘…,2(0) be fulfilled, πœ€π‘…,1(0)βŠ‚π·, π‘₯βˆˆπœ•πœ€π‘…,1(0) or 𝑦=0, πœ€π‘…,1(0)βŠ‚π·, π‘₯βˆˆπœ•πœ€π‘…,1(0). Then, for fundamental solution 𝐺(π‘₯,𝑦), the estimations 𝐢3capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έξ‚„(0)βˆ’1≀𝐺(π‘₯,𝑦)≀𝐢4capβ„’(πœ€)ξ€·πœ€π‘…,1ξ€Έξ‚„(0)βˆ’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 β„’, conditions (1.2)–(1.6) be fulfilled. Then, for removability of the compact πΈβŠ‚π· relative to the first boundary value problem for the operator β„’ in the space β„³(𝐷), it is necessary and sufficient that capβ„’(𝐸)=0.(3.1)

Proof. Let the ellipsoid πœ€ have the same sense as above. It is easy to see that if condition (3.1) is fulfilled, then capβ„’(πœ€)(𝐸)=0.(3.2) Not losing generality, we can limit the case, when the coefficients of the operator β„’ is continuously differentiable in πœ€. Let us fix an arbitrary πœ€>0 and π‘₯0βŠ‚π·β§΅πΈ. By virtue of (3.1), there exists the neighbourhood 𝐻 of the compact 𝐸 such that capβ„’(πœ€)𝐻<πœ€.(3.3) At this, we can assume that πœ€ is such small that ξ‚€π‘₯dist0,𝐻β‰₯12ξ€·π‘₯dist0ξ€Έ.,𝐸(3.4) Denote by 𝑉𝐻(π‘₯) and πœ‡π» the β„’-capacity potential of the compact 𝐻 relative to the ellipsoid πœ€ and β„’-capacity of the distribution 𝐻, respectively. According to above proved, 𝑉𝐻(ξ€œπ‘₯)=πœ€π‘”(π‘₯,𝑦)π‘‘πœ‡π»(𝑦),(3.5) moreover, the function 𝑉𝐻(π‘₯) is the generalized solution of the equation ℒ𝑉𝐻=0 in πœ€β§΅π», vanishing on 0 and in πœ•πœ€ on 1 in πœ•π» in the sense π‘Š12,𝛼(πœ€). Let now 𝑒(π‘₯)βˆˆβ„³(𝐷) be an arbitrary solution of the equation ℒ𝑒=0 in 𝐷⧡𝐸, vanishing on πœ•π·, 𝑀=sup𝐷|𝑒|. It is easy to see that the function 𝑉𝐻(π‘₯) is nonnegative on πœ•π·, in the sense π‘Š12,𝛼(𝐷). Hence, it follows, that the function 𝑒(π‘₯)βˆ’π‘€π‘‰π»(π‘₯) is generalized solution of the equation ℒ𝑒=0 in 𝐷⧡𝐻, is non-positive on πœ•(𝐷⧡𝐻). According to Lemma 2.4,  𝑒(π‘₯)βˆ’π‘€π‘‰π»(π‘₯)≀0 and 𝐷⧡𝐻 in particular 𝑒π‘₯0≀𝑀𝑉𝐻π‘₯0≀𝑀supπ‘¦βˆˆπœ•π»π‘”ξ€·π‘₯0ξ€Έπœ‡,𝑦𝐻𝐻=𝑀supπ‘¦βˆˆπœ•π»π‘”ξ€·π‘₯0ξ€Έ,𝑦capβ„’(πœ€)𝐻.(3.6) By virtue of continuity of the function 𝑔(π‘₯,𝑦) at π‘₯≠𝑦 and (3.4), we obtain supπ‘¦βˆˆπœ•π»π‘”ξ€·π‘₯0ξ€Έ,𝑦≀𝐢6𝛾,𝛼,𝑛,π‘₯0ξ€Έ.,𝐸(3.7) Thus, from (3.3) and (3.6), we conclude 𝑒π‘₯0≀𝑀𝐢6πœ€.(3.8) Using an arbitrariness πœ€, we have 𝑒π‘₯0≀0.(3.9) Making analogous considerations with the function 𝑒(π‘₯)+𝑀𝑉𝐻(π‘₯), we obtain 𝑒π‘₯0ξ€Έβ‰₯0.(3.10) From (3.8)-(3.9) and an arbitrariness of the point π‘₯0 it follows that 𝑒(π‘₯)≑0 in 𝐷⧡𝐸. Thereby, the sufficiency of condition (3.1) is proved. Let us prove its necessity. Let us assume that capβ„’(𝐸)>0. Denote by πœ€ξ…ž the ellipsoid such that πœ€ξ…žβŠ‚π›Ώ, πΈβŠ‚πœ€ξ…ž. Assume 𝐷=πœ€β€‰β€‰potential of the compact 𝐸 relative to the ellipsoid πœ€ξ…ž and β„’-capacity distribution 𝐸, respectively. Following to [11], we can give the equivalent definition of Vallee-Poussin type of β„’-capacity of the compact 𝐸, relative to the ellipsoid πœ€ξ…ž. Let 𝑔(π‘₯,𝑦) be a Green function of the operator β„’ in πœ€ξ…ž. Let us call the measure πœ‡ on 𝐸, β„’-admissible if πœ‡βŠ‚πΈ and π‘‰πΈπœ‡(ξ€œπ‘₯)=πœ€β€²π‘”(π‘₯,𝑦)π‘‘πœ‡(𝑦)≀1forπ‘₯∈supπ‘πœ‡.(3.11) The value supπœ‡(𝐸)=cap(πœ€β€²)β„’(𝐸), where an exact upper boundary is taken on all β„’-admissible measures and is called β„’-capacity of the compact 𝐸 relative to the ellipsoid πœ€ξ…ž.
Analogously, the β„’-capacity capβ„’(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 πœ‡(𝐸) is reached by the set of all β„’-admissible measures πœ‡. This measure is β„’-capacity distribution of the compact 𝐸.
According to the above proved, the function 𝑒𝐸(π‘₯) is generalized solution of the equation ℒ𝑒𝐸=0 in πœ€ξ…žβ§΅πΈ, equaling to zero on πœ•πœ€ξ…ž. Besides, from (3.10) and maximum principle, it follows that 𝑒𝐸(π‘₯)βˆˆπ‘€(πœ€ξ…ž). On the other side, 𝑒𝐸(π‘₯)β‰’0, as 𝑉𝐻(𝐸)>0. Theorem is proved.

Lemma 3.2. Let relative to the coefficients of the operator β„’ condition (1.2) be fulfilled. Then, if π‘¦βˆˆπœ•πœ€π‘…,2(0), then 𝐢7(𝛾,𝛼,𝑛)𝑅𝑛+(βŸ¨π›ΌβŸ©/2)βˆ’2≀capβ„’ξ€·πœ€π‘…,1ξ€Έ(𝑦)≀𝐢8(𝛾,𝛼,𝑛)𝑅𝑛+(βŸ¨π›ΌβŸ©/2)βˆ’2.(3.12)

Proof. Let β„’0=π‘›βˆ‘π‘–=1(πœ•/πœ•π‘₯𝑖)(πœ†π‘–(π‘₯)(πœ•/πœ•π‘₯𝑖)). Then, according to (1.2), 𝛾capβ„’0ξ€·πœ€π‘…,1ξ€Έ(𝑦)≀capβ„’ξ€·πœ€π‘…,1ξ€Έ(𝑦)β‰€π›Ύβˆ’1capβ„’0ξ€·πœ€π‘…,1ξ€Έ.(𝑦)(3.13) Let 𝑒(π‘₯)∈𝐢∞0(πœ€π‘…,(3/2)(𝑦)), 𝑒(π‘₯)=1 for πœ€π‘…,1(𝑦); moreover, ||𝑒𝑖(||≀𝐢π‘₯)9(πœ†,𝑛)𝑅1+(𝛼𝑖/2);𝑖=1,…,𝑛.(3.14) Then, capβ„’0ξ€·πœ€π‘…,1ξ€Έβ‰€ξ€œ(𝑦)πœ€π‘…,(3/2)𝑛(𝑦)𝑖=1πœ†π‘–(π‘₯)𝑒2𝑖𝑑π‘₯.(3.15) On the other side, as π‘¦βˆˆπœ•πœ€π‘…,2(0), then βˆ‘π‘›π‘–=1(𝑦2𝑖/𝑅𝛼𝑖)=4𝑅2, and thereby ||𝑦𝑖||≀2𝑅1+(𝛼𝑖/2);𝑖=1,…,𝑛.(3.16) Besides, as π‘₯βˆˆπœ€π‘…,(3/2)(𝑦), then ||π‘₯π‘–βˆ’π‘¦π‘–||≀32𝑅1+(𝛼𝑖/2);𝑖=1,…,𝑛.(3.17) Thus, ||π‘₯𝑖||≀||𝑦𝑖||+||π‘₯π‘–βˆ’π‘¦π‘–||≀72𝑅1+(𝛼𝑖/2);𝑖=1,…,𝑛.(3.18) Hence, it follows that |π‘₯|𝛼≀𝑅𝑛𝑖=1𝑧22/(2+πœ†π‘–).(3.19) Therefore, πœ†π‘–(π‘₯)≀𝐢𝛼𝑖10𝑅𝛼𝑖≀𝐢𝛼+10𝑅𝛼𝑖;𝑖=1,…,𝑛.(3.20) where 𝛼+=max{𝛼1,…,𝛼𝑛}.
Allowing for (3.14) and (3.20) in (3.15), we obtain capβ„’0ξ€·πœ€π‘…,1ξ€Έ(𝑦)≀𝐢10(𝛼,𝑛)π‘…βˆ’2ξ€·πœ€mes𝑅,(3/2)ξ€Έ(𝑦)=𝐢11(𝛼,𝑛)𝑅𝑛+(βŸ¨π›ΌβŸ©/2)βˆ’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 capβ„’0ξ€·πœ€π‘…,1ξ€Έ(𝑦)β‰₯capβ„’0ξ‚€πœ€βˆšπ‘…,(1/2𝑛)(𝑦).(3.22) Besides, considering the same as in [8], we conclude capβ„’0ξ‚€πœ€βˆšπ‘…,(1/2𝑛)(𝑦)β‰₯𝐢12(𝛼,𝑛)cap(πœ€0)β„’0ξ‚€πœ€βˆšπ‘…,(1/2𝑛)(𝑦),(3.23) where πœ€0=πœ€βˆšπ‘…,(1/𝑛)(𝑦).
Let π‘Š={𝑒(π‘₯)βˆΆπ‘’(π‘₯)𝐢∞0(πœ€0),𝑒(π‘₯)=1forπ‘₯βˆˆπœ€βˆšπ‘…,(1/2𝑛)(𝑦)}. Then, cap(πœ€0)β„’0ξ‚€πœ€βˆšπ‘…,(1/2𝑛)(𝑦)=infπ‘’βˆˆπ‘Šξ€œπœ€0𝑛𝑖=1πœ†π‘–(π‘₯)𝑒2𝑖𝑑π‘₯.(3.24) On the other side, if π‘¦βˆˆπœ•πœ€π‘…,2(0), then we can find 𝑖0, 1≀𝑖0≀𝑛 such that 𝑦2𝑖0β‰₯4𝑅2+𝛼𝑖0/𝑛, that is; ||𝑦𝑖0||β‰₯4𝑅1+(𝛼𝑖0/2)βˆšπ‘›.(3.25) Besides, as π‘₯βˆˆπœ€0, then ||π‘₯𝑖0βˆ’π‘¦π‘–0||≀𝑅1+(𝛼𝑖0/2)βˆšπ‘›.(3.26) Therefore, ||π‘₯𝑖0||β‰₯||𝑦𝑖0||βˆ’||π‘₯𝑖0βˆ’π‘¦π‘–0||β‰₯𝑅1+(𝛼𝑖0/2)βˆšπ‘›.(3.27) Thereby, πœ†π‘–(π‘₯)β‰₯π‘›βˆ’1/(2+𝛼𝑖0)𝑅;𝑖=1,…,𝑛,(3.28) where π›Όβˆ’=min{𝛼1,…,𝛼𝑛}.
Allowing for (3.28) in (3.24), we obtain cap(πœ€0)β„’0ξ‚€πœ€βˆšπ‘…,(1/2𝑛)(𝑦)=𝐢13(𝛼,𝑛)infπ‘’βˆˆπ‘Šξ€œπœ€0𝑛𝑖=1𝑅𝛼𝑖𝑒2𝑖𝑑π‘₯.(3.29) Denote by 𝐡𝑅(𝑧) the ball {π‘₯∢|π‘₯βˆ’π‘§|<𝑅}. Let us make in (3.30) the substitution of the variables 𝑣𝑖=π‘₯𝑖/𝑅1+(𝛼𝑖/2); 𝑖=1,…,𝑛, and let ̃𝑦 be an image of the point 𝑦, where ξ‚‹π‘Š={̃𝑒(𝑣)βˆΆΜƒπ‘’(𝜏)𝐢∞0(𝐡0),̃𝑒(𝜏)=1forπ‘£βˆˆπ΅βˆš(1/2𝑛)(̃𝑦)}. Then, from (3.30), we deduce 𝐡0=𝐡√(1/2𝑛)(̃𝑦) where by (3.30), cap(πœ€0)β„’0ξ‚€πœ€βˆšπ‘…,(1/2𝑛)(𝑦)β‰₯𝐢13𝑅𝑛+(βŸ¨π›ΌβŸ©/2)βˆ’2infΜƒπ‘’βˆˆξ‚‹π‘Šξ€œπ΅0𝑛𝑖=1ξ‚΅πœ•Μƒπ‘’πœ•π‘£π‘–ξ‚Ά2π‘‘πœ=𝐢13𝑅𝑛+(βŸ¨π›ΌβŸ©/2)βˆ’2cap(𝐡0)ξ‚€π΅βˆš(1/2𝑛)(,̃𝑦)(3.30) we will denote by cap(𝐡0)(𝐡√(1/2𝑛)(̃𝑦)) Wiener capacity of the compact 𝐡√(1/2𝑛)(̃𝑦), relative to the ball 𝐡0. Now, it is sufficient to note that cap(𝐡0)(𝐡√(1/2𝑛)(̃𝑦))=𝐢14(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 β„’ condition (1.2) be fulfilled. Then, 𝐢15(𝛾,𝛼,𝑛)𝑅𝑛+(βŸ¨π›ΌβŸ©/2)βˆ’2≀capβ„’ξ€·πœ€π‘…,1ξ€Έ(𝑦)≀𝐢16(𝛾,𝛼,𝑛)𝑅𝑛+(βŸ¨π›ΌβŸ©/2)βˆ’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 πœ€π‘…,(1/4)(𝑦)βŠ‚πœ€π‘…,1(0), that is, capβ„’ξ€·πœ€π‘…,(1/4)𝑦<capβ„’ξ€·πœ€π‘…,1ξ€Έ,(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) π‘¦βˆˆπœ•πœ€π‘…,2(0) be fulfilled, then for any 𝜌∈(0,𝑅], the estimation capβ„’ξ€·πœ€πœŒ,1𝑦≀𝐢17(𝛾,𝛼,𝑛)πœŒπ‘›+(βŸ¨π›ΌβŸ©/2)βˆ’21+𝑛𝑖=1ξ‚΅π‘…πœŒξ‚Άπ›Όπ‘–ξƒͺ(3.33) is true.
Then, 𝑣(π‘₯)∈𝐢∞0(πœ€πœŒ,(3/2)(𝑦)), 𝑣(π‘₯)=1 for π‘₯βˆˆπœ€πœŒ,1(𝑦)||𝑣𝑖(||≀𝐢π‘₯)18(𝛼,𝑛)𝜌1+(𝛼𝑖/2);𝑖=1,…,𝑛,capβ„’0ξ€·πœ€πœŒ,1𝑦=π›Ύβˆ’1𝐢218πœŒβˆ’2ξ€œπœ€πœŒ,(3/2)𝑛(𝑦)𝑖=1πœ†π‘–(π‘₯)πœŒβˆ’π›Όπ‘–π‘‘π‘₯.(3.34) On the other side, arguing the same, as well as in the proof of Lemma 3.2, we have πœ†π‘–(π‘₯)<𝐢19(𝛼,𝑛)(𝑅+𝜌)𝛼𝑖,π‘₯βˆˆπœ€πœŒ,(3/2)(𝑦);𝑖=1,…,𝑛.(3.35) Now, it is sufficient to take into account that 𝑛𝑖=1𝑅1+πœŒξ‚Άπ›Όπ‘–β‰€π‘›ξ“π‘–=1𝑅1+πœŒξ‚Άπ›Όπ‘–ξ‚Ήξƒ©β‰€π‘›1+𝑛𝑖=1ξ‚΅π‘…πœŒξ‚Άπ›Όiξƒͺ,(3.36) and from (3.34)-(3.35) follows the required estimation (3.33).

Corollary 3.5. If conditions (1.2)– (1.6) 𝑦≠0 are fulfilled, then at π‘₯βˆˆπœ€π‘‘|𝑦|𝑑,1(𝑦), π‘₯≠𝑦 for the fundamental solution 𝐺(π‘₯,𝑦), the estimation 𝐺(π‘₯,𝑦)β‰₯𝐢20(ξ€·||||𝛾,𝛼,𝑛)π‘₯βˆ’π‘¦π›Όξ€Έ2βˆ’π‘›βˆ’(βŸ¨π›ΌβŸ©/2)1+π‘›βˆ‘π‘–=1ξ€·||𝑦||𝛼/||||π‘₯βˆ’π‘¦π›Όξ€Έπ›Όπ‘–(3.37) is true.
If 𝑦=0, then estimation (3.37) is true for all π‘₯β‰ 0. Here, 𝑑=1/𝑛22/(2+𝛼).
For proving, at first, let us show that if 𝑦≠0, then π‘¦βˆ‰πœ€π‘‘|𝑦|𝑑,2(0). Really, as ||𝑦||𝛼=𝑛𝑖=1||𝑦𝑖||2/(2+𝛼𝑖),(3.38) then there exists 𝑖0, 1≀𝑖0≀𝑛 such that ||𝑦0||2/(2+𝛼𝑖0)β‰₯||𝑦||𝛼𝑛.(3.39) Thus, ||𝑦2𝑖0||ξ€·||𝑦||𝛼𝛼𝑖0β‰₯ξ€·||𝑦||𝛼2𝑛2+𝛼𝑖.(3.40) Thereby, 𝑛𝑖=1𝑦2𝑖𝑑||𝑦||𝛼𝛼𝑖β‰₯𝑦2𝑖0𝑑||𝑦||𝛼𝛼𝑖0β‰₯𝑑||𝑦||𝛼2(𝑑𝑛)2+𝛼𝑖0=4𝑑||𝑦||𝛼2ξ€·22/(2+𝛼𝑖0)𝑑𝑛2+𝛼𝑖0.(3.41) Now, it is sufficient to note that 22/(2+𝛼𝑖0)𝑑𝑛≀22/(2+𝛼)𝑑𝑛=1, and the required assertion is proved. On the other side from (3.38), it follows that for all 𝑖, 1≀𝑖≀𝑛||𝑦𝑖||2/(2+𝛼𝑖)≀||𝑦||𝛼,(3.42) that is, 𝑛𝑖=1𝑦2𝑖||𝑦||𝛼𝛼𝑖||𝑦||≀𝑛𝛼2.(3.43) So, one will show that πœ€βˆš|𝑦|𝛼,𝑛(0) if only 𝑦≠0.
Let now for 𝑦≠0, π‘₯βˆˆπœ€π‘‘|𝑦|𝑑,1(𝑦) and π‘₯≠𝑦. Denote by |π‘₯βˆ’π‘¦|𝛼 the 𝜌. It is easy to see that there exists 𝑖1, 1≀𝑖1≀𝑛 such that ||π‘₯𝑖1βˆ’π‘¦π‘–1||2/(2+𝛼𝑖1)β‰₯πœŒπ‘›.(3.44) Hence, it follows that 𝑛𝑖=1ξ€·π‘₯π‘–βˆ’π‘¦π‘–ξ€Έ2πœŒπ›Όπ‘–β‰₯ξ€·π‘₯𝑖1βˆ’π‘¦π‘–1ξ€Έ2πœŒπ›Ό1β‰₯𝜌2𝑛2+𝛼𝑖1β‰₯𝜌2𝑛2+𝛼.(3.45) Thus, π‘₯βˆ‰πœ€πœŒ;𝑑1(𝑦), where 𝑑1=1/𝑛1+(𝛼/2). Analogously, it is proved that π‘₯βˆˆπœ€βˆšπœŒ,𝑛(𝑦). Now, the required estimation (3.37) at 𝑦≠0 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(π‘₯,𝑦) be a positive function, determined in 𝐸𝑛×𝐸𝑛, continuous at π‘₯≠𝑦, moreover limπ‘₯→𝑦𝐹(π‘₯,𝑦)=∞ (condition (A)).
Further, let πΈβŠ‚πΈπ‘› be some compact. Let us call the measure πœ‡ on 𝐸[𝐹] admissible if supπ‘πœ‡βŠ‚πΈ and π‘‰πΈπœ‡βˆ«(π‘₯)=𝐸𝐹(π‘₯,𝑦)π‘‘πœ‡(𝑦)≀1, for π‘₯∈supπ‘πœ‡.
The value supπœ‡(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 β„’ conditions (1.2)–(1.6) be fulfilled. Then, for removability of the compact πΈβŠ‚π· relative to the first boundary-value problem for the operator β„’ in the space β„³(𝐷) it is sufficient that cap[𝐹1](𝐸)=0,(3.46) where 𝐹1βˆ‘(π‘₯,𝑦)=[1+𝑛𝑖=1(|𝑦|𝛼/|π‘₯βˆ’π‘¦|𝛼)𝛼𝑖]βˆ’1(|π‘₯βˆ’π‘¦|𝛼)2βˆ’π‘›βˆ’(βŸ¨π›ΌβŸ©/2).

Proof. We will use the following assertion, which is proved in [11]. Let the function 𝐹(π‘₯,𝑦) be satisfied condition (A), the compact 𝐸 has zero [𝐹]-capacity, πœ‡ zero measure concentrated on 𝐸. Then, there exists the point π‘₯∘∈supπ‘πœ‡, such that π‘‰πΈπœ‡(π‘₯∘)=∞. At this the potential of the measure supπ‘πœ‡ cannot be bounded on any portion 𝐡, that is, for any open set 𝐡 at πΈξ…žβˆˆsupπ‘πœ‡βˆ©π΅; the potential π‘‰πΈβ€²πœ‡(π‘₯) is not bound 𝐡. In particular, if 𝐡 is an arbitrary neighborhood of the point π‘₯∘ that π‘‰πΈβ€²πœ‡(π‘₯∘)=∞.
Let the condition (3.46) be fulfilled, πœ‡ an arbitrary measure, concentrated on 𝐸, π‘₯∘∈supπ‘πœ‡ is a point, corresponding to the above-stated assertion at 𝐹=𝐹1. Let us assume at first that π‘₯βˆ˜β‰ 0. Then, |π‘₯∘|𝛼=𝑣>0. Further, let 𝐡 be such small neighborhood of the point π‘₯∘ that if πΈξ…žβˆˆsupπ‘πœ‡βˆ©π΅, then supπ‘¦βˆˆπΈβ€²||𝑦||𝛼≀(1+πœ€)π‘Ÿ,infπ‘¦βˆˆπΈβ€²||𝑦||𝛼β‰₯(1+πœ€)π‘Ÿ,(3.47) where the number πœ€>0 will be chosen later. Let us consider the ellipsoids πœ€π‘‘|𝑦|𝑑,1(𝑦) at π‘¦βˆˆπΈξ…ž. Let us choose πœ€ such small than π‘₯0βˆˆπœ€π‘‘|𝑦|𝑑,1(𝑦) for all π‘¦βˆˆπΈξ…ž. Then, according to Corollary 3.5 from Lemma 2.7, we obtain π‘‰πΈπœ‡ξ€·π‘₯0ξ€Έ=ξ€œπΈπΊξ€·π‘₯0ξ€Έξ€œ,π‘¦π‘‘πœ‡(𝑦)β‰₯𝐸′𝐺π‘₯0ξ€Έ,π‘¦π‘‘πœ‡(𝑦)β‰₯𝐢20ξ€œπΈπΉ1ξ€·π‘₯0ξ€Έ,π‘¦π‘‘πœ‡(𝑦)=𝐢20π‘‰πΈπœ‡ξ€·π‘₯0ξ€Έ=∞.(3.48) Hence, it follows that any zero measure πœ‡, concentrated on 𝐸 cannot be β„’ admissible. Thus, capβ„’(𝐸)=0, and the required assertion follows from Theorem 3.1.
Let now π‘₯∘=0. Then, using the equality 𝐺(π‘₯,𝑦)=𝐺(𝑦,π‘₯) and Corollary 3.5 from Lemma 2.7, we conclude π‘‰πΈπœ‡(ξ€œ0)=πΈξ€œπΊ(0,𝑦)π‘‘πœ‡(𝑦)=𝐸𝐺(𝑦,0)π‘‘πœ‡(𝑦)β‰₯𝐢20ξ€œπΈπΉ1(𝑦,0)π‘‘πœ‡(𝑦)=𝐢20ξ€œπΈπΉ1(0,𝑦)π‘‘πœ‡(𝑦)=𝐢20π‘‰πΈπœ‡(0)=∞.(3.49) Theorem is proved.

Remark 3.7. Let conditions of the real theorem be fulfilled and the compact πΈβŠ‚π· removable relative to the first boundary-value problem for the operator β„’ in the space β„³(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 π‘₯βˆˆπœ€π‘‘|𝑦|𝑑,1(𝑦), π‘₯≠𝑦(𝑦≠0), and at π‘₯≠𝑦(𝑦=0) the estimations 𝐺(π‘₯,𝑦)≀𝐢21ξ€·||||(𝛾,𝛼,𝑛)π‘₯βˆ’π‘¦π‘‘ξ€Έ2βˆ’π‘›βˆ’(βŸ¨π›ΌβŸ©/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[βˆ’πΉ2](𝐸)=0, where 𝐹2(π‘₯,𝑦)=(|π‘₯βˆ’π‘¦|𝑑)2βˆ’π‘›βˆ’(βŸ¨π›ΌβŸ©/2).
Hence, it follows that if mes (𝐸)>0, then there exists the point π‘₯2∈𝐸, such that 𝑉𝐸(π‘₯1)=∞, where 𝑉𝐸(ξ€œπ‘₯)=𝐸𝐹