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

Authors: Tair S. Gadjiev; Asghar I. Shariffar; Rafig A. Rasulov
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

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


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## Body

## 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.LetEn be n dimensional Euclidean space of the points x=(x1,…,xn). Denote the ball {x:|x-x0|<R} by BR(xR0) for R>0 and the cylinder BR(x0)∪(0,T) by QTR(xR0). Further, let for x0∈En,R>0 and k>0εr,k(x0) be an ellipsoid {x:∑i=1n((xi-xi0)2/Rαi)<(kR)2}. Let D be an bounded domain En with the domain ∂D,0∈D. ε is a such king of ellipsoid that D¯⊂ε,𝔅(ε) is a set of all functions, satisfying in ε¯ the uniform Lipschitz condition and having zero near the ∂ε.Denote byα and (α1,…,αn) the vector 〈α〉=α1,…,αn.Denote byW2,α1(D) the Banach space of the functions u(x) given on D with the finite norm(1.1)‖u‖W2,α1(D)=(∫D(u2+∑i=1nλi(x)ui2)dx)1/2,
where(1.2)ui=∂u∂xi,i=1,…,n.λi(x)=(|x|λ)αi,|x|α=∑i=1n|xi|22+αi,0≤αi<2n-1.
Further, let W∘2,α1(D) be a degenerated set of all functions from C0∞(D) by the norm of the space W2,α1(D). Denote by ℳ(D) the set of all bounded in D functions.LetE⊂D 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 compactE is called the removable relative to the first boundary value problem for the operator L in the space ℳ(D) if all generalized solution of the equation ℒu=0 in ∂/E formed in zero on ∂D and belonging to the space ℳ(D) identically equal to zero. We will say that the function u(x)∈W∘2,α1(ε) is nonnegative on the set H⊂ε, in the sense W∘2,α1(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 x∈H and lim⁡m→∞∥u(m)-u∥W2,α1(ε)=0.The functionu(x)∈W2,α1(D) is nonnegative and ∂D in sense W2,α1(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 lim⁡m→∞∥u(m)-u∥W2,α1(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 W∘2,α1(ε) if at the same time u(x)≥1 and u(x)≤1 on H, in the sense W∘2,α1(ε).Letω(x) be measurable function in D, finite and positive for a.e. x∈D. Denote by ℒp,ω(D) the Banach space of the functions given on D, with the norm(1.3)‖u‖Lp,ω(D)=(∫D(ω(x))p/2|u|pdx)1/p,1<p<∞.LetWp,α1(D) be a Banach space of the functions given on u(x), with the finite norm D(1.4)‖u‖Wp,α1(D)=(∫D(|u|p+∑i=1n(λi(x))p/2|ui|p)dx)1/p,1<p<∞.Analogously toW∘2,α1(D), it is introduced the subspace W∘p,α1(D) for 1<p<∞. The space conjugated to W∘p,α1(D) we will denote by W*p,α1(D).We will consider the elliptic operator in the bounded domainD⊂En(1.5)L=∑i,j=1n∂∂xi(aij(x)∂∂xj).
In assumption that ∥aij(x)∥ is a real symmetric matrix with measurable in D elements, moreover, for all ξ∈En and a.e. x∈D, the condition(1.6)γ∑i=1nλi(x)ξi2≤∑i,j=1naij(x)ξiξj≤γ-1∑i=1nλi(x)ξi2.
Here, γ∈(0,1] is a constant.The functionu(x)∈W2,α1(D) is called the generalized solution of the equation ℒu=f(x) in D, if for any function η(x)∈W∘2,α1(D) the integral identity
(1.7)∫D∑i,j=1naij(x)uxiηxjdx=∫Dfηdx
be fulfilled.Here,f(x) is a given function from ℒ2(D).LetE⊂D be some compact. The function u(x)∈W2,α1(D∖E) is called generalized solution of the equation ℒu=f(x) in D∖E 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 operatorℒ continued in En∖D with saving condition (1.2), (1.6). For this, it is sufficient, for example, to assume that aij(x)=δijλi(x) for x∈En∖D, i,j=1,…,n, where δij is a Croneker symbol.Leth(x)∈W2,α1(D), f0(x)∈h2(D), fi(x)∈ℒ2,λ-1(D), i=1,2,…,n, be a given functions. Let us consider the first boundary value problem(1.8)Lu=f0(x)+∑i=1n∂fi(x)∂xi,x∈D,(u(x)-h(x))∈W∘2,α1(D).
The function u(x)∈W2,α1(D) we will call generalized solution of problem (1.8) if for any function η(x)∈W∘2,α1(D), the integral identity(1.9)∫D∑i,j=1naij(x)uxiηxjdx=∫D(-f0η+∑i=1nfiηxi)dx
is fulfilled.Our aim to get the necessary and sufficient condition of compact removabilityE 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 operatorℒ condition (1.2), (1.6) are fulfilled, then the first boundary value problem (1.8) has a unique generalized solution  u(x) at any  h(x)∈W2,α1(D),  f0(x)∈h2(D),  fi(x)∈L2,λi-1(D),  i=1,2,…,n. At this, there exists  P0(α,n) such that if  p>p0,  h(x)∈Wp,α1(D),  f0(x)∈hp(D),  fi(x)∈L2,λi-1(D),  i=1,2,…,n,  ∂D∈C1, then solution  u(x) is continuous in  D¯.Lemma 2.2.
Let relative to the coefficients of the operatorℒ conditions (1.2), (1.6) be fulfilled. Then, any generalized solution of the equation  ℒu=0 in  D is continuous by Holder at each strictly internal domain  ∂.Lemma 2.3.
Let relative to the coefficients of the operatorℒ, conditions (1.2), (1.6) be fulfilled and  εR,1¯<D. Then, for any positive generalized solution  u(x), the equation  ℒu=0 in  D the Harnack inequality is true
(2.1)sup⁡εR,1(0)u≤C1(γ,α,n)inf⁡εR,1(0)u.
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 operatorℒ 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  h(x) is bounded on  ∂D in the sense  W2,α1(D), then for solution  u(x) the following maximum principle is true:
(2.2)inf⁡∂Dh≤inf⁡Du≤sup⁡D≤sup⁡∂Dh,
where  inf⁡∂Dh(sup⁡∂Dh) is an exact lower (upper) bound those numbers  a, for which  h(x)≥a(h(x)≤a)  on  ∂D in the sense  W2,α1(D).
These lemmas are proved analogously to paper [7]. Therefore, the proof of these lemmas is not given.
LetH⊂ε be some compact and VH a set of all functions φ(x)∈W∘2,α1(ε) such that φ(x)≥1 on H, in the sense W∘2,α1(ε). Let one considers the functional
(2.3)Jθ(φ)=∫ε∑i,j=1naij(x)φiφjdx,φ(x)∈VH,ℒ is a H compact capacity relative to ellipsoid ε and is called the value inf⁡φ∈VHJθ(u) and denoted by capℒ(ε)(H). In case ε=En, the corresponding value is called ℒ capacity of the compact H and denoted by capℒ(H).Lemma 2.5.
There exists the unique functionu(x)∈W∘2,α1(ε) such that  u(x)≥1 on  H in the sense  W∘2,α1(ε) and  capℒ(ε)(H)=Jℒ(u).Proof.
It is easy to see thatVH is convex closed set in W∘2,α1(ε). From the fact that W∘2,α1(ε) is a Hilbert space, it follows the existence of unique function u(x)∈VH, which achieved an exact lower bound of the functional Jℒ(φ). Next,
(2.4){u(x)}1={u(x)ifu(x)≤1,1ifu(x)>1.
It is clear that{u(x)}1∈W∘2,α1(ε). Moreover, {u(x)}1∈VH. Denote by A+={x:x∈ε,u(x)>1}. We have
(2.5)JL{u(x)1}=(∫A++∫ε∖A+)∑i,j=1naij(x){u}i1{u}j1dx=∫ε∖A+∑i,j=1naij(x)uiujdx.
On the other side, according to (1.2),
(2.6)∫A+∑i,j=1naij(x)uiujdx≥0.
From (2.5) and (2.6), we conclude
(2.7)JL{u(x)1}≤JL(u)=inf⁡φ∈VHJL(φ),
that is, Jℒ{u(x)1}=Jℒ(u). From uniqueness extreme function, it follows that {u(x)}1=u(x), and lemma is proved.
The functionu(x), which achieved an exact lower bound of the functional Jℒ(φ) on the set VH is called ℒ capacity of the compact potential H relative to the ellipsoid ε.Lemma 2.6.
Letℒ be a capacity potential  u(x) of the compact  H relative to  ε which is a generalized solution of the equation  ℒu=0 in  ε∖H, vanishing on  0 and  ∂ε in  1 on  ∂H in the sense  W2,α1(ε).Proof.
It is sufficient to show the truth of the first part of assertion of lemma. Letη(x)∈W∘2,α1(ε) and η(x)≥0 on H in the sense W∘2,α1(ε). Then, for any ε>0(u(x)+εη(x))∈VH. Therefore,
(2.8)JL(u+εη)≥JL(u).
Thus,
(2.9)JL(u)+ε2JL(η)+2ε∫ε∑i,j=1naij(x)uiηjdx≥JL(u),
that is,
(2.10)JL(u)+2ε∫ε∑i,j=1naij(x)uiηjdx≥0.
Tending ε to zero, we conclude
(2.11)∫ε∑i,j=1naij(x)uiηjdx≥0.
It is easy to see as η(x) in (2.11), we can take any function from C1(ε¯) with compact support in ε∖H. Then,
(2.12)∫ε∖H∑i,j=1naij(x)uiηjdx≥0.
Substituting η(x) on -η(x), we arrive to
(2.13)∫ε∖H∑i,j=1naij(x)uiηjdx=0.
Lemma is proved.Letμ 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 ℒu=-μ, equaling to zero on ∂ε if for any function φ(x)∈W∘2,α1(ε)∩C(ε¯), the integral identity(2.14)∫εuLφdx=∫εφdμ
is fulfilled.According to Lemma2.1 (at h=0), there exists the continuous linear operator H from W*2,α1(ε) in W∘2,α1(ε) such that for any functional T∈W*2,α1(ε), the function u=H(T) is unique in W∘2,α1(ε) generalized solution of the equation ℒu=T.The operatorH is called Green operator.By Lemma2.1, this operator at p>p0 we transform W*2,α1(ε) to C(ε¯). It is easy to see that the function u(x) is weak solution of the equation ℒu=-μ, equaling to zero on ∂ε if and only if for any function ψ(x)∈C(ε¯) the integral identity(2.15)∫εuψdx=∫εH(ψ)dμ
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 ℒu=-μ equaling to zero on ∂ε.Let us say that the chargeμ∈W*2,α1(ε) if there exists the vector f¯(x)=(f∘(x),f1(x),…,fn(x))f0(x)∈h2(ε), fi(x)∈L2,λi(ε), i=1,2,…,n, for any function φ(x)∈W∘2,α1(ε)∩C(ε¯), the integral identity(2.16)μ(φ)=∫εφdμ=∫ε(f∘φ-∑i=1nfiφi)dx
is true.At this, it is evident that(2.17)|∫εφdμ|≤C2(f¯)‖φ‖W2,α1(ε).Lemma 2.7.
The weak solutionu(x) of the equation  ℒu=-μ, equaling to zero on  ∂ε, belongs to  W∘2,α1(ε), if and only if  μ∈W*2,α1(ε).Proof.
At first, we will show that if the functionφ(x)∈W∘2,α1(ε) satisfies the integral identity
(2.18)∫ε∑i,j=1naij(x)uiφjdx=-∫εφdμ,
for any function φ(x)∈W∘2,α1(ε)∩C(ε¯)zero on ∂ε, then it is weak solution of the equation ℒu=-μ, equaling to zero on ∂ε. Really, assuming that φ=H(ψ), ψ(x)∈C(ε¯), we obtain
(2.19)∫εH(ψ)dμ=∫εφdμ=-∫ε∑i,j=1naij(x)uiφjdx=∫εu∑i,j=1n(aij(x)φj)idx=∫εuLφdx=∫εuψdx,
and now, it is sufficient to use the identity (2.15). We will show that μ∈W*2,α1(ε). For this, it is sufficient to prove that if fi(x)=∑i=1naij(x)ui(x), then fi(x)∈L2,λi-1(ε), i=1,2,…,n,. Assume in condition (2.18) that ξ1=⋯=ξi-1=ξi+1=⋯=ξn=0, ξi=1/λi(x).
We will obtain(2.20)γ≤aii(x)λi(x)≤γ-1;i=1,…,n.
Leti≠j. Assuming that ξk=0 at k≠j and k≠i, ξi=1/λi(x), ξj=1/λj(x), we will obtain
(2.21)2γ≤aii(x)λi(x)+ajj(x)λj(x)+2aij(x)λi(x)λj(x)≤2γ-1.
Using (2.20), we conclude
(2.22)|aij(x)|λi(x)λj(x)≤γ-1-γ;i,j=1,…,n;i≠j.
From (2.20) and (2.22), it follows that
(2.23)|aij(x)|λi(x)λj(x)≤γ-1;i,j=1,…,n.
Thus, from (2.23), take out for j=1,…,n(2.24)∫ε1λj(x)(fj)2dx=∫ε1λj(x)(∑i=1naij(x)ui)2dx≤γ-2n∑i=1n∫ελi(x)ui2dx<α.
So, μ∈W*2,α1(ε). Inversely, if u(x) is a weak solution of the equation ℒu=-μ, vanishing on ∂ε, then there exists μ∈W*2,α1(ε) such that
(2.25)(f∘φ-∑i=1nfiφi)dx=∫εφdμ=∫εuLφdx=∫εu∑i,j=1n(aij(x)φj)idx=-∫ε∑i,j=1naij(x)uiφjdx,
for any function φ(x)∈W*2,α1(ε)∩C(ε¯), ℒφ(x)∈C(ε¯).
Then, from Lemma2.1, we obtain that u(x)∈W∘2,α1(ε). The lemma is proved.Let nowδ(x) be Dirac measure, concentrated at the point 0, and y an arbitrary fixed point ε.The weak solutiong(x,y) of the equation ℒy=-δ(x-y), vanishing on ∂ε, is called Green function of the operator ℒ in ε.In caseε=En the corresponding function is called the fundamental solution of the operator ℒ and denoted by G(x,y).According to the above proved, ifψ(x) is an arbitrary function from C(ε¯), then the generalized solution φ(x)∈W∘2,α1(ε) of the equation ℒφ=-ψ can be introduced in the following form:(2.26)φ(y)=∫εg(x,y)ψ(x)dx.
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 onε, the integral
(2.27)u(x)=∫εg(x,y)dμ(y)
exists, finite a.e. in  ε and is weak solution of the equation  ℒu=-μ, equaling to zero on  ∂ε.Proof.
Without losing generality, we will assume that the chargeμ is the measure in ε. Let φ(x)∈C(ε¯), ψ(x)≥0 in ε. Denote by φ(x)∈W∘2,α1(ε) the generalized solution of the equation ℒφ=-ψ(x). Then, φ(x)∈C(ε¯) according to Lemma 2.1 and ψ(x)≥0 according to Lemma 2.4. At this,
(2.28)φ(y)=∫εg(x,y)ψ(x)dx.
Then, by the Fubini theorem, we conclude that the integral ∫εg(x,y)dμ(y) there exists for almost all x∈ε; moreover,
(2.29)∫εH(ψ)dμ(y)=∫εφ(y)dμ(y)=∬ε×εg(x,y)ψ(x)dxdμ(y)=∫εψ(x)u(x)dx.
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 nowℒ-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
(2.30)∫ε∑i,j=1naij(x)uiηjdx=∫εηdμ.
Further, since u=1 on H in the sense W∘2,α1(ε), then the carrier of the measure μ is situated on ∂H. The measure μ is called ℒ-capacity of the compact H.
According to Lemma2.8,  ℒ-capacity potential u(x) is weak solution of the equation ℒu=-μ, equaling to zero on ∂ε and can be represented in the following form:
(2.31)u(x)=∫εg(x,z)dμ(z).
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 x∈H and lim⁡m→∞∥η(m)-u∥W2,α1(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 capℒ(ε)(H) as m→∞. Thus,
(2.32)capL(ε)(H)=μ(H).Lemma 2.9.
Let relative to coefficients of the operatorℒ conditions (1.2)–(1.6), y∈∂εR,2(0),  ε¯R,1(0)⊂D,  x∈∂εR,1(y) be fulfilled. Then, for the Green function  g(x,y) the following estimations are true:
(2.33)C3(γ,α,n)[capL(ε)(ε¯R,1(y))]-1≤g(x,y)≤C4(γ,α,n)[capL(ε)(ε¯R,1(y))]-1.
If ε¯R,1(0)⊂D, x∈∂εR,1(0) then,
(2.34)C3[capL(ε)(ε¯R,1(0))]-1≤g(x,0)≤C4[capL(ε)(ε¯R,1(0))]-1.Proof.
Without loss of generality, we can assume that the coefficients of the operatorℒ are continuously differentiable in ε¯. The general case is obtained by means of limit passage. Then, at x≠y, the function g(x,y) is continuous by x and y moreover,
(2.35)lim⁡x→yg(x,y)=∞.
Leta be a positive number, which will be chosen later, Ka={x:g(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, ℒ is capacity potential Ka, represented in the form (2.31), so it means that it equal to zero in it. Thus,
(2.36)1=∫εy(y,z)dμa(z),
where μ is a ℒ-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
(2.37)μa(Ka)=capL(ε)(Ka)=1a.
Let us assume now that a=inf⁡x∈∂εR,1(y)g(x,y). According to maximum principle, ε¯R,1(y)⊂Ka. Therefore, from (2.37), we conclude
(2.38)capL(ε)(ε¯R,1(y))≤capL(ε)(Ka)=1inf⁡x∈∂εR,1(y)g(x,y).
If we will assume b=sup⁡x∈∂εR,1(y)g(x,y), then ε¯R,1(y)⊂Ka; that is,
(2.39)capL(ε)(ε¯R,1(y))≤capL(ε)(Kb)=1sup⁡x∈∂εR,1(y)g(x,y).
From (2.38) and (2.39), follows that
(2.40)inf⁡x∈∂εR,1(y)g(x,y)≤[capL(ε)(ε¯R,1(y))]-1≤sup⁡x∈∂εR,1(y)g(x,y).
On the other side, according to Lemma 2.3,
(2.41)sup⁡x∈∂εR,1(y)g(x,y)≤C5(γ,α,n)inf⁡x∈∂εR,1(y)g(x,y).
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 andy∈∂εR,2(0) be fulfilled, ε¯R,1(0)⊂D, x∈∂εR,1(0) or y=0, ε¯R,1(0)⊂D, x∈∂εR,1(0). Then, for fundamental solution G(x,y), the estimations
(2.42)C3[capL(ε)(ε¯R,1(0))]-1≤G(x,y)≤C4[capL(ε)(ε¯R,1(0))]-1
are true.


## 3. Removability Criterion of the Compact in the SpaceM(D)

Theorem 3.1.
Let relative to the coefficients of the operatorℒ, conditions (1.2)–(1.6) be fulfilled. Then, for removability of the compact  E⊂D relative to the first boundary value problem for the operator  ℒ in the space  ℳ(D), it is necessary and sufficient that
(3.1)capL(E)=0.Proof.
Let the ellipsoidε have the same sense as above. It is easy to see that if condition (3.1) is fulfilled, then
(3.2)capL(ε)(E)=0.
Not losing generality, we can limit the case, when the coefficients of the operator ℒ is continuously differentiable in ε¯. Let us fix an arbitrary ε>0 and x0⊂D∖E. By virtue of (3.1), there exists the neighbourhood H of the compact E such that
(3.3)capL(ε)(H¯)<ε.
At this, we can assume that ε is such small that
(3.4)dist(x0,H¯)≥12dist(x0,E).
Denote by VH(x) and μH the ℒ-capacity potential of the compact H¯ relative to the ellipsoid ε and ℒ¯-capacity of the distribution H¯, respectively. According to above proved,
(3.5)VH(x)=∫εg(x,y)dμH(y),
moreover, the function VH(x) is the generalized solution of the equation ℒVH=0 in ε∖H¯, vanishing on 0 and in ∂ε on 1 in ∂H in the sense W2,α1(ε). Let now u(x)∈ℳ(D) be an arbitrary solution of the equation ℒu=0 in D∖E, vanishing on ∂D, M=sup⁡D|u|. It is easy to see that the function VH(x) is nonnegative on ∂D, in the sense W2,α1(D). Hence, it follows, that the function u(x)-MVH(x) is generalized solution of the equation ℒu=0 in D∖H¯, is non-positive on ∂(D∖H¯). According to Lemma 2.4,  u(x)-MVH(x)≤0 and D∖H¯ in particular
(3.6)u(x0)≤MVH(x0)≤Msup⁡y∈∂Hg(x0,y)μH(H¯)=Msup⁡y∈∂Hg(x0,y)capL(ε)(H¯).
By virtue of continuity of the function g(x,y) at x≠y and (3.4), we obtain
(3.7)sup⁡y∈∂Hg(x0,y)≤C6(γ,α,n,x0,E).
Thus, from (3.3) and (3.6), we conclude
(3.8)u(x0)≤MC6ε.
Using an arbitrariness ε, we have
(3.9)u(x0)≤0.
Making analogous considerations with the function u(x)+MVH(x), we obtain
(3.10)u(x0)≥0.
From (3.8)-(3.9) and an arbitrariness of the point x0 it follows that u(x)≡0 in D∖E. Thereby, the sufficiency of condition (3.1) is proved. Let us prove its necessity. Let us assume that capℒ(E)>0. Denote by ε′ the ellipsoid such that ε¯′⊂δ, E⊂ε′. Assume D=ε  potential of the compact E relative to the ellipsoid ε′ and ℒ-capacity distribution E, respectively. Following to [11], we can give the equivalent definition of Vallee-Poussin type of ℒ-capacity of the compact E, relative to the ellipsoid ε′. Let g(x,y) be a Green function of the operator ℒ in ε′. Let us call the measure μ on E, ℒ-admissible if μ⊂E and
(3.11)VμE(x)=∫ε′g(x,y)dμ(y)≤1forx∈sup⁡pμ.
The value sup⁡μ(E)=capℒ(ε′)(E), where an exact upper boundary is taken on all ℒ-admissible measures and is called ℒ-capacity of the compact E relative to the ellipsoid ε′.
Analogously, theℒ-capacity capℒ(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 ℒ-admissible measures μ. This measure is ℒ-capacity distribution of the compact E.
According to the above proved, the functionuE(x) is generalized solution of the equation ℒuE=0 in ε′∖E, 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 operatorℒ condition (1.2) be fulfilled. Then, if  y∈∂εR,2(0), then
(3.12)C7(γ,α,n)Rn+(〈α〉/2)-2≤capL(ε¯R,1(y))≤C8(γ,α,n)Rn+(〈α〉/2)-2.Proof.
Letℒ0=∑i=1n(∂/∂xi)(λi(x)(∂/∂xi)). Then, according to (1.2),
(3.13)γcapL0(ε¯R,1(y))≤capL(ε¯R,1(y))≤γ-1capL0(ε¯R,1(y)).
Let u(x)∈C0∞(εR,(3/2)(y)), u(x)=1 for εR,1(y); moreover,
(3.14)|ui(x)|≤C9(λ,n)R1+(αi/2);i=1,…,n.
Then,
(3.15)capL0(ε¯R,1(y))≤∫εR,(3/2)(y)∑i=1nλi(x)ui2dx.
On the other side, as y∈∂εR,2(0), then ∑i=1n(yi2/Rαi)=4R2, and thereby
(3.16)|yi|≤2R1+(αi/2);i=1,…,n.
Besides, as x∈εR,(3/2)(y), then
(3.17)|xi-yi|≤32R1+(αi/2);i=1,…,n.
Thus,
(3.18)|xi|≤|yi|+|xi-yi|≤72R1+(αi/2);i=1,…,n.
Hence, it follows that
(3.19)|x|α≤R∑i=1n(z2)2/(2+λi).
Therefore,
(3.20)λi(x)≤C10αiRαi≤C10α+Rαi;i=1,…,n.
where α+=max⁡{α1,…,αn}.
Allowing for (3.14) and (3.20) in (3.15), we obtain
(3.21)capL0(ε¯R,1(y))≤C10(α,n)R-2mes(εR,(3/2)(y))=C11(α,n)Rn+(〈α〉/2)-2,
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
(3.22)capL0(ε¯R,1(y))≥capL0(ε¯R,(1/2n)(y)).
Besides, considering the same as in [8], we conclude
(3.23)capL0(ε¯R,(1/2n)(y))≥C12(α,n)capL0(ε0)(ε¯R,(1/2n)(y)),
where ε0=εR,(1/n)(y).
LetW={u(x):u(x)C0∞(ε0),u(x)=1forx∈εR,(1/2n)(y)}. Then,
(3.24)capL0(ε0)(ε¯R,(1/2n)(y))=inf⁡u∈W∫ε0∑i=1nλi(x)ui2dx.
On the other side, if y∈∂εR,2(0), then we can find i0, 1≤i0≤n such that yi02≥4R2+αi0/n, that is;
(3.25)|yi0|≥4R1+(αi0/2)n.
Besides, as x∈ε0, then
(3.26)|xi0-yi0|≤R1+(αi0/2)n.
Therefore,
(3.27)|xi0|≥|yi0|-|xi0-yi0|≥R1+(αi0/2)n.
Thereby,
(3.28)λi(x)≥n-1/(2+αi0)R;i=1,…,n,
where α-=min⁡{α1,…,αn}.
Allowing for (3.28) in (3.24), we obtain
(3.29)capL0(ε0)(ε¯R,(1/2n)(y))=C13(α,n)inf⁡u∈W∫ε0∑i=1nRαiui2dx.
Denote by BR(z) the ball {x:|x-z|<R}. Let us make in (3.30) the substitution of the variables vi=xi/R1+(αi/2); i=1,…,n, and let ỹ be an image of the point y, where W̃={ũ(v):ũ(τ)C0∞(B0),ũ(τ)=1forv∈B(1/2n)(ỹ)}. Then, from (3.30), we deduce B0=B(1/2n)(ỹ) where by (3.30),
(3.30)capL0(ε0)(ε¯R,(1/2n)(y))≥C13Rn+(〈α〉/2)-2inf⁡ũ∈W̃∫B0∑i=1n(∂ũ∂vi)2dτ=C13Rn+(〈α〉/2)-2cap(B0)(B¯(1/2n)(ỹ)),
we will denote by cap(B0)(B¯(1/2n)(ỹ)) Wiener capacity of the compact B¯(1/2n)(ỹ), relative to the ball B0. Now, it is sufficient to note that cap(B0)(B¯(1/2n)(ỹ))=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 operatorℒ condition (1.2) be fulfilled. Then,
(3.31)C15(γ,α,n)Rn+(〈α〉/2)-2≤capL(ε¯R,1(y))≤C16(γ,α,n)Rn+(〈α〉/2)-2.
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,
(3.32)capL(ε¯R,(1/4)(y¯))<capL(ε¯R,1(0)),
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
(3.33)capL(ε¯ρ,1(y¯))≤C17(γ,α,n)ρn+(〈α〉/2)-2(1+∑i=1n(Rρ)αi)
is true.
Then,v(x)∈C0∞(ερ,(3/2)(y)), v(x)=1 for x∈ερ,1(y)(3.34)|vi(x)|≤C18(α,n)ρ1+(αi/2);i=1,…,n,capL0(ε¯ρ,1(y¯))=γ-1C182ρ-2∫ερ,(3/2)(y)∑i=1nλi(x)ρ-αidx.
On the other side, arguing the same, as well as in the proof of Lemma 3.2, we have
(3.35)λi(x)<C19(α,n)(R+ρ)αi,x∈ερ,(3/2)(y);i=1,…,n.
Now, it is sufficient to take into account that
(3.36)∑i=1n(1+Rρ)αi≤∑i=1n[1+(Rρ)αi]≤n(1+∑i=1n(Rρ)αi),
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 x∈εd|y|d,1(y), x≠y for the fundamental solution G(x,y), the estimation
(3.37)G(x,y)≥C20(γ,α,n)(|x-y|α)2-n-(〈α〉/2)1+∑i=1n(|y|α/|x-y|α)αi
is true.
Ify=0, then estimation (3.37) is true for all x≠0. Here, d=1/n22/(2+α).
For proving, at first, let us show that ify≠0, then y∉εd|y|d,2(0). Really, as
(3.38)|y|α=∑i=1n|yi|2/(2+αi),
then there exists i0, 1≤i0≤n such that
(3.39)|y0|2/(2+αi0)≥|y|αn.
Thus,
(3.40)|yi02|(|y|α)αi0≥(|y|α)2n2+αi.
Thereby,
(3.41)∑i=1nyi2(d|y|α)αi≥yi02(d|y|α)αi0≥(d|y|α)2(dn)2+αi0=4(d|y|α)2(22/(2+αi0)dn)2+αi0.
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(3.42)|yi|2/(2+αi)≤|y|α,
that is,
(3.43)∑i=1nyi2(|y|α)αi≤n(|y|α)2.
So, one will show that ε|y|α,n(0) if only y≠0.
Let now fory≠0, x∈εd|y|d,1(y) and x≠y. Denote by |x-y|α the ρ. It is easy to see that there exists i1, 1≤i1≤n such that
(3.44)|xi1-yi1|2/(2+αi1)≥ρn.
Hence, it follows that
(3.45)∑i=1n(xi-yi)2ραi≥(xi1-yi1)2ρα1≥ρ2n2+αi1≥ρ2n2+α.
Thus, x∉ερ;d1(y), where d1=1/n1+(α/2). Analogously, it is proved that x∈ερ,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.
LetF(x,y) be a positive function, determined in En×En, continuous at x≠y, moreover lim⁡x→yF(x,y)=∞ (condition (A)).
Further, letE⊂En be some compact. Let us call the measure μ on E[F] admissible if sup⁡pμ⊂E and VμE(x)=∫EF(x,y)dμ(y)≤1, for x∈sup⁡pμ.
The valuesup⁡μ(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 operatorℒ conditions (1.2)–(1.6) be fulfilled. Then, for removability of the compact  E⊂D relative to the first boundary-value problem for the operator  ℒ in the space  ℳ(D) it is sufficient that
(3.46)cap[F1](E)=0,
where  F1(x,y)=[1+∑i=1n(|y|α/|x-y|α)αi]-1(|x-y|α)2-n-(〈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 x∘∈sup⁡pμ, such that VμE(x∘)=∞. At this the potential of the measure sup⁡pμ cannot be bounded on any portion B, that is, for any open set B at E′∈sup⁡pμ∩B; the potential VμE′(x) is not bound B. In particular, if B is an arbitrary neighborhood of the point x∘ that VμE′(x∘)=∞.
Let the condition (3.46) be fulfilled, μ an arbitrary measure, concentrated on E, x∘∈sup⁡pμ is a point, corresponding to the above-stated assertion at F=F1. Let us assume at first that x∘≠0. Then, |x∘|α=v>0. Further, let B be such small neighborhood of the point x∘ that if E′∈sup⁡pμ∩B, then
(3.47)sup⁡y∈E′|y|α≤(1+ε)r,inf⁡y∈E′|y|α≥(1+ε)r,
where the number ε>0 will be chosen later. Let us consider the ellipsoids εd|y|d,1(y) at y∈E′. Let us choose ε such small than x0∈εd|y|d,1(y) for all y∈E′. Then, according to Corollary 3.5 from Lemma 2.7, we obtain
(3.48)VμE(x0)=∫EG(x0,y)dμ(y)≥∫E′G(x0,y)dμ(y)≥C20∫EF1(x0,y)dμ(y)=C20VμE(x0)=∞.
Hence, it follows that any zero measure μ, concentrated on E cannot be ℒ admissible. Thus, capℒ(E)=0, and the required assertion follows from Theorem 3.1.
Let nowx∘=0. Then, using the equality G(x,y)=G(y,x) and Corollary 3.5 from Lemma 2.7, we conclude
(3.49)VμE(0)=∫EG(0,y)dμ(y)=∫EG(y,0)dμ(y)≥C20∫EF1(y,0)dμ(y)=C20∫EF1(0,y)dμ(y)=C20VμE(0)=∞.
Theorem is proved.Remark 3.7.
Let conditions of the real theorem be fulfilled and the compactE⊂D removable relative to the first boundary-value problem for the operator ℒ in the space ℳ(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 x∈εd|y|d,1(y), x≠y(y≠0), and at x≠y(y=0) the estimations
(3.50)G(x,y)≤C21(γ,α,n)(|x-y|d)2-n-(〈α〉/2)
are true.
Further, analogously to Theorem3.1, it is shown that if the compact E is removable, then according to cap[-F2](E)=0, where F2(x,y)=(|x-y|d)2-n-(〈α〉/2).
Hence, it follows that if mes(E)>0, then there exists the point x2∈E, such that VE(x1)=∞, where
(3.51)VE(x)=∫EF2(x,y)dy.
Moreover, if B′ is an arbitrary neighborhood of the point E′=B′∩E, then the potential VE′(x) is not bounded on E′. Let us consider the case x1≠0. Choose small neighborhood B′ of the point x1 that at all x∈E′, y∈E′ the inequality |xi-yi|≤1; i=1,…,n are fulfilled. For x∈E′, we have
(3.52)VE′(x)=∫E′(∑i=1n|xi-yi|(2/2+αi))2-n-(〈α〉/2)dy≤∫E′(∑i=1n|xi-yi|)2-n-(〈α〉/2)dy≤∫E′|x-y|2-n-(〈α〉/2)dy≤∫B′′|z|2-n-(〈α〉/2)dy,
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〉/2≤n/(n-1)≤3/2, and the assertion the corollary is proved.



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*Source: 512109-2011-10-17.xml*