On Removable Sets for Degenerated Elliptic Equations
We establish necessary and sufficient conditions of removability of compact sets.
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2014
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508145941479424 |
|---|---|
| author | Bayramova, N. Q. Gadjiev, T. S. Байрамова, Н. Я. Гаджиїв, Т. С. |
| author_facet | Bayramova, N. Q. Gadjiev, T. S. Байрамова, Н. Я. Гаджиїв, Т. С. |
| author_sort | Bayramova, N. Q. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:26:14Z |
| description | We establish necessary and sufficient conditions of removability of compact sets. |
| first_indexed | 2026-03-24T02:20:34Z |
| format | Article |
| fulltext |
UDC 517.9
T. S. Gadjiev, N. Q. Bayramova (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS
ПРО МНОЖИНИ, ЩО УСУВАЮТЬСЯ,
ДЛЯ ВИРОДЖЕНИХ ЕЛIПТИЧНИХ РIВНЯНЬ
We establish the necessary and sufficient conditions of compact removability.
Встановлено необхiднi та достатнi умови компактної усувностi.
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 E. I. Moi-
seev [2]. The compact removability for elliptic and parabolic equations of nondivergent structure is
considered by E. M. Landis [3]. T. S. Gadjiev, V. A. Mamedova [4]. The removability condition
of compact in the space of continuous functions are constructed in the papers Harvey, Polking [5],
T. Kilpelainen [6]. The different questions of qualitative properties of solutions of uniformly dege-
nerated elliptic equations is studied by S. Chanillo, R. Z. Wreeden [7]. Uniform elliptic operator of
the second order of divergent structure is considered in the paper [8].
Let En be n dimensional Euclidean space of the points x = (x1, . . . , xn). Denote by R > 0 for
BR
(
x0R
)
the ball
{
x :
∣∣x− x0∣∣ < R
}
, and by QRT
(
x0R
)
the cylinder BR
(
x0
)
∪(0, T ) . Further let for
x0 ∈ En, R > 0 and k > 0 εr,k
(
x0
)
be an ellipsoid
{
x :
∑n
i=1
(
xi − x0i
)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 by α and (α1, . . . , αn) the vector 〈α〉 = α1, . . . , αn.
Denote by W 1
2,α(D) the Banach space of the functions u(x) given on D with the finite norm
‖u‖W 1
2,α(D) =
∫
D
(
u2 +
n∑
i=1
λi(x)u
2
i
)
dx
1/2 ,
where
ui =
∂u
∂xi
, i = 1, . . . , n, λi(x) = (|x|λ)
αi , |x|α =
n∑
i=1
|xi|
2
2 + αi
,
0 ≤ αi <
2
n− 1
.
(1)
Further, let
◦
W 1
2,α(D) be a degenerated set of all functions from C∞0 (D) by the norm of the space
W 1
2,α(D). Denote byM(D) the set of all bounded in D functions.
Let E ⊂ 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.
c© T. S. GADJIEV, N. Q. BAYRAMOVA, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8 1041
1042 T. S. GADJIEV, N. Q. BAYRAMOVA
The compact E is called the removable relative to the first boundary-value problem for the
operator L in the spaceM(D), if all generalized solution of the equation Lu = 0 in ∂ /E formed in
zero on ∂D and belonging to the spaceM(D), identically equal to zero. We’ll say that the function
u(x) ∈
◦
W 1
2,α(ε) is nonnegative on the set H ⊂ ε, in sense
◦
W 1
2,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 x ∈ H and
limm→∞
∥∥u(m) − u
∥∥
W 1
2,α(ε)
= 0.
The function u(x) ∈ W 1
2,α(D) is nonnegative and ∂D in sense W 1
2,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 limm→∞
∥∥u(m) − u
∥∥
W 1
2,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 sense
◦
W 1
2,α(ε), if at the same
time u(x) ≥ 1 and u(x) ≤ 1 on H, in sense
◦
W 1
2,α(ε).
Let ω(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
‖u‖Lp,ω(D) =
∫
D
(ω(x))p/2 |u|p dx
1/p , 1 < p <∞.
Let W 1
p,α(D) be a Banach space of the functions given on u(x), with the finite norm D:
‖u‖W 1
p,α(D) =
∫
D
(
|u|p +
n∑
i=1
(λi(x))
p/2 |ui|p
)
dx
1/p , 1 < p <∞.
Analogously to
◦
W 1
2,α(D), it is introduced the subspace
◦
W 1
p,α(D) for 1 < p < ∞. The space,
conjugated to
◦
W 1
p,α(D) we’ll denote by
∗
W 1
p,α(D).
We’ll consider the elliptic operator in the bounded domain D ⊂ En
L =
n∑
i,j=1
∂
∂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
γ
n∑
i=1
λi(x)ξ
2
i ≤
n∑
i,j=1
aij(x)ξiξj ≤ γ−1
n∑
i=1
λi(x)ξ
2
i . (2)
Here γ ∈ (0, 1] is a constant.
The function u(x) ∈ W 1
2,α(D) is called the generalized solution of the equation Lu = f(x) in
D, if for any function η(x) ∈
◦
W 1
2,α(D) the integral identity∫
D
n∑
i,j=1
aij(x)uxiηxjdx =
∫
D
fηdx (3)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1043
be fulfilled. Here f(x) is a given function from L2(D).
Let E ⊂ D be some compact. The function u(x) ∈ W 1
2,α (D \E ) is called generalized solution
of the equation Lu = f(x) in D \E , vanishing on ∂D, if integral identity (3) is fulfilled for any
function η(x) ∈ AE(D).
We’ll assume that the coefficients of the operator L continued in En \D with saving condition (1),
(2). For this, it is sufficient, for example, let’s assume aij(x) = δijλi(x) for x ∈ En\D, i, j =
= 1, . . . , n, where δij is a Kronecker symbol.
Let h(x) ∈W 1
2,α(D), f0(x) ∈ h2(D), f i(x) ∈ L2,λ−1(D), i = 1, 2, . . . , n, are a given functions.
Let’s consider the first boundary-value problem
Lu = f0(x) +
n∑
i=1
∂f i(x)
∂xi
, x ∈ D, (4)
(u(x)− h(x)) ∈
◦
W 1
2,α(D). (5)
The function u(x) ∈ W 1
2,α(D) we’ll call generalized solution of problem (4), (5) if for any function
η(x) ∈
◦
W 1
2,α(D) the integral identity∫
D
n∑
i,j=1
aij(x)uxiηxjdx =
∫
D
(
−f0η +
n∑
i=1
f iηxi
)
dx
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 1. If relative to the coefficients of the operator L, condition (1), (2) be fulfilled, then the
first boundary-value problem (4), (5) has a unique generalized solution u(x) at any h(x) ∈W 1
2,α(D),
f0(x) ∈ h2(D), f i(x) ∈ L2,λ−1
i
(D), i = 1, 2, . . . , n. At this there exists P0 (α, n) such that, if p > p0,
h(x) ∈W 1
p,α(D), f0(x) ∈ hp(D), f i(x) ∈ L2,λ−1
i
(D), i = 1, 2, . . . , n, ∂D ∈ C1, then solution u(x)
is continuous in D.
Lemma 2. Let relative to the coefficients of the operator L conditions (1), (2) be fulfilled. Then
any generalized solution of the equation Lu = 0 inD is continuous by Hölder at each strictly internal
domain ∂.
Lemma 3. Let relative to the coefficients of the operator L, conditions (1), (2) 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
sup
εR,1(0)
u ≤ C1 (γ, α, n) inf
εR,1(0)
u. (6)
If at this y ∈ ∂εR,2(0) and εR,1(0) ⊂ D, then the inequality of form (6) is true in ellipsoid εR,1(y).
Lemma 4. Let relative to the coefficients of the operator L conditions (1), (2) be fulfilled, and
u(x) be generalized solution of the first boundary-value problem (4), (5) at f i(x) ≡ 0, i = 0, . . . , n.
Then if h(x) is bounded on ∂D in sense W 1
2,α(D), then for solution u(x) the following maximum
principle is true:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1044 T. S. GADJIEV, N. Q. BAYRAMOVA
inf
∂D
h ≤ inf
D
u ≤ sup
D
≤ sup
∂D
h,
where inf∂D h
(
sup∂D h
)
is an exact lower (upper) bound those numbers a, for which h(x) ≥ a
(h(x) ≤ a) on ∂D in sense W 1
2,α(D).
These lemmas are proved analogously to paper [7]. Therefore, we don’t give the proof of these
lemmas.
Let H ⊂ ε be some compact, VH be a set of all functions ϕ(x) ∈
◦
W 1
2,α(ε), such that ϕ(x) ≥ 1
on H, in sense
◦
W 1
2,α(ε). Let’s consider the functional
Jθ (ϕ) =
∫
ε
n∑
i,j=1
aij(x)ϕiϕjdx, ϕ(x) ∈ VH ,
L is a H compact capacity relative to ellipsoid ε is called the value infϕ∈VH Jθ(u) and denoted by
cap
(ε)
L (H). In case ε = En, the corresponding value is called L capacity of the compact H and
denoted by capL(H).
Lemma 5. There exists the unique function u(x) ∈
◦
W 1
2,α(ε) such that u(x) ≥ 1 on H in sense
◦
W 1
2,α(ε) and cap
(ε)
L (H) = JL (u). Moreover, u(x) = 1 on H in sense
◦
W 1
2,α(ε).
Proof. It is easy to see that VH is convex closed set in
◦
W 1
2,α(ε). From the fact that
◦
W 1
2,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 JL(ϕ). Let’s next {u(x)}1 =
{
u(x) if u(x) ≤ 1,
1 if u(x) > 1.
It is clear, that {u(x)}1 ∈
◦
W 1
2,α(ε). Moreover, {u(x)}1 ∈ VH . Denote by A+ =
{
x : x ∈
∈ ε, u(x) > 1
}
. We have
JL
{
u(x)1
}
=
( ∫
A+
+
∫
ε\A+
)
n∑
i,j=1
aij(x) {u}1i {u}
1
j dx =
∫
ε\A+
n∑
i,j=1
aij(x)uiujdx. (7)
On the other side, according to (1) ∫
A+
n∑
i,j=1
aij(x)uiujdx ≥ 0. (8)
From (7) and (8) we conclude
JL
{
u(x)1
}
≤ JL (u) = inf
ϕ∈VH
JL (ϕ) ,
i.e., JL
{
u(x)1
}
= JL(u). From uniqueness extreme function it follows, that {u(x)}1 = u(x).
Lemma 5 is proved.
The function u(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 6. L be a capacity potential u(x) of the compact H relative to ε is a generalized
solution of the equation Lu = 0 in ε\H, vanishing on 0 and ∂ε in 1 on ∂H sense W 1
2,α(ε).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1045
Proof. It is sufficient to show the truthiness of the first part of assertion of lemma. Let η(x) ∈
∈
◦
W 1
2,α(ε) and η(x) ≥ 0 on H in sense
◦
W 1
2,α(ε). Then for any ε > 0 (u(x) + εη(x)) ∈ VH .
Therefore
JL (u+ εη) ≥ JL (u) .
Thus
JL (u) + ε2JL (η) + 2ε
∫
ε
n∑
i,j=1
aij(x)uiηjdx ≥ JL (u) ,
i.e.,
JL (u) + 2ε
∫
ε
n∑
i,j=1
aij(x)uiηjdx ≥ 0.
Tending ε to zero, we conclude ∫
ε
n∑
i,j=1
aij(x)uiηjdx ≥ 0. (9)
It is easy to see as η(x) in (9) we can take any function from C1 (ε) with compact support in ε\H.
Then ∫
ε\H
n∑
i,j=1
aij(x)uiηjdx ≥ 0.
Substituting η(x) on −η(x), we arrive to the equality∫
ε\H
n∑
i,j=1
aij(x)uiηjdx = 0.
Lemma 6 is proved.
Let µ be a charge of bounded variation, given on ε. We’ll 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 1
2,α(ε) capC (ε) the integral identity∫
ε
uLϕdx =
∫
ε
ϕdµ
is fulfilled.
According to Lemma 1 (at h = 0) there exists the continuous linear operator H from
∗
W 1
2,α(ε) in
◦
W 1
2,α(ε), such that for any functional T ∈
∗
W 1
2,α(ε), the function u = H (T ) is unique in
◦
W 1
2,α(ε)
generalized solution of the equation Lu = T.
The operator H is called Green operator.
By Lemma 1 this operator at p > p0 we transform
∗
W 1
2,α(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, iff for any function
ψ(x) ∈ C (ε) the integral identity ∫
ε
uψdx =
∫
ε
H(ψ)dµ (10)
is fulfilled.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1046 T. S. GADJIEV, N. Q. BAYRAMOVA
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’s say, that the charge µ ∈
∗
W 1
2,α(ε) if there exists the vector f(x) =
(
f0(x), f1(x), . . . , fn(x)
)
,
f0(x) ∈ h2(ε), f i(x) ∈ L2,λi(ε), i = 1, 2, . . . , n, for any function ϕ(x) ∈
◦
W 1
2,α(ε) capC (ε) the
integral identity
µ (ϕ) =
∫
ε
ϕdµ =
∫
ε
(
f0ϕ−
n∑
i=1
f iϕi
)
dx
is true.
At this, it is evident that ∣∣∣∣∣∣
∫
ε
ϕdµ
∣∣∣∣∣∣ ≤ C2
(
f
)
‖ϕ‖W 1
2,α(ε)
.
Lemma 7. The weak solution u(x) of the equation Lu = −µ, equaling to zero on ∂ε, belongs
to
◦
W 1
2,α(ε), iff µ ∈
∗
W 1
2,α(ε).
Proof. At first, we’ll show that if the function ϕ(x) ∈
◦
W 1
2,α(ε) satisfies the integral identity
∫
ε
n∑
i,j=1
aij(x)uiϕjdx = −
∫
ε
ϕdµ (11)
for any function ϕ(x) ∈
◦
W 1
2,α(ε) capC (ε), then it is weak solution of the equation Lu = −µ,
equaling to zero on ∂ε. Really, assuming ϕ = H(ψ), ψ(x) ∈ C (ε) we obtain∫
ε
H(ψ)dµ =
∫
ε
ϕdµ = −
∫
ε
n∑
i,j=1
aij(x)uiϕjdx =
=
∫
ε
u
n∑
i,j=1
(aij(x)ϕj)i dx =
∫
ε
uLϕdx =
∫
ε
uψdx,
and now it is sufficient to use the identity (10). We’ll show that µ ∈
∗
W 1
2,α(ε). For this, it is sufficient
to prove, that if f i(x) =
∑n
i=1
aij(x)ui(x), then f i(x) ∈ L2,λ−1
i
(ε), i = 1, 2, . . . , n. Assume in
condition (11) ξ1 = . . . = ξi−1 = ξi+1 = . . . = ξn = 0, ξi =
1√
λi(x)
.
We have
γ ≤ aii(x)
λi(x)
≤ γ−1, i = 1, . . . , n. (12)
Let i 6= j. Assuming ξk = 0 at k 6= j and k 6= i, ξi =
1√
λi(x)
, ξj =
1√
λj(x)
, we get
2γ ≤ aii(x)
λi(x)
+
ajj(x)
λj(x)
+
2aij(x)√
λi(x)λj(x)
≤ 2γ−1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1047
Using (12), we conclude
|aij(x)|√
λi(x)λj(x)
≤ γ−1 − γ, i, j = 1, . . . , n, i 6= j. (13)
From (12) and (13) it follows that
|aij(x)|√
λi(x)λj(x)
≤ γ−1, i, j = 1, . . . , n. (14)
Thus, from (14) take out for j = 1, . . . , n∫
ε
1
λj(x)
(
f j
)2
dx =
∫
ε
1
λj(x)
(
n∑
i=1
aij(x)ui
)2
dx ≤ γ−2n
n∑
i=1
∫
ε
λi(x)u
2
i dx < α.
So, µ ∈
∗
W 1
2,α(ε). Inversely, if u(x) is a weak solution of the equation Lu = −µ, vanishing on ∂ε,
then there exists µ ∈
∗
W 1
2,α(ε), such that(
f0ϕ−
n∑
i=1
f iϕi
)
dx =
∫
ε
ϕdµ =
∫
ε
uLϕdx =
=
∫
ε
u
n∑
i,j=1
(aij(x)ϕj)i dx = −
∫
ε
n∑
i,j=1
aij(x)uiϕjdx
for any function ϕ(x) ∈
◦
W 1
2,α(ε) capC (ε), Lϕ(x) ∈ C (ε) .
Then, from Lemma 1 we obtain that u(x) ∈
◦
W 1
2,α(ε).
Lemma 7 is proved.
Let now δ(x) be Dirac measure, concentrated at the point 0, y is an arbitrary fixed point ε.
The weak solution g(x, y) of the equation Ly = −δ(x − y), vanishing on ∂ε is called Green
function of the operator L in ε.
In case ε = En the corresponding function is called the fundamental solution of the operator L
and denoted by G(x, y).
According to above proved, if ψ(x) is an arbitrary function from C (ε) , then the generalized
solution ϕ(x) ∈
◦
W 1
2,α(ε) of the equation Lϕ = −ψ can be introduced in the following from:
ϕ(y) =
∫
ε
g(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 8. For any charge, of bounded variation on ε the integral
u(x) =
∫
ε
g(x, y)dµ(y)
exists, finite a.e. in ε and is weak solution of the equation Lu = −µ, equaling to zero on ∂ε.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1048 T. S. GADJIEV, N. Q. BAYRAMOVA
Proof. Without losing generality, we’ll assume that the charge µ is the measure in ε. Let
ϕ(x) ∈ C (ε), ψ(x) ≥ 0 in ε. Denote by ϕ(x) ∈
◦
W 1
2,α(ε) the generalized solution of the equation
Lϕ = −ψ(x). Then ϕ(x) ∈ C (ε) according to Lemma 1 and ψ(x) ≥ 0 according to Lemma 4. At
this
ϕ(y) =
∫
ε
g(x, y)ψ(x)dx.
Then, by Fubini theorem we conclude, that the integral
∫
ε
g(x, y)dµ(y) there exists for almost all
x ∈ ε, moreover∫
ε
H(ψ)dµ(y) =
∫
ε
ϕ(y)dµ(y) =
∫∫
ε×ε
g(x, y)ψ(x)dxdµ(y) =
∫
ε
ψ(x)u(x)dx. (15)
Let’s note, that the equality (15) is fulfilled for weak nonnegative and continuous in ε function
ψ(x). Now, it is sufficient to remember the identity (10).
Lemma 8 is proved.
Let’s consider now L-capacity of the potential u(x) of the compact H relative to the ellipsoid
ε. Before, it was proved that u(x) satisfies the inequality (9) at any nonnegative on H the function
η(x) ∈ C∞0 (ε). By the Schwartz theorem [9] there exists the measure µ on H such that∫
ε
n∑
i,j=1
aij(x)uiηjdx =
∫
ε
ηdµ. (16)
Further, since u = 1 on H in sense
◦
W 1
2,α(ε), then the carrier of the measure µ is situated on ∂H.
The measure µ is called L -capacity distribution of the compact H.
According to Lemma 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:
u(x) =
∫
ε
g (x, z) dµ(z). (17)
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 limm→∞
∥∥η(m) − u
∥∥
W 1
2,α(ε)
= 0. Assuming in
equality (10) 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)
L (H) as m→∞. Thus,
cap
(ε)
L (H) = µ(H). (18)
Lemma 9. Let relative to coefficients of the operator L conditions (1), (2), 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)
L (εR,1(y))
]−1
≤ g(x, y) ≤ C4 (γ, α, n)
[
cap
(ε)
L (εR,1(y))
]−1
. (19)
If εR,1(0) ⊂ D, x ∈ ∂εR,1(0), then
C3
[
cap
(ε)
L (εR,1(0))
]−1
≤ g (x, 0) ≤ C4
[
cap
(ε)
L (εR,1(0))
]−1
. (20)
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ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1049
Proof. Without loss of generality, we can assume that the coefficients of the operator L are
continuously differentiable in ε. The general case is obtained by means of limit passage. Then at
x 6= y the function g(x, y) is continuous by x and y, moreover
lim
x→y
g(x, y) =∞. (21)
Let a 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 (21) it follows that y is internal point y of the compact Ka.
Then L is capacity potential Ka, represented in the form (17), so it means it equal to zero in it. Thus,
1 =
∫
ε
y (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) = a and using (18) , we obtain
µa (Ka) = cap
(ε)
L (Ka) =
1
a
. (22)
Let’s assume now, a = infx∈∂εR,1(y) g(x, y). According to maximum principle εR,1(y) ⊂ Ka.
Therefore from (22) we conclude
cap
(ε)
L (εR,1(y)) ≤ cap
(ε)
L (Ka) =
1
inf
x∈∂εR,1(y)
g(x, y)
. (23)
If we’ll assume b = supx∈∂εR,1(y) g(x, y), then εR,1(y) ⊂ Ka, i.e.,
cap
(ε)
L (εR,1(y)) ≤ cap
(ε)
L (Kb) =
1
sup
x∈∂εR,1(y)
g(x, y)
. (24)
From (23) and (24) follows that
inf
x∈∂εR,1(y)
g(x, y) ≤
[
cap
(ε)
L (εR,1(y))
]−1
≤ sup
x∈∂εR,1(y)
g(x, y). (25)
On the other side, according to Lemma 3
sup
x∈∂εR,1(y)
g(x, y) ≤ C5 (γ, α, n) inf
x∈∂εR,1(y)
g(x, y). (26)
Now, the required estimations (19) follows from (25) and (26). Absolutely analogously the truthiness
of equalities (20) is proved.
Corollary 1. 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
C3
[
cap
(ε)
L (εR,1(0))
]−1
≤ G(x, y) ≤ C4
[
cap
(ε)
L (εR,1(0))
]−1
(27)
are true.
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1050 T. S. GADJIEV, N. Q. BAYRAMOVA
3. Removability criterion of the compact in the space M(D).
Theorem 1. Let relative to the coefficients of the operator L, conditions (1), (2) be fulfilled.
Then for removability of the compact E ⊂ D relative to the first boundary-value problem for the
operator L in the spaceM(D) it is necessary and sufficient, that
capL(E) = 0. (28)
Proof. Let the ellipsoid ε has the same sense, that above. It is easy to see that if condition (28)
be fulfilled, then
cap
(ε)
L (E) = 0.
Not losing generality, we can limited with case, when the coefficients of the operator L is continuously
differentiable in ε. Let’s fixed an arbitrary ε > 0 and x0 ⊂ D\E. By virtue of (28) there exists the
neighbourhood H of the compact E, such that
cap
(ε)
L
(
H
)
< ε. (29)
At this, we can assume that ε is such small, that
dist
(
x0, H
)
≥ 1
2
dist
(
x0, E
)
. (30)
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
VH(x) =
∫
ε
g(x, y)dµH(y),
moreover the function VH(x) is generalized solution of the equation LVH = 0 in ε\H, vanishing
on 0 and in ∂ε on 1 in ∂H sense W 1
2,α(ε). Let now, u(x) ∈ M(D) is an arbitrary solution of the
equation Lu = 0 in D\E, vanishing on ∂D, M = supD |u| . It is easy to see, that the function VH(x)
is nonnegative on ∂D, in sense W 1
2,α(D). Hence, it follows, that the function u(x) −MVH(x) is
generalized solution of the equation Lu = 0 in D
∖
H , is nonpositive on ∂
(
D
∖
H
)
. According to
Lemma 4 u(x)−MVH(x) ≤ 0 and D
∖
H in particular
u
(
x0
)
≤MVH
(
x0
)
≤M sup
y∈∂H
g
(
x0, y
)
µH
(
H
)
=M sup
y∈∂H
g
(
x0, y
)
cap
(ε)
L
(
H
)
. (31)
By virtue of continuity of the function g(x, y) at x 6= y and inequality (30) we obtain
sup
y∈∂H
g
(
x0, y
)
≤ C6
(
γ, α, n, x0, E
)
.
Thus, from (29) and (31) we conclude
u
(
x0
)
≤MC6ε. (32)
Using an arbitrariness ε, we lead to the inequality
u
(
x0
)
≤ 0. (33)
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ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1051
Making analogous considerations with the function u(x) +MVH(x), we have
u
(
x0
)
≥ 0. (34)
From (32), (33) and an arbitrariness of the point x0 it follows, that u(x) ≡ 0 in D\E. Thereby, the
sufficiency of condition (28) is proved. Let’s prove its necessarily. Let’s assume that capL(E) > 0.
Denote by ε′ the ellipsoid, such that ε′ ⊂ δ, E ⊂ ε′. Assume D = ε. Further, let uE(x) be VE-
L capacity potential of the compact E relative to the ellipsoid ε′ and L-capacity distribution E,
respectively. Following to [10], 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’s call the measure µ on E, L-admissible, if µ ⊂ E and
V E
µ (x) =
∫
ε′
g(x, y)dµ(y) ≤ 1 for x ∈ sup pµ. (35)
The value supµ(E) = cap
(ε′)
L (E), where an exact upper boundary is taken on all L-admissible
measures, is called L-capacity of the compact E, relative to the ellipsoid ε′.
Analogously, the L-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 function uE(x) is generalized solution of the equation
LuE = 0 in ε′ \E , equaling to zero on ∂ε′. Besides, from (34) and maximum principle it follows
that uE(x) ∈M (ε′) . On the other side uE(x) 6≡ 0, as VH(E) > 0.
Theorem 1 is proved.
Lemma 10. Let relative to the coefficients of the operator L condition (1) be fulfilled. Then, if
y ∈ ∂εR,2(0), then C7 (γ, α, n)R
n+
〈α〉
2
−2 ≤ capL (εR,1(y)) ≤ C8 (γ, α, n)R
n+
〈α〉
2
−2.
Proof. Let L0 =
∑n
i=1
∂
∂xi
(
λi(x)
∂
∂xi
)
. Then, according to (1)
γ capL0 (εR,1(y)) ≤ capL (εR,1(y)) ≤ γ−1 capL0 (εR,1(y)) . (36)
Let u(x) ∈ C∞0
(
εR, 3
2
(y)
)
, u(x) = 1 for εR,1(y), moreover
|ui(x)| ≤
C9 (λ, n)
R1+
αi
2
, i = 1, . . . , n. (37)
Then
capL0 (εR,1(y)) ≤
∫
ε
R, 32
(y)
n∑
i=1
λi(x)u
2
i dx. (38)
On the other side, as y ∈ ∂εR,2(0), then
∑n
i=1
y2i
Rαi
= 4R2 and thereby
|yi| ≤ 2R1+
αi
2 , i = 1, . . . , n.
Besides, as x ∈ εR, 3
2
(y), then
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1052 T. S. GADJIEV, N. Q. BAYRAMOVA
|xi − yi| ≤
3
2
R1+
αi
2 , i = 1, . . . , n.
Thus
|xi| ≤ |yi|+ |xi − yi| ≤
7
2
R1+
αi
2 , i = 1, . . . , n.
Hence, it follows that
|x|α ≤ R
n∑
i=1
(z
2
) 2
2+λi .
Therefore
λi(x) ≤ Cαi10R
αi ≤ Cα+
10 R
αi , i = 1, . . . , n. (39)
where α+ = max {α1, . . . , αn} .
Allowing for (37) and (39) in (38) we obtain
capL0 (εR,1(y)) ≤ C10 (α, n)R
−2mes
(
εR, 3
2
(y)
)
= C11 (α, n)R
n+
〈α〉
2
−2
and by virtue of (36), the estimation from upper in (35) is proved.
For showing the truthiness of the estimations from lower in (35), we note that
capL0 (εR,1(y)) ≥ capL0
(
εR, 1
2
√
n
(y)
)
. (40)
Besides, considering the same as in [8], we conclude
capL0
(
εR, 1
2
√
n
(y)
)
≥ C12 (α, n) cap
(ε0)
L0
(
εR, 1
2
√
n
(y)
)
, (41)
where ε0 = εR, 1√
n
(y).
Let W =
{
u(x) : u(x)C∞0 (ε0) , u(x) = 1 for x ∈ εR, 1
2
√
n
(y)
}
. Then
cap
(ε0)
L0
(
εR, 1
2
√
n
(y)
)
= inf
u∈W
∫
ε0
n∑
i=1
λi(x)u
2
i dx. (42)
On the other side, if y ∈ ∂εR,2(0), then we can find i0, 1 ≤ i0 ≤ n, such that y2i0 ≥
4R2+αi0
n
, i.e.,
|yi0 | ≥
4R1+
αi0
2
√
n
.
Besides, as x ∈ ε0, then
|xi0 − yi0 | ≤
R1+
αi0
2
√
n
.
Therefore
|xi0 | ≥ |yi0 | − |xi0 − yi0 | ≥
R1+
αi0
2
√
n
.
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ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1053
Thereby
λi(x) ≥ n
− 1
2+αi0 R, i = 1, . . . , n, (43)
where α− = min {α1, . . . , αn} .
Allowing for (43) in (42) we have
cap
(ε0)
L0
(
εR, 1
2
√
n
(y)
)
= C13 (α, n) inf
u∈W
∫
ε0
n∑
i=1
Rαiu2i dx. (44)
Denote by BR(z) the ball {x : |x− z| < R}. Let’s make in (44) the substitution of the vari-
ables vi =
xi
R1+
αi
2
, i = 1, . . . , n, and let ỹ is an image of the point y, where W̃ =
{
ũ(v) :
ũ(τ)C∞0 (B0) , ũ(τ) = 1 for v ∈ B 1
2
√
n
(ỹ)
}
. Then from (44) we deduce B0 = B 1
2
√
n
(ỹ) where by
cap
(ε0)
L0
(
εR, 1
2
√
n
(y)
)
≥ C13R
n+
〈α〉
2
−2 inf
ũ∈W̃
∫
B0
n∑
i=1
(
∂ũ
∂vi
)2
dτ =
= C13R
n+
〈α〉
2
−2 cap(B0)
(
B 1
2
√
n
(ỹ)
)
, (45)
we’ll denote by cap(B0)
(
B 1
2
√
n
(ỹ)
)
Wiener capacity of the compact B 1
2
√
n
(ỹ), relative to the ball
B0. Now, it is sufficient to note that cap(B0)
(
B 1
2
√
n
(ỹ)
)
= C14(n) and required estimation follows
from (40), (41) and (45).
Lemma 10 is proved.
Lemma 11. Let relative to the coefficients of the operator L condition (1) be fulfilled. Then
C15 (γ, α, n)R
n+
〈α〉
2
−2 ≤ capL (εR,1(y)) ≤ C16 (γ, α, n)R
n+
〈α〉
2
−2. (46)
Upper estimation in (46) is proved analogously to the estimation in (35). For the proofing of the
lower estimation, it is sufficient to note that εR, 1
4
(y) ⊂ εR,1(0), i.e.,
capL
(
εR, 1
4
(y)
)
< capL (εR,1(0)), (47)
where y =
(
1
2
R1+α
2 , 0, . . . , 0
)
and repeat the consideration of the proofing of the previous lemma.
Corollary 2. If conditions (1), (2) y ∈ ∂εR,2(0) be fulfilled, then for any ρ ∈ (0, R] the estimation
capL (ερ,1 (y)) ≤ C17 (γ, α, n) ρ
n+
〈α〉
2
−2
(
1 +
n∑
i=1
(
R
ρ
)αi)
(48)
is true.
Then v(x) ∈ C∞0
(
ερ, 3
2
(y)
)
, v(x) = 1 for x ∈ ερ,1(y)
|vi(x)| ≤
C18 (α, n)
ρ1+
αi
2
, i = 1, . . . , n,
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1054 T. S. GADJIEV, N. Q. BAYRAMOVA
capL0 (ερ,1 (y)) = γ−1C2
18ρ
−2
∫
ε
ρ, 32
(y)
n∑
i=1
λi(x)ρ
−αidx. (49)
On the other side, arguing the same, as well as in the proof of Lemma 10 we came lead to the
inequality
λi(x) < C19 (α, n) (R+ ρ)αi , x ∈ ερ, 3
2
(y), i = 1, . . . , n. (50)
Now, it is sufficient to take into account that
n∑
i=1
(
1 +
R
ρ
)αi
≤
n∑
i=1
[
1 +
(
R
ρ
)αi]
≤ n
(
1 +
n∑
i=1
(
R
ρ
)αi)
,
and from (49), (50) follows the required estimation (48).
Corollary 3. If conditions (1), (2) y 6= 0, be fulfilled, then at x ∈ εd|y|d,1(y), x 6= y for the
fundamental solution G(x, y) the estimation
G(x, y) ≥ C20 (γ, α, n)
(|x− y|α)
2−n− 〈α〉
2
1 +
∑n
i=1
(
|y|α
|x− y|α
)αi (51)
is true.
If y = 0, then estimation (51) is true for all x 6= 0. Here d =
1
n2
2
2+α
.
For proving, at first let’s show, that if y 6= 0, then y /∈ εd|y|d,2(0). Really, as
|y|α =
n∑
i=1
|yi|
2
2+αi , (52)
then there exists i0, 1 ≤ i0 ≤ n, such that
|y0|
2
2+αi0 ≥
|y|α
n
.
Thus ∣∣y2i0∣∣
(|y|α)
αi0
≥
(|y|α)
2
n2+αi
.
There by
n∑
i=1
y2i
(d |y|α)
αi ≥
y2i0
(d |y|α)
αi0
≥
(d |y|α)
2
(dn)2+αi0
=
4 (d |y|α)
2(
2
2
2+αi0 dn
)2+αi0 .
Now, it is sufficient to note that 2
2
2+αi0 dn ≤ 2
2
2+αdn = 1 and the required assertion is proved. On
the other side from (52) it follows that for all i, 1 ≤ i ≤ n,
|yi|
2
2+αi ≤ |y|α ,
i.e.,
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ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1055
n∑
i=1
y2i
(|y|α)
αi ≤ n (|y|α)
2 .
So, we’ll show that ε|y|α,√n(0), if only y 6= 0.
Let now, for y 6= 0, x ∈ εd|y|d,1(y) and x 6= y. Denote by |x− y|α the ρ. It is easy to see that
there exists i1, 1 ≤ i1 ≤ n, such that
|xi1 − yi1 |
2
2+αi1 ≥ ρ
n
.
Hence, it follows that
n∑
i=1
(xi − yi)2
ραi
≥ (xi1 − yi1)
2
ρα1
≥ ρ2
n2+αi1
≥ ρ2
n2+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 (51) at y 6= 0 follows from (27) and Corollary 1 from Lemma 10. If y = 0, then (51), it
immediately follows from (27) and Lemma 7.
Let F (x, y) be a positive function, determined in En × En, continuous at x 6= y, moreover
limx→y F (x, y) =∞ (condition (A)).
Further, let E ⊂ En be some compact. Let’s call the measure µ on E [F ] admissible, if
sup pµ ⊂ E and V E
µ (x) =
∫
E
F (x, y)dµ(y) ≤ 1, for x ∈ sup pµ.
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 2. Let relative to the coefficients of the operator L conditions (1), (2) be fulfilled. Then
for removability of the compact E ⊂ D relative to the first boundary-value problem for the operator
L in the spaceM(D) it is sufficient that
cap[F1](E) = 0, (53)
where
F1(x, y) =
[
1 +
n∑
i=1
(
|y|α
|x− y|α
)αi]−1
(|x− y|α)
2−n− 〈α〉
2 .
Proof. We’ll use the following assertion, which is proved in [10]. 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 V E
µ (x◦) = ∞. At this the potential of the
measure sup pµ can’t be bounded on any portion B, i.e., for any open set B at E′ ∈ sup pµ capB,
the potential V E′
µ (x) is not bound B. In particular, if B is n arbitrary neighborhood of the point x◦
that V E′
µ (x◦) =∞.
Let the condition (53) be fulfilled, µ be an arbitrary measure, concentrated on E, x◦ ∈ sup pµ is a
point, corresponding to the above-stated assertion at F = F1. Let’s assume at first, that x◦ 6= 0. Then
|x◦|α = v > 0. Further, let B be such small neighborhood of the point x◦, that if E′ ∈ sup pµ capB,
then
sup
y∈E′
|y|α ≤ (1 + ε) r, inf
y∈E′
|y|α ≥ (1 + ε) r,
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1056 T. S. GADJIEV, N. Q. BAYRAMOVA
where the number ε > 0 will be chosen later. Let’s consider the ellipsoids εd|y|d,1(y) at y ∈ E′.
Let’s choose ε such small, than x0 ∈ εd|y|d,1(y) for all y ∈ E′. Then according to Corollary 2 from
Lemma 7 we obtain
V E
µ
(
x0
)
=
∫
E
G
(
x0, y
)
dµ(y) ≥
∫
E′
G
(
x0, y
)
dµ(y) ≥
≥ C20
∫
E
F1
(
x0, y
)
dµ(y) = C20V
E
µ
(
x0
)
=∞.
Hence, it follows that any zero measure µ, concentrated on E can’t be L admissible. Thus capL(E) =
= 0 and the required assertion is follows from Theorem 1.
Let now x◦ = 0. Then, using the equality G(x, y) = G (y, x) and Corollary 2 from Lemma 7 we
conclude
V E
µ (0)
∫
E
G (0, y) dµ(y) =
∫
E
G (y, 0) dµ(y) ≥ C20
∫
E
F1 (y, 0) dµ(y) =
= C20
∫
E
F1 (0, y) dµ(y) = C20V
E
µ (0) =∞.
Theorem 2 is proved.
Remark. Let conditions of the real theorem be fulfilled, and the compact E ⊂ D is removable
relative to the first boundary-value problem for the operator L in the spaceM(D). Then mes(E) = 0.
At first, let’s note for proofing that the discussion are the same, as at conclusion of estimation (51),
we can show that at x ∈ εd|y|d,1(y), x 6= y (y 6= 0) and at x 6= y (y = 0) the estimations
G(x, y) ≤ C21 (γ, α, n) (|x− y|d)
2−n− 〈α〉
2 (54)
are true.
Further, analogously to Theorem 6, 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 V E
(
x1
)
=∞,
where
V E(x) =
∫
E
F2(x, y)dy.
Moreover, if B′ is an arbitrary neighborhood of the point E′ = B′ capE, then the potential V E′(x)
is not bounded on E′. Let’s consider the case x1 6= 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
V E′(x) =
∫
E′
(
n∑
i=1
|xi − yi|
2
2+αi
)2−n− 〈α〉
2
dy ≤
∫
E′
(
n∑
i=1
|xi − yi|
)2−n− 〈α〉
2
dy ≤
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
ON REMOVABLE SETS FOR DEGENERATED ELLIPTIC EQUATIONS 1057
≤
∫
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 (2)
〈α〉
2
≤ n
n− 1
≤ 3
2
and the assertion the corollary is proved.
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Received 08.01.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
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| id | umjimathkievua-article-2198 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:34Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/05/11b1c62bdc760b4b98c74c66745e9005.pdf |
| spelling | umjimathkievua-article-21982019-12-05T10:26:14Z On Removable Sets for Degenerated Elliptic Equations Про множини, що усуваються, для вироджених еліптичних рівнянь Bayramova, N. Q. Gadjiev, T. S. Байрамова, Н. Я. Гаджиїв, Т. С. We establish necessary and sufficient conditions of removability of compact sets. Встановлено нєо6хідні та достатні умови компактної усувності Institute of Mathematics, NAS of Ukraine 2014-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2198 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 8 (2014); 1041–1057 Український математичний журнал; Том 66 № 8 (2014); 1041–1057 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2198/1397 https://umj.imath.kiev.ua/index.php/umj/article/view/2198/1398 Copyright (c) 2014 Bayramova N. Q.; Gadjiev T. S. |
| spellingShingle | Bayramova, N. Q. Gadjiev, T. S. Байрамова, Н. Я. Гаджиїв, Т. С. On Removable Sets for Degenerated Elliptic Equations |
| title | On Removable Sets for Degenerated Elliptic Equations |
| title_alt | Про множини, що усуваються, для вироджених еліптичних рівнянь |
| title_full | On Removable Sets for Degenerated Elliptic Equations |
| title_fullStr | On Removable Sets for Degenerated Elliptic Equations |
| title_full_unstemmed | On Removable Sets for Degenerated Elliptic Equations |
| title_short | On Removable Sets for Degenerated Elliptic Equations |
| title_sort | on removable sets for degenerated elliptic equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2198 |
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