On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form
We consider nondivergent elliptic and parabolic equations of the second order whose leading coefficients satisfy the uniform Lipschitz condition. We find a sufficient condition for the removability of a compact set with respect to these equations in the space of Hölder functions.
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| author | Gadjiev, T. S. Mamedova, V. A. Гаджиїв, Т. С. Мамедова, В. А. |
| author_facet | Gadjiev, T. S. Mamedova, V. A. Гаджиїв, Т. С. Мамедова, В. А. |
| author_sort | Gadjiev, T. S. |
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| datestamp_date | 2020-03-18T19:45:43Z |
| description | We consider nondivergent elliptic and parabolic equations of the second order whose leading coefficients satisfy the uniform Lipschitz condition. We find a sufficient condition for the removability of a compact set with respect to these equations in the space of Hölder functions. |
| first_indexed | 2026-03-24T02:36:32Z |
| format | Article |
| fulltext |
UDC 517.9
T. S. Gadjiev, V. A. Mamedova (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
ON REMOVABLE SETS OF SOLUTIONS
OF SECOND-ORDER ELLIPTIC AND PARABOLIC
EQUATIONS IN NONDIVERGENT FORM
ПРО УСУВНI МНОЖИНИ РОЗВ’ЯЗКIВ ЕЛIПТИЧНИХ
ТА ПАРАБОЛIЧНИХ РIВНЯНЬ ДРУГОГО ПОРЯДКУ
У НЕДИВЕРГЕНТНIЙ ФОРМI
We consider nondivergent elliptic and parabolic equations of the second order whose leading coefficients
satisfy the uniform Lipschitz condition. We find the sufficient condition of removability of compact with
respect to these equations in the space of Hölder functions.
Розглянуто недивергентнi елiптичнi та параболiчнi рiвняння другого порядку, у яких коефiцiєнти при
старших членах задовольняють однорiдну умову Лiпшиця. Знайдено достатню умову усувностi компа-
кту вiдносно цих рiвнянь у просторi функцiй Гельдера.
Introduction. The subject of this paper is finding the sufficient condition of removabi-
lity of compact for nondivergent elliptic and parabolic equations in the space C0,λ
(
D
)
.
This problem have been investigated by many researchers. For the Laplace equation the
corresponding result was found by L. Carleson [1]. Concerning the second-order elliptic
equations of divergent structure, we show in this direction the papers T. S. Gadjiev,
V. A. Mamedova [2], E. I. Moiseev [3]. For a class of nondivergent elliptic equations
of the second order with discontinuous coefficients of the removability condition was
considered by I. T. Mamedov [4]. Note also the papers E. M. Landis [5], T. S. Gadjiev,
V. A.Mamedova [6], in which the conditions of removability have been obtained for a
compact in the space of continuous functions. In [7], T. Kilpelainen and X. Zhong have
studied the divergent quasilinear equation without minor members proved the removabi-
lity of compact. Removable sets for pointwise solutions of elliptic partial differential
equations was found by J. Diederich [8]. Removable singularities of solutions of linear
partial differential equations were considered in R. Harvey, J. Polking [9]. Exceptional
sets at the boundary for subharmonic functions were investigated by B. Dahlberg [10].
The aim of our paper is to consider the removability question from the single point of
view for nondivergent elliptic and parabolic equations. The paper consists of three parts:
in the first part, we consider the Dirichlet problem for nondivergent elliptic equation of
the second order; in the second part, we consider the Neumann problem for nondivergent
parabolic equation of the second order; in the third part, we consider the mixed problem
for nondivergent parabolic equation of the second order.
As opposed to previous works, in this paper, in terms of Hausdorff measures, more
exact geometrical characteristics of removability are given. Note that in most cases
in previous papers, characteristics of removability were basically presented for narrow
class of equations in terms of capacities. The value of our paper is that for the first
time in this work, we are considering the wide classes of the nondivergent elliptic
and parabolic equations with minor members. Besides, the removability conditions of
compact is obtained in terms of Hausdorff measure.
c© T. S. GADJIEV, V. A. MAMEDOVA, 2009
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11 1485
1486 T. S. GADJIEV, V. A. MAMEDOVA
1. Let’s consider Dirichlet problem for nondivergent elliptic equation of the second
order. Let D be a bounded domain situated in n dimensional Euclidean space Rn of
points x = (x1, . . . , xn) , n ≥ 3, ∂D be its boundary. Consider in D the elliptic equation
Lu =
n∑
i,j=1
aij(x)u
ij
+
n∑
i=1
bi(x)u
i
+ c(x)u = 0, (1)
in supposition that {aij(x)} is a real symmetric matrix, moreover,
γ|ξ|2 ≤
n∑
i,j=1
aij(x)ξiξj ≤ γ−1|ξ|2, ξ ∈ Rn, x ∈ D, (2)
aij(x) ∈ C1
(
D
)
, i, j = 1, . . . , n, (3)
|bi(x)| ≤ b0, −b0 ≤ c(x) ≤ 0, i = 1, . . . , n, x ∈ D. (4)
Here, ui =
∂u
∂xi
, uij =
∂2u
∂xi∂xj
, i, j = 1, . . . , n, γ ∈ (0, 1] and b0 ≥ 0 are constants.
Besides we’ll suppose that the lower coefficients of the operator L are measurable
functions in D. Let λ ∈ (0, 1) be a number. Denote by C0,λ
(
D
)
a Banach space of the
functions u(x) defined in D with the finite norm
‖u‖C0,λ(D) = sup
x∈D
|u(x)|+ sup
x,y∈D
x6=y
|u(x)− u (y)|
|x− y|λ
.
The compact E ⊂ D is called exceptional with respect to the equation (1) in the
space C0,λ
(
D
)
if from
Lu = 0, x ∈ D\E, u|∂D\E = 0, u(x) ∈ C0,λ
(
D
)
(5)
it follows that u(x) ≡ 0 in D.
Denote by BR (z) and SR (z) the ball {x : |x− z| < R} and the sphere
{
x : |x −
− z| = R
}
of radius R with the center at the point z ∈ Rn respectively. We’ll need
the following generalization of mean value theorem belonging to E. M. Landis and
M. L. Gerver [11].
Let the domain G be considered between the spheres SR (0) and S2R (0) and let the
part of the boundary of this domain, which is located strictly inside of lair R < |x| < 2R,
be a smooth surface. If we specify it in this way, it shows ∂G ∩ {x : R < |x| < 2R}
should not be ∂G. Further, let in G the uniformly positive definite matrix {aij(x)} ,
i, j = 1, . . . , n, and the function u(x) ∈ C2 (G) ∩ C1
(
G
)
be given. Then there exists
the piecewise smooth surface Σ dividing in G the spheres SR (0) and S2R (0) such that∫
Σ
∣∣∣∣∂u
∂ν
∣∣∣∣ ds ≤ Kosc
G
u
mesnG
R2
.
Here K > 0 is a constant, depending only on the matrix {aij(x)} and n, and
∂u
∂ν
is a
derivative by a conormal determined by the equality
∂u(x)
∂ν
=
n∑
i,j=1
aij(x)
∂u(x)
∂xi
cos (n̄, xj),
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
ON REMOVABLE SETS OF SOLUTIONS OF SECOND-ORDER ELLIPTIC AND PARABOLIC . . . 1487
where cos (n̄, xj) , j = 1, . . . , n, are directing cosines of unit external normal vector
to Σ.
Remark 1. We say that Σ divides the spheres SR (0) and S2R (0) in G, if there
exists such ε > 0 that each broken, laying in G and connecting the points belonging to
ε neighbourhood of SR (0) and ε neighbourhood S2R (0) has not an empty intersection
with ε.
Denote by W 1
2 (D) a Banach space of the functions u(x) given in D with the finite
norm
‖u‖W 1
2 (D) =
∫
D
(
u2 +
n∑
i,j=1
u2
i
)
dx
1/2
,
and let W̊ 1
2 (D) be a completion C∞
0 (D) by the norm of the space W 1
2 (D) .
By ms
H (A) we’ll denote the Hausdorff measure of the set A of order
s > 0. Further everywhere the notation C (. . .) means that the positive constant C
depends only on content of brackets.
Theorem 1. Let D be a bounded domain in Rn, E ⊂ D be a compact. If with
respect to the coefficients of the operator L the conditions (2) – (4) are fulfilled, then for
exceptionality of the compact E with respect to the equation (1) in the space C0,λ
(
D
)
it sufficies that
mn−2+λ
H (E) = 0. (6)
Proof. At first we show that without loss of generality we can suppose the condition
∂D ∈ C1 to be fulfilled. Suppose that the condition (6) provides the exceptionability
of the compact E for the domains, whose boundary is the surface of the class C1, but
∂D /∈ C1 and when fulfilling (6) the compact E is not exceptional. Then the problem
(5) has nontrivial solution u(x), moreover u|E = f(x) and f(x) 6= 0. We always can
suppose the lowest coefficients of the operator L to be infinitely differentiable in D.
Moreover, without loss of generality, we’ll suppose that the coefficients of the operator
L are extended to a ball B ⊃ D with saving the conditions (2) – (4). Let f+(x) =
= max {f(x), 0} , f−(x) = min {f(x), 0} , and u±(x) be solutions of the boundary-
value problems generalized by Wiener (see [12])
Lu± = 0, x ∈ D\E, u±
∣∣
∂D\E = 0, u±
∣∣
E
= f±.
It is evident, that u(x) = u+(x) + u−(x). Further, let D′ be a domain such that
∂D′ ∈ C1, D ⊂ D′, D
′ ⊂ B, and ϑ±(x) be solutions of the problems
Lϑ±(x) = 0, x ∈ D′\E,
ϑ±
∣∣
∂D′ = 0, ϑ±
∣∣
E
= f±, ϑ±(x) ∈ C0,λ (D′) .
By the maximum principle for x ∈ D
0 ≤ u+(x) ≤ ϑ+(x), ϑ−(x) ≤ u−(x) ≤ 0.
But according to our supposition ϑ+(x) ≡ ϑ−(x) ≡ 0. Hence, it follows that u(x) ≡ 0.
So, we’ll suppose that ∂D ∈ C1. Now, let u(x) be a solution of the problem (5), and
the condition (6) be fulfilled. Give an arbitrary ε > 0. Then there exists a sufficiently
small positive number δ and a system of the balls
{
Brk
(xk)
}
, k = 1, 2, . . . , such that
rk < δ, E ⊂
∞
∪
k=1
Brk
(xk) and
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
1488 T. S. GADJIEV, V. A. MAMEDOVA
∞∑
k=1
rn−2+λ
k < ε. (7)
Consider a system of the spheres
{
B2rk
(xk)
}
, and let Dk = D ∩ B2rk
(xk), k =
= 1, 2, . . . . Without loss of generality, we can suppose that the cover
{
B2rk
(xk)
}
has
a finite multiplicity a0(n). By the Landis – Gerver theorem, for every k there exists a
piecewise smooth surface Σk dividing in Dk the spheres Srk
(xk) and S2rk
(xk) such
that ∫
Σk
∣∣∣∣∂u
∂ν
∣∣∣∣ ds ≤ Kosc
Dk
u
mesnDk
r2
k
. (8)
Since u(x) ∈ C0,λ
(
D
)
, there exists a constant H1 > 0, depending only on the function
u(x), such that
osc
Dk
u ≤ H1 (2rk)λ
. (9)
Besides
mesnDk ≤ mesnB2rk
(xk) = Ωn2nrn
k , k = 1, 2, . . . , (10)
where Ωn = mesnB1 (0) . Considering (9), (10) in (8), we get∫
Σk
∣∣∣∣∂u
∂ν
∣∣∣∣ ds ≤ C1r
n−2+λ
k , k = 1, 2, . . . , (11)
where C1 = KH12n+λ.
Let DΣ be an open set, arranged in D\E, whose boundary consists on unifi-
cation of Σ and Γ, where Σ =
∞
∪
k=1
Σk, Γ = ∂D\
∞
∪
k=1
D+
k , D+
k be a part of Dk,
remained after the partition of points, arranged between the Σk and S2rk
(xk), k =
= 1, 2, . . . . Denote by D′
Σ an arbitrary connected component of DΣ, and by M -elliptic
operator of a divergent structure
B =
n∑
i,j=1
∂
∂xi
(
aij(x)
∂
∂xj
)
.
According to the Green formula for any functions z(x) and ω(x) belonging to the
intersection C2 (D′
Σ) ∩ C1
(
D
′
Σ
)
, we have∫
D′
Σ
(zBω − ωBz) dx =
∫
∂D′
Σ
(
z
∂ω
∂ν
− ω
∂z
∂ν
)
ds. (12)
Since ∂D ∈ C1, we have u(x) ∈ C2 (D′
Σ)∩C1
(
D
′
Σ
)
(see [13]). Supposing in (12)
z = 1, ω = u2, we get ∫
DΣ
B(u2)dx =
∫
∂D′
Σ
u
∂u
∂ν
ds.
But |u(x)| ≤ M < ∞ for x ∈ D. Therefore allowing for (11) and (7) we conclude∫
D′
Σ
B(u2)dx ≤ 2Ma0
∞∑
k=1
∫
Σk
∣∣∣∣∂u
∂ν
∣∣∣∣ ds ≤ 2Ma0C1
∞∑
k=1
rn−2+λ
k < C2ε, (13)
where C2 = 2Ma0C1.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
ON REMOVABLE SETS OF SOLUTIONS OF SECOND-ORDER ELLIPTIC AND PARABOLIC . . . 1489
On the other hand
B(u2) = 2uBu + 2
n∑
i,j=1
aij (x) u
i
uj ,
and besides
B(x) = Lu+
n∑
i=1
di(x)ui − c(x)u,
where di =
∑n
i=1
∂aij(x)
∂xj
− bi(x), i = 1, . . . , n. It is clear that by virtue of conditi-
ons (3), (4)
∣∣di(x)
∣∣ ≤ d0 < ∞, i = 1, . . . , n. Thus from (13) we obtain
2
∫
D′
Σ
u
n∑
i=1
di(x)uxi
dx− 2
∫
D′
Σ
u2c(x)dx + 2
∫
D′
Σ
n∑
i,j=1
aij(x)uxi
uxj
dx < C2ε.
Hence, it follows that for any α > 0
2γ
∫
D′
Σ
|∇u|2 dx < 2d0
∫
D′
Σ
|u| |ui | dx + C2ε ≤
≤ d0λ
∫
D′
Σ
|∇u|2 dx +
d0n
λ
∫
D′
Σ
u2dx + C2ε ≤
≤ d0λ
∫
D′
Σ
|∇u|2 dx +
d0nM2mesnD
λ
+ C2ε. (14)
Supposing λ =
γ
d0
from (14) we conclude∫
D′
Σ
|∇u|2 dx ≤ C3,
where C3 =
d0nM2mesnD
λ
+
C2
γ
(without loss of generality, we suppose that ε ≤ 1).
Hence, it follows that ∫
D
|∇u|2 dx ≤ C4 (C3, E, D).
Thus u(x) ∈ W 1
2 (D) . From the boundary condition and mesn−1 (∂D∩E) = 0
we get u (x) ∈ W̊ 1
2 (D) . Now, let σ ≥ 2 be a number, which will be chosen later,
D+
Σ = {x : x ∈ D′
Σ, u(x) > 0} . Without loss of generality, we suppose that the set D+
Σ
isn’t empty. Supposing in (12) z = 1, ω = uσ we get∫
D+
Σ
M (uσ) dx = σ
∫
∂D+
Σ
uσ−1
∣∣∣∣∂u
∂ν
∣∣∣∣ ds < C5 (a0,M, σ, C1) ε.
But, on the other hand
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
1490 T. S. GADJIEV, V. A. MAMEDOVA
M (uσ) = σuσ−1Mu + σ(σ − 1)uσ−2
n∑
i,j=1
aij(x)uiuj =
= σuσ−1
n∑
i=1
di(x)ui − σuσc(x) + σ(σ − 1)uσ−2
n∑
i,j=1
aij(x)uiuj .
Hence, we conclude
σ(σ − 1)
∫
D+
Σ
uσ−2
n∑
i,j=1
aij(x)uiujdx + σ
∫
D+
Σ
uσ−1
n∑
i=1
di(x)uidx < C5ε. (15)
Let D+ =
{
x : x ∈ D,u(x) > 0
}
, D+
1 be an arbitrary connected component of D+
1 .
Subject to the arbitrariness of ε from (13) we get
(σ − 1)γ
∫
D+
1
uσ−2 |∇u|2 dx ≤ d0
∫
D+
1
uσ−1
n∑
i=1
|ui| dx.
Thus, for any µ > 0
(σ − 1)γ
∫
D+
1
uσ−2 |∇u|2 dx ≤ d0µ
2
∫
D+
1
uσ−2
(
n∑
i=1
|ui|
)2
dx +
d0
2µ
∫
D+
1
uσdx. (16)
But, on the other hand
I = −σ
n∑
i=1
∫
D+
1
xiu
σ−1uidx = −
n∑
i=1
∫
D+
1
xi (uσ)i dx = n
∫
D+
1
uσdx,
and besides, for any β > 0
I =
σβ
2
∫
D+
1
r2uσdx +
σ
2β
∫
D+
1
uσ−2
(
n∑
i=1
xi
r
ui
)2
dx ≤
≤ σβ
2
∫
D+
1
r2uσdx +
σ
2β
∫
D+
1
uσ−2 |∇u|2 dx,
where r = |x|. Denote by ω(D) the quantity sup
x∈D
|x|. Without loss of generality, we’ll
suppose that ω(D) = 1. Then
I ≤ σ
2β
∫
D+
1
uσdx +
σ
2β
∫
D+
1
uσ−2 |∇u|2 dx.
Thus (
n− σβ
2
)∫
D+
1
uσdx ≤ σ
2β
∫
D+
1
uσ−2 |∇u|2 dx.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
ON REMOVABLE SETS OF SOLUTIONS OF SECOND-ORDER ELLIPTIC AND PARABOLIC . . . 1491
Now, choosing β =
n
σ
, we get finally
∫
D+
1
uσdx ≤ σ2
n2
∫
D+
1
uσ−2 |∇u|2 dx. (17)
Subject to (17) in (16) we conclude
(σ − 1)γ
∫
D+
1
uσ−2 |∇u|2 dx ≤
(
d0µn
2
+
d0σ
2
2µn2
) ∫
D+
1
uσ−2 |∇u|2 dx. (18)
Now, choose µ such that
(σ − 1)γ >
d0µn
2
+
d0σ
2
2µn2
. (19)
Then from (17) – (19) it will follow that u(x) ≡ 0 in D+
1 , and thus u(x) ≡ 0 in D.
Suppose that µ =
(σ − 1)γ
d0n
. Then (19) is equivalent to the condition
n >
(
σ
σ − 1
)2(
d0
γ
)2
. (20)
At first, suppose, that
n >
(
d0
γ
)2
. (21)
Let’s choose and fix such big σ ≥ 2, that by fulfilling (21) the inequality (20) was
true. Thus the theorem is proved, if with respect to n the condition (21) is fulfilled. Show
that it is true for any n. For this, at first, note, that if n ≥ 3, then condition (21) will
take the form
n >
(
d0ω(D)
γ
)2
.
Besides, the assertion of the theorem remains valid if in the problem (5) we replace
the condition u|∂D\E = 0 by the conditions u|Γ1
= 0 and
∂u
∂v
∣∣∣∣
Γ2
= 0, where Γ1∪Γ2 =
= ∂D\E.
Now, let the condition (21) be not fulfilled. Denote by k the least natural number,
for which
n + k >
(
d0
γ
)2
. (22)
Consider (n + k)-dimensional semi-cylinder D′ = D × (−δ0, δ0)× . . .× (−δ0, δ),
where the number δ0 > 0 will be chosen later. Since ω(D) = 1, we have ω(D′) ≤
≤ 1 + δ0
√
k. Let’s choose and fix δ0 such small that, along with the condition (22), the
condition
n + k >
(
d0ω(D′)
γ
)2
(23)
is fulfilled too.
Let
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
1492 T. S. GADJIEV, V. A. MAMEDOVA
y = (x1, . . . , xn, xn+1, . . . , xn+k), E′ = E × [−δ0, δ0]× . . .× [−δ0, δ0]︸ ︷︷ ︸
k times
.
Consider on the domain D′ the equation
L′ϑ =
n∑
i,j=1
aij(x)ϑij +
k∑
i=1
∂2ϑ
∂x2
n+i
+
n∑
i=1
bi(x)ϑi + c(x)ϑ = 0. (24)
It is easy to see that the function ϑ (y) = u (x) is a solution of the equation (24) in
D′\E′. Besides, mn+k−2+λ
H (E′) = (2δ0)kmn−2+λ
H (E) = 0, the function ϑ (y) vanishes
on
(
∂D × [−δ0, δ0]× . . .× [−δ0, δ0]︸ ︷︷ ︸
k times
)
\E′ and
∂ϑ
∂ν′
= 0 at xn+i = ±δ0, i = 1, . . . , k,
where
∂
∂ν′
is a derivative by the conormal, generated by the operator L′. Noting that
γ (L′) = γ (L) , d0 (L′) = d0 (L) and subject to the condition (23), from the proved
above we conclude that ϑ (y) ≡ 0, i.e., D′.
The theorem is proved.
Remark 2. As is seen from the proof, the assertion of the theorem remains valid
if, instead of the condition (3), it is required that the coefficients aij(x), i, j = 1, . . . , n,
have to satisfy in domain D the uniform Lipschitz condition.
2. Let’s consider Neumann problem for nondivergent parabolic equation of the
second order. In the case of Laplace operator, the question on removability sets relati-
ve to the Neumann problem was studied in the papers [2] and [3]. The questions of
removability for solutions of the first boundary-value problem for elliptic and parabolic
equations were considered in the papers [5] and [14]. In the paper [15] the analogous
questions of boundary-value problems are considered for linear and quasilinear elliptic
equations.
Let’s consider cylindrical domain QT = Ω × (0, T ) , 0 < T < ∞, in (n + 1)-
dimensional Euclidean space of the points (x1, . . . , xn, t) in Rn+1, n ≥ 2, where
Ω ⊂ Rn is a bounded domain, ∂Ω is its boundary. Let E0 be some compact set lying on
∂Ω, E = E0 × (0, T ) , Q0 =
{
(x, t): x ∈ Ω, t = 0
}
. Γ (QT ) = Q0 ∪ (∂Ω× (0, T )) be
a parabolic boundary QT . Let’s consider the following boundary-value problem in QT :
Lu =
n∑
i,j=1
aij(x, t)uxixj +
n∑
i=1
bi(x, t)uxi + c(x, t)u− ut = 0 in QT , (25)
∂u
∂ν
∣∣∣∣
Γ(QT )\E
= 0, (26)
where
∂u
∂ν
is a derivative by conormal. ∂Ω is a sufficiently smooth surface.
Let’s call the set E removable relative to the second boundary-value problem (25),
(26) in C0,λ
(
QT
)
, 0 < λ < 1, if from
Lu = 0,
∂u
∂ν
∣∣∣∣
Γ(QT )\E
= 0, u(x, t) ∈ C0,λ
(
QT
)
, (27)
it follows that u(x, t) ≡ 0 in QT , i.e., problem (25), (26) has only trivial solution.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
ON REMOVABLE SETS OF SOLUTIONS OF SECOND-ORDER ELLIPTIC AND PARABOLIC . . . 1493
Relative to the coefficients we assume the fulfilment of the following conditions:
γ|ξ|2 ≤
n∑
i,j=1
aij(x, t)ξiξj ≤ γ−1 |ξ|2 , ξ ∈ En, (28)
∣∣aij(x, t)− aij (y, t)
∣∣ ≤ k1 |x− y| , (29)∣∣bi(x, t)
∣∣ ≤ b0, −b0 ≤ c(x, t) ≤ 0. (30)
Here γ ∈ (0, 1] , i, j = 1, n, b0 > 0, k1 are constants. Besides, the lower coefficients are
the functions measurable in QT .
Denote by BR (z) and SR (z) the ball {x : |x− z| < R} and the sphere
{
x : |x −
−z| = R
}
of radius R with the center at the point z ∈ Rn.
We assume that u(x, t) is a solution of the first boundary-value problem for heat
conduction equation and consider the function z(x) =
∫ t
0
u2(x, t)dt. Let’s fix an arbi-
trary t0, 0 < t0 < T. At above mentioned conditions on coefficients, for an arbitrary
ε > 0 we can find the surfaces Σi, isolating the ball of radius ri from the ball of radius
2ri in the cylinder QT and isolating the singular points Γ (QT ) so that∫
Σi
∣∣∣∣∂u
∂ν
∣∣∣∣ ds ≤ C1 osc
ri<r<2ri
urn−2
i . (31)
The existence of such surfaces follows from [11].
Let DΣ be an open set, situated in QT \E, whose boundary consists of unification of
Σ and Γ, where Σ =
∞
∪
k=1
Σk, Dk = D∩B2rk
(xk), k = 1, 2, Γ = ∂D\
∞
∪
k=1
Dk, D+
k be a
part of Dk remained after elimination of points, arranged between the Σk and S2rk
(xk),
k = 1, 2, . . . . Denote by D′Σ an arbitrary connected component DΣ.
Further,
∂z
∂ν
=
n∑
i=1
∂z
∂xi
νi =
n∑
i=1
2νi
t∫
0
uuxidt =
t∫
0
2u
n∑
i=1
νiuxidt = 2
t∫
0
u
∂u
∂ν
dt,
where νi are directive cosines. Here, we are to take into account that by virtue of cylinder
property of QT νi remain fixed at any t. By Green formula
∫
DΣ
2
t∫
0
|∇xu|2 dt
dx +
∫
DΣ
2
t∫
0
uutdt
dx =
m∑
j=1
∫
Σj
∂z
∂ν
ds
or
2
t∫
0
|∇xu|2 dtdx +
∫
DΣ
[
u2(x, t)− u2 (x, 0)
]
dx ≤
m∑
j=1
∫
Σj
∣∣∣∣∂z
∂ν
∣∣∣∣ ds
and allowing for u|Q0
= 0 we have
∫
DΣ
t∫
0
|∇xu|2 dtdx ≤ C2
2
m∑
j=1
osc
rj≤r≤2rj
zrn−2
j .
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
1494 T. S. GADJIEV, V. A. MAMEDOVA
Since
∣∣u(x1, t)− u(x2, t)
∣∣ ≤ C3 |x1 − x2|λ we have
∣∣z(x1)− z(x2)
∣∣ ≤ t∫
0
∣∣u(x1, t) + u(x2, t)
∣∣∣∣u(x2, t)− u(x1, t)
∣∣dt ≤
≤ 2C3sup
QT
|u| |x1 − x2|λ |t| ≤ C4 |x1 − x2|λ ,
and so ∫
DΣ
t∫
0
|∇xu|2 dt dx ≤ C5
2
4λ
m∑
j=1
rn−2+λ
j ≤ C6ε.
Hence, by virtue of arbitrariness of ε we obtain that
∫
DΣ
t∫
0
∣∣∇xu
∣∣2dtdx = 0,
or
∣∣∇xu(x, t)
∣∣ = 0. Hence, allowing for ut = ∆u = 0, we have u(x, t) ≡ const . But
u|Q0
= 0, therefore u(x, t) ≡ 0.
Now let u(x, t) be a solution of problem (25), (26). Taking the function z(x) and
treating as in the work [15], allowing for the above mentioned estimations z(x) we’ll
obtain u (x, t) ≡ 0.
So, the following theorem is proved.
Theorem 2. Let QT = Ω × (0, T ) be a cylindrical domain in Rn+1, n ≥ 2,
E ⊂ QT be some compact, and let conditions (28) – (30) be fulfilled relative to the
coefficients. Then for removability of the compact E relative to problem (25), (26) in the
space C0,λ
(
QT
)
, it suffices that
mn−2+λ
H (E) = 0.
3. Let’s consider the mixed boundary-value problem for the second order nondi-
vergent parabolic equation. Let Γ1 and Γ2 be such two sets that Γ (QT ) \E = Γ1 ∪ Γ2
and Γ1 ∩ Γ2 = ∅.
Let’s consider the following mixed problem:
Lu =
n∑
i,j=1
aij(x, t)uxixj
+
n∑
i=1
bi(x, t)uxi
+ c(x, t)u− ut = 0 in QT ,
u|Γ1
= 0,
∂u
∂ν
∣∣∣∣
Γ2
= 0.
(32)
We find solution of problem (32) from the class C2,1 (QT ) ∩ C0
(
QT \E
)
.
Theorem 3. Let QT ⊂ Rn+1, n ≥ 2, be a cylinder, E ⊂ QT be a compact, and
let conditions (28) – (30) be fulfilled relative to the coefficients. Then for removability of
the compact E relative to problem (32) in the space C0,λ
(
QT
)
it suffices that
mn−2+λ
H (E) = 0.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
ON REMOVABLE SETS OF SOLUTIONS OF SECOND-ORDER ELLIPTIC AND PARABOLIC . . . 1495
The Theorem 3 is proved by the same ideas that in Theorem 1.
Let’s consider the following equation in QT :
Lu =
n∑
i,j=1
∂
∂xi
(
aij(x, t)
∂u
∂xj
)
+
+
n∑
i=1
bi(x, t)uxi
+ c(x, t)u + b (x, t, u,∇u)− ut = 0. (33)
Assume that aij(x, t) are bounded, measurable functions satisfying condition (28),
bi(x, t), c(x, t) satisfy condition (30) and
∣∣b (x, t, u,∇u)
∣∣ ≤ g(u) · |∇u|2 ,
k∫
0
g(u)du < ∞, k < ∞. (34)
For equation (33) we consider the problem
Lu = 0 in QT \E,
∂u
∂ν
∣∣∣∣
Γ(QT )\E
= 0. (35)
We try to find a solution of this problem in the class{
W 1
2 (QT ) ∩ C0,λ
(
QT
)
, 0 ≤ u(x, t) ≤ k
}
.
Theorem 4. Let QT be a cylindrical domain in Rn+1, n ≥ 2, E ⊂ QT be some
compact, and let relative to the coefficients of equation (33) conditions (28), (30), (34)
be fulfilled. Then for removability of the compact E relative to problem (35) it suffices
that
mn−2+λ
H (E) = 0.
Before we pass to the proof, let’s note that if the solutions are sought in the class{
W 1
2 (QT ) ∩ C0
(
QT
)
, 0 ≤ u(x, t) ≤ k
}
, then the set E is removable if
mn−2
H (E) < ∞.
Proof of Theorem 3. The function ϑ(x, t) =
∫ u(x,t)
0
exp
(
1
λ1
∫ t
0
g (g(τ)dτ)
)
dt
is a subsolution of the linear operator
L1 =
n∑
i,j=1
∂
∂xi
(
aij(x, t)
∂
∂xi
)
− ∂
∂t
.
Further, analogously to the proof of Theorem 1, we obtain that ϑ(x, t) ≡ 0, which
proves the theorem.
1. Carleson L. Selected problems on exceptional sets. – Toronto etc.: D. Van. Nostrand Comp., 1967. –
126 p.
2. Gadjiev T. S., Mamedova V. A. On removable sets of solutions of boundary-value problems for quasilinear
elliptic equations // Trudi Voron. Zimney Mat. Shkoly. – 2006. – P. 12 – 17.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
1496 T. S. GADJIEV, V. A. MAMEDOVA
3. Moiseev E. I. On exceptional and unexceptional boundary sets of Neuman problem // Differents. Uravne-
niya. – 1973. – 9, № 5. – P. 901 – 911 (in Russian).
4. Mamedov I. T. On exceptional sets of solutions of Gilbarg – Serrin equation in the space of Hölder
functions // Izv. AN Azerb. Ser. phys.-tech., math. Issue math. and mech. – 1998. – 18, № 2. – P. 46 – 51
(in Russian).
5. Landis E. M. To question on uniqueness of solution of the first boundary-value problem for elliptic and
parabolic equations of the second order // Uspekhi Mat. Nauk. – 1978. – 33, № 3. – P. 151 (in Russian).
6. Gadjiev T. S., Mamedova V. A. Removable sets of solutions of the second order boundary-value problem
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1972. – 165. – P. 333 – 352.
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Received 07.04.09
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 11
|
| id | umjimathkievua-article-3116 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:32Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ca/b8ffdcc30315513fc8e96f05dd0203ca.pdf |
| spelling | umjimathkievua-article-31162020-03-18T19:45:43Z On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form Про усувні множини розв'язків еліптичних та параболічних рiвнянь другого порядку у недивергентній формi Gadjiev, T. S. Mamedova, V. A. Гаджиїв, Т. С. Мамедова, В. А. We consider nondivergent elliptic and parabolic equations of the second order whose leading coefficients satisfy the uniform Lipschitz condition. We find a sufficient condition for the removability of a compact set with respect to these equations in the space of Hölder functions. Розглянуто недивергентні єліптичні та параболiчнi рівняння другого порядку, у яких коєФіцієнти при старших членах задовольняють однорідну умову Ліпшиця. Знайдено достатню умову усувності компакту відносно цих рівнянь у просторі функцій Гельдера. Institute of Mathematics, NAS of Ukraine 2009-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3116 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 11 (2009); 1485-1496 Український математичний журнал; Том 61 № 11 (2009); 1485-1496 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3116/2980 https://umj.imath.kiev.ua/index.php/umj/article/view/3116/2981 Copyright (c) 2009 Gadjiev T. S.; Mamedova V. A. |
| spellingShingle | Gadjiev, T. S. Mamedova, V. A. Гаджиїв, Т. С. Мамедова, В. А. On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form |
| title | On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form |
| title_alt | Про усувні множини розв'язків еліптичних та параболічних рiвнянь другого порядку у недивергентній формi |
| title_full | On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form |
| title_fullStr | On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form |
| title_full_unstemmed | On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form |
| title_short | On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form |
| title_sort | on removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3116 |
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