On a classical solvability of a Florin problem
There is considered the multidimensional two-phase Stefan problem with a small parameter k at the velocity of a free boundary in a Stefan condition. The unique solvability and coercive uniform with respect to k estimate of the solution for t ≤ T₀, T₀ - independent on k, are proved and on the basis o...
Збережено в:
| Дата: | 2008 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2008
|
| Назва видання: | Український математичний вісник |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/124294 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On a classical solvability of a Florin problem / G.I. Bizhanova // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 16-31. — Бібліогр.: 21 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-124294 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1242942025-02-09T21:13:09Z On a classical solvability of a Florin problem Bizhanova, G.I. There is considered the multidimensional two-phase Stefan problem with a small parameter k at the velocity of a free boundary in a Stefan condition. The unique solvability and coercive uniform with respect to k estimate of the solution for t ≤ T₀, T₀ - independent on k, are proved and on the basis of this the existence, uniqueness and estimate of the solution of a Florin problem (Stefan problem with k = 0) are obtained in the Holder spaces. 2008 Article On a classical solvability of a Florin problem / G.I. Bizhanova // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 16-31. — Бібліогр.: 21 назв. — англ. 1810-3200 2000 MSC. 35R35, 35B25, 35K60 https://nasplib.isofts.kiev.ua/handle/123456789/124294 en Український математичний вісник application/pdf Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
There is considered the multidimensional two-phase Stefan problem with a small parameter k at the velocity of a free boundary in a Stefan condition. The unique solvability and coercive uniform with respect to k estimate of the solution for t ≤ T₀, T₀ - independent on k, are proved and on the basis of this the existence, uniqueness and estimate of the solution of a Florin problem (Stefan problem with k = 0) are obtained in the Holder spaces. |
| format |
Article |
| author |
Bizhanova, G.I. |
| spellingShingle |
Bizhanova, G.I. On a classical solvability of a Florin problem Український математичний вісник |
| author_facet |
Bizhanova, G.I. |
| author_sort |
Bizhanova, G.I. |
| title |
On a classical solvability of a Florin problem |
| title_short |
On a classical solvability of a Florin problem |
| title_full |
On a classical solvability of a Florin problem |
| title_fullStr |
On a classical solvability of a Florin problem |
| title_full_unstemmed |
On a classical solvability of a Florin problem |
| title_sort |
on a classical solvability of a florin problem |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2008 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/124294 |
| citation_txt |
On a classical solvability of a Florin problem / G.I. Bizhanova // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 16-31. — Бібліогр.: 21 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT bizhanovagi onaclassicalsolvabilityofaflorinproblem |
| first_indexed |
2025-11-30T20:52:00Z |
| last_indexed |
2025-11-30T20:52:00Z |
| _version_ |
1850250027845287936 |
| fulltext |
Український математичний вiсник
Том 5 (2008), № 1, 16 – 31
On a classical solvability of a Florin problem
Galina I. Bizhanova
(Presented by E. Ya. Khruslov)
Abstract. There is considered the multidimensional two-phase Stefan
problem with a small parameter κ at the velocity of a free boundary in
a Stefan condition. The unique solvability and coercive uniform with
respect to κ estimate of the solution for t ≤ T0, T0 — independent
on κ, are proved and on the basis of this the existence, uniqueness and
estimate of the solution of a Florin problem (Stefan problem with κ = 0)
are obtained in the Hölder spaces.
2000 MSC. 35R35, 35B25, 35K60.
Key words and phrases. Free boundary problem, parabolic equa-
tion, small parameter, Hölder space, existence, uniqueness, estimate of
solution.
1. Statement of the problems. Main results
Let Ω be a bounded domain in R
n, n ≥ 2, with a boundary Σ. In
Ω there is a closed surface γ(t), t ∈ [0, t0], which divides Ω into two
sub-domains Ω1(t) and Ω2(t) with the boundaries ∂Ω1(t) = Σ ∪ γ(t),
∂Ω2(t) = γ(t). Denote γ(0) := Γ ⊂ Ω and Ωj(0) := Ωj , j = 1, 2. We
assume dist(Γ, Σ) ≥ d0 = const > 0, diam Ω2 ≥ d0 to guarantee that a
surface γ(t) will not touch Σ and a domain Ω2(t) will not degenerate for
small time.
Let Γ ∈ C2+α, α ∈ (0, 1), then we can represent γ(t) for small t ≤ t0
by an equation [8, 9]
x = ξ + ρ(ξ, t)N(ξ), ξ = ξ(x) ∈ Γ, t ∈ [0, t0], (1.1)
where ρ
∣∣
t=0
= 0, N(ξ) = (N1, . . . , Nn) ∈ C2+α(Γ; Rn) is a unit vector
field on Γ satisfying condition ν0(ξ)N
T (ξ) ≥ d1 = const > 0, ν0(ξ) is a
unit normal to Γ directed into Ω2
Received 8.02.2008
ISSN 1810 – 3200. c© Iнститут математики НАН України
G. I. Bizhanova 17
Here and further by symbol “T ” we denote transposed matrix AT and
column-vector NT ; dk, Ck, k = 1, 2, . . . , are positive constants.
Let ΩT = Ω × (0, T ), ΣT = Σ × [0, T ], ΓT = Γ × [0, T ], ΩjT =
Ωj × (0, T ), QjT =
{
(x, t) : x ∈ Ωj(t), t ∈ (0, T )
}
, j = 1, 2.
Consider two-phase Stefan problem with the unknown functions
uj(x, t), j = 1, 2, and ρ(ξ, t) satisfying the parabolic equations, initial
and boundary conditions
∂tuj − aj ∆uj = 0 in QjT , j = 1, 2, (1.2)
γ(t)
∣∣
t=0
= Γ, uj
∣∣
t=0
= u0j(x) in Ωj , j = 1, 2, (1.3)
u1
∣∣
Σ
= p(x, t), t ∈ (0, T ), (1.4)
and conditions on a free boundary γ(t), t ∈ (0, T ),
u1 = u2 = 0, (1.5)
λ1 ∂νu1 − λ2 ∂νu2 = −κ νNT ∂tρ, (1.6)
where aj , λj , j = 1, 2, are positive constants; κ > 0 — small parameter,
ν(x, t) — a unit normal to γ(t) directed into Ω2(t), νN
T∂tρ=Vν is a
velocity of a free boundary on the direction of ν due to (1.1); ∂t = ∂/∂t,
∂ν = ∂/∂ν = ν∇T is the normal derivative, ∇ = ∂x1 , . . . , ∂xn .
Letting κ to zero in the condition (1.6) we shall have degenerate Stefan
or Florin [14] problem with unknown functions uj , j = 1, 2, ρ:
∂tuj − aj ∆uj = 0 in QjT , j = 1, 2, (1.7)
γ(t)
∣∣
t=0
= Γ, uj
∣∣
t=0
= u0j(x) in Ωj , j = 1, 2, (1.8)
u1
∣∣
Σ
= p(x, t), t ∈ (0, T ), (1.9)
u1 = u2 = 0, λ1 ∂νu1 − λ2 ∂νu2 = 0 on γ(t), t ∈ (0, T ). (1.10)
Classical solvability of the multidimensional Stefan problem was stud-
ied by A. Friedman and D. Kinderlehrer [15], L. A. Caffarelli [11, 12],
D. Kinderlehrer and L. Nirenberg [17], A. M. Meirmanov [19], E. I. Han-
zawa [16], B. V. Bazaliy [1], E. V. Radkevich [20], B. V. Bazaliy and
S. P. Degtyarev [2], M. A. Borodin [10], G. I. Bizhanova [5,6], G. I. Bizha-
nova and V. A. Solonnikov [9]. In [21] J. F. Rodrigues, V. A. Solonnikov
and F. Yi have obtained the existence of the multidimensional one-phase
Florin problem locally in time in the Hölder space C2+β,1+β/2, 0 < β < α,
with the help of the imbedding theorem applied to the solution from
C2+α,1+α/2, α ∈ (0, 1) of the corresponding Stefan problem with the
small parameter.
18 On a classical solvability...
Solvability in C2+α,1+α/2, α ∈ (0, 1), for small time of the multidimen-
sional one-phase Florin problem was established by A. Fasano, M. Prim-
icerio and E. V. Radkevich [13]. In [5, 6] G. I. Bizhanova has proved
existence, uniqueness and estimates of the solution of multidimensional
two-phase Florin problem in the classical and weighted Hölder spaces
with time power weights [3], when free boundary is a graph of function
on the plane xn = 0 and on the unit sphere.
We are considering (1.2)–(1.6) as a problem with a small parameter
κ at the principle term — velocity of a free boundary in the condition
(1.6). Comparing Theorems 1.1 and 1.2 we can see that the smoothness
of a free boundary in the Stefan and Florin problems is different and it
is higher in the Stefan problem. That is the problem (1.2)–(1.6) with a
small parameter is singularly perturbed.
We note that applying of the method of a small parameter permits us
to obtain required results for the solutions of the problems, in which one
of the unknowns is given in the implicit form, like in the Florin problem
a free boundary is set.
Using the solution of the Stefan problem (1.2)–(1.6) and letting κ to
zero we shall prove existence, uniqueness and estimate of the solution
of the Florin problem (1.7)–(1.10) without loss of a smoothness of this
solution. We can not apply for that available results on the solvability of
Stefan problem, because the time T0 of an existence of the solution and
a constant in the estimate for it depend on a small parameter κ.
In Chapter 2 we prove Theorem 1.1 for the solution of Stefan problem
with T0 and a constant in the estimate of a solution independent on κ
and in Chapter 3 on the basis of Theorem 1.1 we obtain Theorem 1.2 on
the solvability of a Florin problem.
The problems are considering in the classical Hölder spaces C
l,l/2
x t (Ω̄T ),
l is positive non–integer, of the functions u(x, t) with the norm [18]
|u|
(l)
ΩT
:=
∑
2k+|m|<l
|∂k
t ∂
m
x u|ΩT
+
∑
2k+|m|=[l]
[∂k
t ∂
m
x u]
(l−[l])
ΩT
+
∑
2k+|m|=[l]−1
[∂k
t ∂
m
x u]
(
1+l−[l]
2
)
t,ΩT
,
where the last term is omitted, if [l] = 0, | v|ΩT
= max(x,t)∈ΩT
|v|,
[v]
(α)
ΩT
= [v]
(α)
x,ΩT
+ [v]
(α/2)
t,ΩT
,
[v]
(α)
x,ΩT
= max
(x,t),(z,t)∈ΩT
∣∣v(x, t) − v(z, t)
∣∣ |x− z|−α,
G. I. Bizhanova 19
[v]
(α)
t,ΩT
= max
(x,t), (x,t1)∈ΩT
∣∣v(x, t) − v(x, t1)
∣∣ |t− t1|
−α, α ∈ (0, 1).
◦
C
l,l/2
x t (ΩT ) is a sub-space of the functions u(x, t) ∈ C
l,l/2
x t (ΩT ) satis-
fying the conditions ∂k
t u
∣∣
t=0
= 0, k ≤ [l/2].
We formulate the main results of the paper.
Theorem 1.1. Let Σ, Γ ∈ C2+α, α ∈ (0, 1).
For any functions u0j ∈ C2+α(Ωj), j = 1, 2, p ∈ C
2+α,1+α/2
x t (ΣT )
satisfying the compatibility conditions of zero and the first order on Σ
and Γ and the conditions
0 < κ ≤ κ0, ∂ν0 u0j
∣∣
Γ
≤ −d2 < 0, j = 1, 2, (1.11)
there exists T0 > 0 such that the Stefan problem (1.2)–(1.6) has a unique
solution uj ∈ C
2+α,1+α/2
x t (QjT0
), j = 1, 2, ρ ∈ C
2+α,1+α/2
x t (ΓT0), κ∂tρ ∈
C
1+α,1+α/2
x t (ΓT0) and the following estimate holds for t ∈ (0, T0]:
2∑
j=1
|uj |
(2+α)
Qjt
+ |ρ|
(2+α)
Γt
+ |κ∂tρ|
(1+α)
Γt
≤ C1
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
Σt
)
,
(1.12)
where T0 and a constant C1 do not depend on κ.
Theorem 1.2. Let Σ, Γ ∈ C2+α, α ∈ (0, 1). For any functions u0j ∈
C2+α(Ωj), j = 1, 2, p ∈ C
2+α,1+α/2
x t (ΣT ) satisfying the compatibility
conditions of zero and the first order on Σ and Γ and the condition
∂ν0 u0j
∣∣
Γ
≤ −d2, j = 1, 2, there exists T0 > 0 such that the Florin prob-
lem (1.7)–(1.10) has a unique solution uj ∈ C
2+α,1+α/2
x t (QjT0
), j = 1, 2,
ρ ∈ C
2+α,1+α/2
x t (ΓT0) and the following estimate holds for t ∈ (0, T0]:
2∑
j=1
|uj |
(2+α)
Qjt
+ |ρ|
(2+α)
Γt
≤ C2
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
Σt
)
. (1.13)
We note that the compatibility conditions for a Florin problem are
the compatibility conditions for a Stefan problem with κ = 0.
2. Proof of Theorem 1.1
We apply coordinate transformation [8, 9, 16] to the problem (1.2)–
(1.6) to reduce it to the problem in given domains Ω1 ∪ Ω2
x = y + χ(λ(y)) ρ(ξ, τ)N(ξ), y ∈ O, ξ = ξ(y) ∈ Γ,
x = y, y ∈ Ω\O, t = τ,
(2.1)
20 On a classical solvability...
where O is a 2λ0-neighborhood of Γ, λ0 > 0 is sufficiently small value
depending on Γ and such that γ(t) ⊂ O for ∀ t ∈ [0, t0], λ(y) is the
distance between a point ξ = ξ(y) ∈ Γ and a point y ∈ O lying on
a vector N(ξ) or it’s continuation (see [9]), χ(λ) is a smooth cut-off
function: χ = 1, |λ| < λ0, χ = 0, |λ| ≥ 2λ0.
The mapping (2.1) transforms Γ into γ(t) and the domains Ωj into
the unknown ones Ωj(t), j = 1, 2. We keep the variable t instead of a
new one τ .
We construct auxiliary functions [18] ρ0(ξ, t) ∈ C
3+α, 3+α
2
y t (ΓT ) under
the conditions
ρ0
∣∣
t=0
= 0, ∂tρ0
∣∣
t=0
≡ ∂tρ
∣∣
t=0
= −
aj∆u0j
∣∣
Γ
ν0NT∂ν0u0j
∣∣
Γ
, j = 1, 2,
and Vj(y, t) ∈ C
2+α,1+α/2
y t (Rn
T ), j = 1, 2, as the solutions of the Cauchy
problems
∂tVj − aj ∆Vj−χ∂tρ0N ∇TVj = 0 in R
n
T , (2.2)
Vj
∣∣
t=0
= ũ0j(y) in R
n. (2.3)
These functions satisfy the estimates
|ρ0|
(3+α)
ΓT
≤ C3|u0j |
(2+α)
Ωj
, |Vj |
(2+α)
R
n
T
≤ C4
2∑
j=1
|u0j |
(2+α)
Ωj
, j = 1, 2.
(2.4)
Here symbol “ ˜ ” denotes the smooth extension of a function into R
n,
R
n
T = R
n × (0, T ); ρ
∣∣
t=0
is found, when we reduce the compatibility
conditions. We note also that the functions ρ0, V1, V2 are one and the
same for the Stefan and Florin problems.
In the problem (1.2)–(1.6) we make the following substitutions
ρ(ξ, t) = ρ0(ξ, t)+ψ(ξ, t), uj(y+χρN, t) = vj(y, t)+Vj(y, t), j = 1, 2,
(2.5)
where ψ, vj are the new unknown functions satisfying zero initial condi-
tions ∂k
t vj
∣∣
t=0
= 0, ∂k
t ψ
∣∣
t=0
= 0, k = 0, 1; j = 1, 2.
Jacobian matrix of the transformation (2.1) J = {∂xi/∂yj}1≤i,j≤n
may be represented in the form [8]
J =
{
δij + ∂yj
(
Ni χ(ρ0 + ψ)
)}
1≤i, j≤n
= I +
(
∇TNχ
(
ρ0 + ψ
))T
:= I + J01 + J1 = J0 + J1,
J0 = I + J01, J01 =
(
∇TNχρ0
)T
,
J1 =
(
∇TNχψ
)T
= NTχ∇ψ + ψ
(
∇T (Nχ)
)T
:= J11 + J12,
(2.6)
G. I. Bizhanova 21
where δij is a Kronecker delta, I is identity matrix, ∇ = (∂y1 , . . . , ∂yn).
With the help of the expansion formulae of the inverse Jacobian ma-
trix J−1 and J−1
0 : J−1 ≡ (I + B)−1 = I − BJ−1, B = J01 + J1,
J−1
0 ≡ (I + J01)
−1 = I − J01J
−1
0 , we extract linear principal terms with
respect to unknown functions, known functions and remainder terms con-
taining the rests after separating linear terms and known functions. Then
we obtain the problem in a given domain Ω1 ∩Ω2 for the unknown func-
tions vj , j = 1, 2, ψ satisfying zero initial data
∂tvj − aj ∆vj − (∂tψ − aj ∆ψ)χNJ−T
0 ∇TVj = fj(y, t) + Fj(vj , ψ)
in ΩjT , j = 1, 2, (2.7)
v1
∣∣
Σ
= p1(y, t), t ∈ (0, T ), (2.8)
vj
∣∣
Γ
= ηj(y, t), t ∈ (0, T ), j = 1, 2, (2.9)
(
λ1 ∂ν0v1 − λ2 ∂ν0v2 + κ ν0N
T ∂tψ
− ν0N
T
[
(λ1 ∇V1 − λ2 ∇V2)J
−1
0 J−T
0 + κNJ−T
0 ∂tρ0
]
∇Tψ
) ∣∣∣
Γ
= ϕ(y, t;κ) + Φ(v1, v2, ψ;κ)
∣∣∣
Γ
, t ∈ (0, T ), (2.10)
where the symbol “T ” means transposed matrix and column-vector, ν0N
T
≥ d1 > 0,
fj = χ∂tρ0NJ
−T
0 ∇TVj − ∂tVj + aj(J
−T
0 ∇T )TJ−T
0 ∇TVj , j = 1, 2,
(2.11)
Fj = χ∂t(ρ0 + ψ)NJ−T (∇T vj − JT
1 J
−T
0 ∇TVj)
+ aj
[
∇BT +
(
BT J−T ∇T
)T
J−TJT
11
− (J−T
0 JT
1 J
−T ∇T )T + (J−T ∇T )TJ−TJT
12
]
J−T
0 ∇TVj
− aj
[
∇BT +
(
BT J−T∇T
)T ]
J−T∇T vj
− aj(∇ψ)∇T (χNJ−T
0 ∇TVj), j = 1, 2, (2.12)
p1 =
(
p(y, t) − V1(y, t)
)∣∣∣
Σ
, ηj = −Vj(y, t)
∣∣∣
Γ
, j = 1, 2, (2.13)
ϕ = −ν0 J
−1
0
[
J−T
0 ∇T (λ1V1 − λ2V2)
∣∣
Γ
+ κNT∂tρ0
]
, (2.14)
Φ = ν0
(
BT + J−1B
)
J−T∇T (λ1 v1 − λ2 v2)
− ν0 M∇T (λ1 V1 − λ2 V2)
+ κ ν0J
−1
(
BNT∂tψ +
(
J12 −B J11
)
J−1
0 NT∂tρ0
)
, (2.15)
22 On a classical solvability...
M = J−1
[
B JT
11+JT
01J
−T
0 JT
11−J
−T
0 JT
12
]
J−T +J−1
(
B J11−J12
)
J−1
0 J−T
0 .
In the same manner we reduce Florin problem (1.7)–(1.10) to the
problem with unknown functions vj , j = 1, 2, ψ satisfying zero initial
conditions
∂tvj − aj ∆vj − (∂tψ − aj ∆ψ)χNJ−T
0 ∇TVj = fj(y, t) + Fj(vj , ψ)
in ΩjT , j = 1, 2, (2.16)
v1
∣∣
Σ
= p1(y, t), t ∈ (0, T ), vj
∣∣
Γ
= ηj(y, t), j = 1, 2, (2.17)
(
λ1 ∂ν0v1 − λ2 ∂ν0v2 − ν0N
T (λ1 ∇V1 − λ2 ∇V2)J
−1
0 J−T
0 ∇T ψ
) ∣∣∣
Γ
= ϕ(y, t; 0) + Φ(v1, v2, ψ; 0)
∣∣∣
Γ
, t ∈ (0, T ), (2.18)
where functions fj , Fj , p1, ηj , ϕ, Φ are determined by formulae (2.11)–
(2.15).
Theorem 2.1. Let the assumptions of Theorem 1.1 be fulfilled. Then
there exists T0 > 0, such that the Stefan problem (2.7)–(2.10) has a
unique solution vj ∈
◦
C
2+α,1+α/2
y t (ΩjT0), j = 1, 2, ψ ∈
◦
C
2+α,1+α/2
y t ( ΓT0),
κ∂tψ ∈
◦
C
1+α, 1+α
2
y t ( ΓT0) and this solution satisfies an estimate for t ≤ T0
2∑
j=1
|vj |
(2+α)
Ωjt
+ |ψ|
(2+α)
Γt
+ |κ∂tψ|
(1+α)
Γt
≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p |
(2+α)
Σt
)
,
(2.19)
where T0 and a constant C5 do not depend on κ.
Consider the functions fj , p1, ηj , j = 1, 2, ϕ determined by (2.11),
(2.13), (2.14).
Lemma 2.1. Let Σ, Γ ∈ C2+α, α ∈ (0, 1). For any functions u0j ∈
C2+α(Ωj), j = 1, 2, p ∈ C
2+α,1+α/2
y t (ΣT ) satisfying the compatibility
conditions of zero and the first order on Σ and Γ there exists t1 > 0,
such that fj ∈
◦
C
α,α/2
y t (Ωjt1), ηj ∈
◦
C
2+α,1+α/2
y t ( Γt1), j = 1, 2, p1 ∈
◦
C
2+α,1+α/2
y t (Σt1), ϕ ∈
◦
C
1+α, 1+α
2
y t ( Γt1) and an estimate holds
2∑
j=1
(
|fj |
(α)
Ωjt
+ |ηj |
(2+α)
Γt
)
+ |p1|
(2+α)
Σt
+ |ϕ|
(1+α)
Γt
≤ C6
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
Σt
)
, (2.20)
for t ≤ t1, κ ∈ (0, κ0], where constant C6 does not depend on κ.
G. I. Bizhanova 23
Proof. This estimate is derived with the help of the estimates (2.4) for
the functions ρ0, V1, V2 and an estimate ‖J−1
0 ‖
(α+ν)
Γt
≤ 1/(1 − q), ν =
0, 1, q ∈ (0, 1), of the inverse matrix J−1
0 existing for t ≤ t1 under
the conditions ρ0(ξ(y), t) ∈ C
3+α, 3+α
2
y t (ΓT ), ρ0
∣∣
t=0
= 0 (see [8]) (here
‖{aij}1≤i,j≤n‖
(l)
ΓT
:= n maxi,j |aij |
(l)
ΓT
). The functions fj satisfy zero initial
data by (2.2), (2.3), the functions p1, ηj , j = 1, 2, ϕ — due to the
compatibility conditions.
Consider a linear problem with the unknowns satisfying zero initial
data
∂tZj − aj ∆Zj − αj(x, t)
(
∂tΨ − aj ∆Ψ
)
= fj(x, t) in ΩjT , j = 1, 2,
(2.21)
Z1
∣∣
Σ
= p1(x, t), t ∈ (0, T ), (2.22)
Zj
∣∣
Γ
= ηj(x, t), t ∈ (0, T ), j = 1, 2, (2.23)
(
λ1 ∂ν0Z1 − λ2 ∂ν0Z2
)∣∣
Γ
+ κ ∂tΨ + d(x, t)∇T Ψ = ϕ(x, t), t ∈ (0, T ),
(2.24)
where λj , aj are positive constants, j = 1, 2, d = (d1, . . . , dn).
Theorem 2.2. Let Σ, Γ ∈ C2+α, α ∈ (0, 1), αj(x, t) ∈ C
α,α/2
x t (ΩjT ),
di(x, t) ∈ C
1+α,1+α/2
x t (ΓT ), j = 1, 2, i = 1, . . . , n, and
0 < κ ≤ κ0, αj(x, 0)
∣∣
Γ
≤ −d3 < 0, j = 1, 2. (2.25)
Then for every functions fj ∈
◦
C
α,α/2
x t (ΩjT ), p1 ∈
◦
C
2+α,1+α/2
x t (ΣT ),
ηj ∈
◦
C
2+α,1+α/2
x t ( ΓT ), j = 1, 2, ϕ ∈
◦
C
1+α, 1+α
2
x t ( ΓT ) the problem (2.21)–
(2.24) has a unique solution Zj ∈
◦
C
2+α,1+α/2
x t (ΩjT ), j = 1, 2, Ψ ∈
◦
C
2+α,1+α/2
x t ( ΓT ), κ∂tΨ ∈
◦
C
1+α, 1+α
2
x t ( ΓT ) and it satisfies an estimate
2∑
j=1
|Zj |
(2+α)
Ωjt
+ |Ψ|
(2+α)
Γt
+ |κ∂tΨ|
(1+α)
Γt
≤ C7
( 2∑
j=1
(
|fj |
(α)
Ωjt
+ |ηj |
(2+α)
Γt
)
+ |p1|
(2+α)
Σt
+ |ϕ|
(1+α)
Γt
)
,
t ≤ T, (2.26)
where T and constant C7 do not depend on κ.
24 On a classical solvability...
This theorem is proved by standard technique. The proof is based on
the following model problem with unknown functions ψ(x′, t), uj(x, t),
j = 1, 2,
∂tuj − aj ∆uj = 0 in DjT , j = 1, 2,
uj
∣∣
t=0
= 0 in Dj , j = 1, 2;
ψ
∣∣
t=0
= 0 on R;
uj + αjψ = 0 on RT , j = 1, 2,
(2.27)
b∇Tu1 − c∇Tu2 + h′∇′Tψ + κ ∂tψ = g(x′, t) on RT ,
where all coefficients are constant; D1 := R
n
−, D2 := R
n
+, DjT := Dj ×
(0, T ); R is a plane xn = 0 in R
n, RT := R× [0, T ]; b = (b1, . . . , bn), c =
(c1, . . . , cn), h′ = (h1, . . . , hn−1); αj , j = 1, 2, are coefficients αj(ξ0, 0),
ξ0 ∈ Γ in the equations (2.21).
In the Hölder spaces this problem with arbitrary κ was studied by
B. V. Bazaliy [1], E. V. Radkevich [20], G. I. Bizhanova [4]. J. F. Ro-
drigues, V. A. Solonnikov, F. Yi [21] have established the uniform on κ
estimates of the solution of a one-phase problem.
In [7] the following theorem was proved.
Theorem 2.3. Let αj < 0, j = 1, 2, bn > 0, cn > 0, 0 < κ ≤ κ0. For
every function g ∈
◦
C
1+α, 1+α
2
x′ t (RT ), α ∈ (0, 1), the problem (2.27) has a
unique solution uj ∈
◦
C
2+α,1+α/2
x t (DjT ), j = 1, 2, ψ ∈
◦
C
2+α,1+α/2
x′ t (RT ),
κ ∂tψ ∈
◦
C
1+α, 1+α
2
x′ t (RT ), and it satisfies the estimate
2∑
j=1
|uj |
(2+α)
DjT
+ |ψ|
(2+α)
RT
+ |κ∂tψ|
(1+α)
RT
≤ C8|g|
(1+α)
RT
,
where T and a constant C8 do not depend on κ.
Proof of Theorem 2.1. We introduce the Hölder spaces. Let
◦
D2+α(ΓT )
be the space of functions ψ(ξ, t) such that ψ(ξ, t) ∈
◦
C
2+α,1+α/2
y t (ΓT ),
κ∂tψ ∈
◦
C
1+α, 1+α
2
y t (ΓT ). Let
B(ΩT ) :=
◦
C
2+α,1+α/2
y t (Ω1T ) ×
◦
C
2+α,1+α/2
y t (Ω2T ) ×
◦
D2+α(ΓT ),
H(ΩT ) :=
◦
C
α,α/2
y t (Ω1T ) ×
◦
C
α,α/2
y t (Ω2T ) ×
◦
C
2+α,1+α/2
y t (ΣT )
×
◦
C
2+α,1+α/2
y t (ΓT ) ×
◦
C
2+α,1+α/2
y t (ΓT ) ×
◦
C
1+α, 1+α
2
y t (ΓT )
G. I. Bizhanova 25
be the spaces of the functions w = (v1, v2, ψ) and h = (f1, f2, p1, η1, η2, ϕ)
respectively with the norms
‖w‖B(ΩT ) :=
2∑
j=1
|vj |
(2+α)
ΩjT
+ |ψ|
(2+α)
ΓT
+ |κ∂tψ|
(1+α)
ΓT
,
‖h‖H(ΩT ) :=
2∑
j=1
|fj |
(α)
ΩjT
+ |p1|
(2+α)
ΣT
+
2∑
j=1
|ηj |
(2+α)
ΓT
+ |ϕ|
(1+α)
ΓT
.
We write the problem (2.7)–(2.10) in the operator form
A[w] = h+ N [w], (2.28)
where w = (v1, v2, ψ) is unknown vector, h = (f1, f2, p1, η1, η2, ϕ) —
given one, A is a linear operator determined by all the terms in the left-
hand sides of the equations and conditions of the problem (2.7)–(2.10),
N = (F1, F2, 0, 0, 0, Φ) — nonlinear operator, and A : B(ΩT ) → H(ΩT ),
N : B(ΩT ) → H(ΩT ).
In the left-hand sides of the equations and conditions of the problem
(2.7)–(2.10) there are the same linear terms as in the problem (2.21)–
(2.24). The condition (2.25): αj(x, 0)
∣∣
Γ
≤ −d3 < 0 with αj(x, 0)
∣∣
Γ
=
χNJ−T
0 ∇TVj
∣∣
Γ, t=0
= ∂Nu0j
∣∣
Γ
= ν0N
T∂ν0u0j
∣∣
Γ
is fulfilled by ν0N
T ≥
d1 > 0 and (1.11). So due to Theorem 2.2 and an estimate (2.26) we can
represent the problem (2.28) in the form
w = A−1[h+ N [w]] (2.29)
and obtain an estimate
‖w‖B(ΩT ) ≡ ‖A−1
[
h+ N [w]
]
‖B(ΩT )
≤ C9
(
‖h‖H(ΩT ) +
2∑
j=1
|Fj(vj , ψ)|
(α)
ΩjT
+ |Φ(v1, v2, ψ;κ)|
(1+α)
ΓT
)
. (2.30)
Let B(M) ⊂ B(ΩT0) be a closed ball with the center at zero: B(M) :=
{w | vj ∈
◦
C
2+α,1+α/2
y t (ΩjT0), j = 1, 2, ψ ∈
◦
C
2+α,1+α/2
y t (ΓT0), κ∂tψ ∈
◦
C
1+α, 1+α
2
y t (ΓT0), ‖w‖B(ΩT0
) ≤ M, t ≤ T0}, M = C9‖h‖H(ΩT0
)(1 − q)−1,
q ∈ (0, 1).
To prove that an operator A−1[h + N [w]] acts from the closed ball
B(M) into itself and is a contractive one we estimate the norm (2.30)
26 On a classical solvability...
and the following one
‖A−1[h+ N [w]] −A−1[h+ N [w̃]]‖B(Ωt)
≡ ‖A−1
[
N [w] −N [w̃]
]
‖B(Ωt)
≤ C9
( 2∑
j=1
|Fj(vj , ψ) − Fj(ṽj , ψ̃)|
(α)
Ωjt
+ |Φ(v1, v2, ψ;κ) − Φ(ṽ1, ṽ2, ψ̃;κ)|
(1+α)
Γt
)
(2.31)
for ∀w, w̃ ∈ B(M).
With the help of the estimates ‖J−1‖
(α+ν)
Γt
≤ C10
(
1 + t
1−ν
2 |ψ|
(2+α)
Γt
)
,
t ≤ t2; ‖J−1
0 ‖
(α+ν)
Γt
≤ 1/(1 − q), q ∈ (0, 1), t ≤ t1; ‖J11‖
(α+ν)
Γt
≤
C11 t
1−ν
2 |ψ|
(2+α)
Γt
, ‖J12‖
(α+ν)
Γt
≤C12 t
2−ν
2 |ψ|
(2+α)
Γt
, ‖J01‖
(α+ν)
Γt
≤C13 t
2+α−ν
2 ×
|ρ0|
(3+α)
Γt
, ν = 0, 1, of the inverse Jacobian matrix J−1 and J−1
0 and the
matrices J1 = J11 + J12, J01 determined by (2.6) we evaluate the norms
(2.30), (2.31) containing the functions (2.12) Fj , j = 1, 2, and (2.15) Φ,
then we derive
‖A−1[h+ N [w]]‖B(Ωt) ≤ C9 ‖h‖H(Ωt) + r1(t, |ψ|
(2+α)
Γt
) ‖w‖B(Ωt), (2.32)
‖A−1
[
N [w] −N [w̃]
]
‖B(Ωt)
≤ r2(t, |v1|
(2+α)
Ω1t
, |v2|
(2+α)
Ω2t
, |ψ|
(2+α)
Γt
) ‖w − w̃‖B(Ωt), (2.33)
where r1(0,M) = 0, r2(0,M,M,M) = 0.
We find T1 from the inequalities r1(t,M) ≤ q, r2(t,M,M,M) ≤ q,
q ∈ (0, 1), then from (2.32) and (2.33) we shall have the estimates
‖A−1[h+ N [w]]‖B(Ωt) ≤ C9 ‖h‖H(Ωt) + q‖w‖B(Ωt)
≤ C9 ‖h‖H(Ωt) + qM ≤M ≡ C9 ‖h‖H(ΩT0
)(1 − q)−1, (2.34)
‖A−1[h+ N [w]] −A−1[h+ N [w̃]]‖B(Ωt) ≤ q ‖w − w̃‖B(Ωt) (2.35)
for all w, w̃ ∈ B(M), ∀ t ≤ T0 = min(t0, t1, t2, T1) (the parametrization
of a free boundary (1.1) is valid for t ≤ t0; for t ≤ t1 and t ≤ t2 the
inverse matrices J−1
0 and J−1 exist).
From (2.34) and (2.35) by contraction mapping principle it follows
that the problem (2.28) or (2.7)–(2.10) has a unique solution w = (v1, v2,
ψ) ∈ B(ΩT0). We can see that T0 and a constant C9(1 − q)−1 do not
depend on κ.
G. I. Bizhanova 27
From (2.29) by (2.34) it follows ‖w‖B(Ωt) ≤ C9 (1 − q)−1 ‖h‖H(Ωt).
Applying an estimate (2.20) for the vector h we find an estimate (2.19)
‖w‖B(Ωt) ≤ C9 (1 − q)−1 ‖h‖H(Ωt)
≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
Σt
)
, t ≤ T0, (2.36)
with a constant C5 = C6C9(1 − q)−1 independent on κ.
From the formulae (2.5) with x = y + χ (ρ0 + ψ) N and estimates
(2.4) for Vj , j = 1, 2, and ρ0 we shall have Theorem 1.1 and estimate
(1.12).
3. Proof of Theorem 1.2
We write down an index κ at the functions vj , j = 1, 2, ψ of the
Stefan problem (2.7)–(2.10). Due to Theorem 2.1 this problem has a
unique solution vjκ ∈
◦
C
2+α,1+α/2
y t (ΩjT0), j = 1, 2, ψκ ∈
◦
C
2+α,1+α/2
y t (ΓT0),
κ∂tψκ ∈
◦
C
1+α, 1+α
2
y t (ΓT0) and it satisfies a uniform with respect to κ ∈
(0, κ0] estimate (2.36) ((2.19)) for t ≤ T0:
2∑
j=1
|vjκ|
(2+α)
Ωjt
+ |ψκ|
(2+α)
Γt
+ |κ∂tψκ|
(1+α)
Γt
≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
Σt
)
. (3.1)
From here it follows that the sequences {vjκ}, j = 1, 2, {ψκ}, as κ → 0,
are compact in
◦
C
2,1
y t (ΩjT0),
◦
C
2,1
y t(ΓT0) respectively. We choose the con-
verging subsequences
{vjκn}, j = 1, 2, {ψκn}, κn → 0, (3.2)
and denote
lim
κn→0
vjκn = vj , lim
κn→0
ψκn = ψ, (3.3)
where vj ∈
◦
C
2,1
y t (ΩjT0), ψ ∈
◦
C
2,1
y t(ΓT0). These functions satisfy an esti-
mate
2∑
j=1
|vj |C2,1
y t (Ωjt)
+ |ψ|
C2,1
y t (Γt)
≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
Σt
)
, t ≤ T0,
(3.4)
28 On a classical solvability...
which is derived from an estimate (3.1) due to (3.3). To show that the
functions vj , j = 1, 2, ψ possess higher smoothness we estimate the Hölder
constants
[∂2
yvj ]
(α)
ΩjT0
, [∂tvj ]
(α)
ΩjT0
, [∂yvj ]
( 1+α
2
)
t,ΩjT0
, [∂2
yψ]
(α)
ΓT0
, [∂tψ]
(α)
ΓT0
, [∂yψ]
( 1+α
2
)
t,ΓT0
.
(3.5)
We evaluate, for instance, the difference ∂tψ(y, t) − ∂tψ(z, t)
|∂tψ(y, t) − ∂tψ(z, t)| ≤ |∂tψ(y, t) − ∂tψκn(y, t)|
+ |∂tψ(z, t) − ∂tψκn(z, t)| + |∂tψκn(y, t) − ∂tψκn(z, t)|. (3.6)
In (3.6) we apply an estimate (3.1) for the function ψκn
|∂tψκn(y, t) − ∂tψκn(z, t)| ≤ [∂tψκ]
(α)
y, ΓT0
|y − z|α
≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
ΣT0
)
|y − z|α
and let κn → 0, then due to (3.3) we obtain an inequality
|∂tψ(y, t) − ∂tψ(z, t)| ≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
ΣT0
)
|y − z|α, t ≤ T0,
which leads to the estimate of the Hölder constant
[∂tψ]
(α)
y, ΓT0
≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
ΣT0
)
. (3.7)
We obtain such estimates for the all other Hölder constants in (3.5).
On the basis of (3.4) and estimates of the Hölder constants, as (3.7)
we shall have for the limit functions (3.3) that vj ∈
◦
C
2+α,1+α/2
y t (ΩjT0),
j = 1, 2, ψ ∈
◦
C
2+α,1+α/2
y t (ΓT0) and
2∑
j=1
|vj |
(2+α)
Ωjt
+ |ψ|
(2+α)
Γt
≤ C5
( 2∑
j=1
|u0j |
(2+α)
Ωj
+ |p|
(2+α)
Σt
)
, t ≤ T0. (3.8)
To show that the limit functions vj , j = 1, 2, ψ satisfy the Florin prob-
lem (2.16)–(2.18) we rewrite the problem (2.7)–(2.10) for the functions
of the subsequences (3.2) and with κn instead of κ in a Stefan condition
(2.10), in this problem we let κn to 0 taking into account (3.3), then we
G. I. Bizhanova 29
obtain that the functions vj , j = 1, 2, ψ are the solution of the problem
(2.16)–(2.18).
We prove a uniqueness of the solution of a Florin problem (2.16)–
(2.18). For that we assume there are two solutions of this problem w =
(v1, v2, ψ) and w̃ = (ṽ1, ṽ2, ψ̃) and let {wκn} and {w̃κn} be subsequences
converging to w and w̃ as κn → 0 respectively. We consider Stefan
problem (2.29) written for the functions of the subsequences wκn and
w̃κn and estimate the difference wκn − w̃κn = A−1[h+N [wκn ]]−A−1[h+
N [w̃κn ]]= A−1
[
N [w] −N [w̃]
]
using (2.31)
2∑
j=1
|vjκn − ṽjκn |
(2+α)
Ωjt
+ |ψκn − ψ̃κn |
(2+α)
Γt
≤ C9
( 2∑
j=1
|Fj(vjκn , ψκn) − Fj(ṽjκn , ψ̃κn)|
(α)
Ωjt
+ |Φ(v1, v2, ψκn ;κn) − Φ(ṽ1κn , ṽ2κn , ψ̃κn ;κn)|
(1+α)
Γt
)
.
We let κn to zero and apply the estimates (2.33), (2.35)
2∑
j=1
|vj − ṽj |
(2+α)
Ωjt
+ |ψ − ψ̃|
(2+α)
Γt
≤ C9
( 2∑
j=1
|Fj(vj , ψ) − Fj(ṽj , ψ̃)|
(α)
Ωjt
+ |Φ(v1, v2, ψ; 0) − Φ(ṽ1, ṽ2, ψ̃; 0)|
(1+α)
Γt
)
≤ r2(t,M, M, M)
( 2∑
j=1
|vj − ṽj |
(2+α)
Ωjt
+ |ψ − ψ̃|
(2+α)
Γt
)
,
2∑
j=1
|vj − ṽj |
(2+α)
Ωjt
+ |ψ − ψ̃|
(2+α)
Γt
≤ q
( 2∑
j=1
|vj − ṽj |
(2+α)
Ωjt
+ |ψ − ψ̃|
(2+α)
Γt
)
, t ∈ (0, T0],
where q ∈ (0, 1). This inequality leads to the identity w ≡ w̃ and to the
uniqueness of the solution of Florin problem (2.16)–(2.18).
From the formulae (2.5) with x = y + χNρ
ρ := ρ0 + ψ, uj(x, t) := vj(x− χNρ, t) + Vj(x− χNρ, t), j = 1, 2,
(3.9)
30 On a classical solvability...
we obtain that ρ ∈ C
2+α,1+α/2
x t (ΓT0), uj ∈ C
2+α,1+α/2
x t (QjT0
), j = 1, 2,
and with the help of the estimates (2.4) for the functions ρ0, Vj ; (3.8) for
vj , ψ , we have got an estimate (1.13) for the functions uj(x, t), j = 1, 2,
and ρ.
Obtained functions uj , j = 1, 2, and ρ (3.9) are the solution of the
Florin problem (1.7)–(1.10). Really, we substitute them into equations
and conditions (1.7)–(1.10), make coordinate transformation (2.1) and
substitutions (2.5) with ρ and uj , determined by (3.9), then we obtain
for the functions vj , j = 1, 2, and ψ the Florin problem (2.16)–(2.18).
As it was proved, these functions are the unique solution of the problem
(2.16)–(2.18), that is the functions uj(x, t), j = 1, 2, and ρ determined
by (3.9) are the unique solution of the Florin problem (1.7)–(1.10). �
References
[1] B. V. Bazaliy, The Stefan problem // Dokl. Akad.Nauk Ukrain. SSR. Ser. A.
(1986), N 11, 3–7.
[2] B. V. Bazaliy, S. P. Degtyarev, Classical solvability of the multidimensional Stefan
problem under convective movement of viscose incompressible fluid // Mat. Sb.
132 (1987), N 1, 3–19.
[3] V. S. Belonosov, T. I. Zelenyak, Nonlocal problems in the theory of quasi-linear
parabolic equations // Novosibirsk Gos. Univ., Novosibirsk. 1975, p. 156.
[4] G. I. Bizhanova, Solution in a weighted Hölder space of an initial-boundary value
problem for a second-order parabolic equation with a time derivative in the con-
jugation condition // Algebra i Analiz. 6 (1994), N 1, 64–94 (translation in St.
Petersburg Math. J. 6 (1995), N 1, 51–75).
[5] G. I. Bizhanova, Investigation of the solvability in a weighted Hölder function
space of a multidimensional two-phase Stefan problem and a Florin non-stationary
filtration problem for the second-order parabolic equations (Cauchy-Stefan and
Cauchy-Florin problems) // Zapiski Nauchn. Sem. St. Petersburg Otdel. Mat.
Inst. Steklov (POMI). 213 (1994), 14–47 (translation in J. Math. Sci. (New York).
84(1) (1997), 823–844).
[6] G. I. Bizhanova, Solution in a weighted Hölder function space of multidimensional
two-phase Stefan and Florin problems for the second-order parabolic equations in
a bounded domain // Algebra i Analiz. 7 (1995), N 2, 46–76 (translation in St.
Petersburg Math. J. 7 (1996), N 2, 217–241).
[7] G. I. Bizhanova, Uniform estimates of the solution to the linear two-phase Stefan
problem with a small parameter // Matem. Zhurnal. (2005), N 1, 19–28.
[8] G. I. Bizhanova, J. F. Rodrigues, Classical solutions to parabolic systems with
free boundary of Stefan type // Advances in Differential Equations. (2005), N 12,
1345–1388.
[9] G. I. Bizhanova, V. A. Solonnikov, On problems with free boundaries for second-
order parabolic equations // Algebra i Analiz. 12 (2000), N 6, 98–139 (translation
in St. Petersburg Math. J. 12 (2001), N 6, 949–981.
[10] M. A. Borodin, Existence of a classical solution of the multidimensional Stefan
problem on a finite time interval // Ukrain. Math. Zhurnal. 44, N 12, 1652–1657.
G. I. Bizhanova 31
[11] L. A. Caffarelli, The regularity of the free boundaries in higher dimensions //
Acta Math. 139 (1977), 155–184.
[12] L. A. Caffarelli, Some aspects of one-phase Stefan problem // Indiana Univ. Math.
J. (1978), N 27, 73–77.
[13] A. Fasano, M. Primicerio, E. V. Radkevich, The problems of the mussy region //
Doklady AN SSSR. 320 (1991), N 3, 562–566.
[14] V. A. Florin, Compression of earth medium and filtration under the alternating
porosity with consideration of the influence of latent water // Izv. AN SSSR.
OTN. (1951), N 1, 1625–1649.
[15] A. Friedman, D. Kinderlehrer, A one–phase Stefan problem // Indiana Univ.
Math. J. (1975), N 24, 1005–1035.
[16] E. I. Hanzawa, Classical solutions of the Stefan problem // Tohoku Math. J. 33
(1981), N 3, 297–335.
[17] D. Kinderlehrer, L. Nirenberg, The smoothness of the free boundary in the one-
phase Stefan problem // Comm. Pure Appl. Math. (1978), N 31, 257–282.
[18] O. A. Ladyzhenskaja, V. A. Solonnikov, N. N. Ural’tseva, Linear and quasi-linear
equations of parabolic type. “Nauka”, Moscow. 1967.
[19] A. M. Meirmanov, On the classical solution of the multidimensional Stefan prob-
lem for quasi-linear parabolic equations // Mat. Sbornik. 112 (1980), N 4, 170–
192.
[20] E. V. Radkevich, On the solvability of general non-stationary problems with free
boundary // Some Applications of Functional Analysis to the Problems of Math-
ematical Physics, Akad. Nauk SSSR, Sibirsk. Otdel., Inst. of Math., Novosibirsk.
(1986), 85–111.
[21] J. F. Rodrigues, V. A. Solonnikov, F. Yi, On a parabolic system with time deriva-
tive in the boundary conditions and related free boundary problems // Math. Ann.
315 (1999), 61–95.
Contact information
Galina I. Bizhanova Institute of Mathematics,
Pushkin Str. 125,
Almaty, 050010,
Kazakhstan
E-Mail: galya@math.kz,
galina_math@mail.ru
|