On the quasilinear Poisson equations in the complex plane
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irk-123456789-1702552020-07-10T01:28:22Z On the quasilinear Poisson equations in the complex plane Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. Математика 2020 Article On the quasilinear Poisson equations in the complex plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2020. — № 1. — С. 3-10. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2020.01.003 http://dspace.nbuv.gov.ua/handle/123456789/170255 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Математика Математика Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. On the quasilinear Poisson equations in the complex plane Доповіді НАН України |
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Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
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Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
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Gutlyanskii, V.Ya. |
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On the quasilinear Poisson equations in the complex plane |
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On the quasilinear Poisson equations in the complex plane |
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On the quasilinear Poisson equations in the complex plane |
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On the quasilinear Poisson equations in the complex plane |
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On the quasilinear Poisson equations in the complex plane |
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on the quasilinear poisson equations in the complex plane |
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Видавничий дім "Академперіодика" НАН України |
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On the quasilinear Poisson equations in the complex plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2020. — № 1. — С. 3-10. — Бібліогр.: 15 назв. — англ. |
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Доповіді НАН України |
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AT gutlyanskiivya onthequasilinearpoissonequationsinthecomplexplane AT nesmelovaov onthequasilinearpoissonequationsinthecomplexplane AT ryazanovvi onthequasilinearpoissonequationsinthecomplexplane |
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2025-07-15T05:22:22Z |
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2025-07-15T05:22:22Z |
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1837689149763616768 |
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3
МАТЕМАТИКА
ISSN 10256415. Допов. Нац. акад. наук Укр. 2020. № 1: 3—12
© V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, 2020
https://doi.org/10.15407/dopovidi2020.01.003
UDC 517.5
V.Ya. Gutlyanskiĭ1, O.V. Nesmelova 1, 2, V.I. Ryazanov 1, 3
1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk
2 Donbas State Pedagogical University, Slov’yansk
3 Bogdan Khmelnytsky National University of Cherkasy
Email: vgutlyanskii@gmail.com, staro@ukr.net, vl.ryazanov1@gmail.com
On the quasilinear Poisson equations
in the complex plane
Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ
First, we study the existence and regularity of solutions for the linear Poisson equations ( ) ( )U z g z∆ = in bounded
domains D of the complex plane £ with charges g in the classes 1
loc( ) ( )pL D L D∩ , 1p > . Then, applying the Le
ray—Schauder approach, we prove the existence of Höldercontinuous solutions U in the class 2,
loc ( )pW D for the
quasilinear Poisson equations of the form ( ) ( ) ( ( ))U z h z f U z∆ = ⋅ with h in the same classes as g and continuous
functions :f →¡ ¡ such that ( ) / 0f t t → as t → ∞. These results can be applied to various problems of mathe
matical physics.
Keywords: potential theory, quasilinear Poisson equations, semilinear equations, anisotropic and inhomogeneous
media, quasiconformal mappings.
ОПОВІДІ
НАЦІОНАЛЬНОЇ
АКАДЕМІЇ НАУК
УКРАЇНИ
МАТЕМАТИКА
1. Introduction. The study of elliptic partial differential equations in two dimensions by the
methods of complex analysis and quasiconformal mappings with applications to nonlinear elas
ticity, gas flow, hydrodynamics, and other sections of natural science has been initiated by
M.A. Lavrentiev, L. Bers, L. Nirenberg, I.N. Vekua, B. Bojarski, and others (see, e.g., [15] and
the references therein). A rather comprehensive treatment of the present state of the theory is
given in the excellent book of K. Astala, T. Iwaniec, and G. Martin [6].
In series of our recent papers (see, e.g., [7, 8]), we have proposed another application of
the theory of quasiconformal mappings to the the study of semilinear partial differential equa
tions of the form
div [ ( ) ( )] ( ), ,A z u z f u z D D∇ = ∈ ⊆£ , (1)
the diffusion term of which is the divergenceform elliptic operator ( )L u , whereas its reac
tion term ( )f u is a nonlinear function. Here, the symmetric matrix function ( ) { ( )},ijA z a z=
4 ISSN 10256415. Dopov. Nac. akad. nauk Ukr. 2020. № 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
det ( ) 1A z = , with measurable realvalued entries satisfies the uniform ellipticity condition
2 21
| | ( ) , | | . . , 1 ,A z K a e in D K
K
ξ 〈 ξ ξ〉 ξ < ∞� � � (2)
for every 2 .ξ ∈¡ The set of all such matrix functions is denoted by 2 2( ).KM D×
It was shown that, by the chain rule for the composition function u U= ωo , where ω is
a quasiconformal mapping that agrees with the matrix function ( )A z , the following basic for
mula holds:
div [ ( ) ( ( ( )))] ( ) ( ( ))A z U z J z U zω∇ ω = ∆ ω , (3)
where ( )J zω stands for the Jacobian of the mapping ω.
Namely, this formula (understood in the sense of distributions) takes place for all 1,2
loc ( ),U W G∈
2 2( )KA M D×∈ and quasiconformal homeomorphisms :D Gω → satisfying the Beltrami equation
( ) ( ) ( ) . . ,z zz z z a e in Dω = µ ω (4)
where its complex dilatation
22 11 122
( )
det( )
a a ia
z
I A
− −
µ =
+
(5)
satisfies the uniform ellipticity condition
1
| ( ) | .
1
K
z
K
+
µ
−
� (6)
Thus, the study of semilinear equations of the form (1) in domains D of a finite area can
be reduced in a suitable way to the corresponding investigations of the quasilinear Poisson
equa tions of the form ( )U h f U∆ = ⋅ . The latter is the subject of the present paper.
Below, D denotes the unit disk { : | | 1}z z∈ <£ in the complex plane £, 0 0( ) : { : | | }R z z z z R= ∈ − <£D
for 0z ∈£ and (0, )R ∈ ∞ , : (0)R R=D D .
2. Basic facts from the potential theory. For the sake of completeness, we repeat the fun da
mental results concerning the potential theory in a plane, given in [8], and strengthen some of them.
First of all, we recall that, by Corollary 1 in [8], if, for every Borel set B in £ ,
( ) : ( ) ( )
B
B g z d m zν = ∫ (7)
where :g →£ ¡ is an integrable function with compact support, then
,gN g∆ = (8)
where
1
( ) : ln | | ( ) ( ),
2gN z z w g w d m w= −
π ∫
£
(9)
5ISSN 10256415. Допов. Нац. акад. наук Укр. 2020. № 1
On the quasilinear Poisson equations in the complex plane
in the distributional sense, i.e.,
0( ) ( ) ( ) ( ) ( ) ( ) ( ) .gN z z d m z z g z d m z C∞∆ψ = ψ ψ ∈ θ∫ ∫
£ £
£ (10)
Here, the function g is called a density of the charge ν , and the function gN is said to be
the Newtonian potential of g .
Recall also the definition of the formal complex derivatives:
1 1
: , : , .
2 2
i i z x iy
z x y x yz
∂ ∂ ∂ ∂ ∂ ∂
= − = + = + ∂ ∂ ∂ ∂ ∂∂
(11)
The elementary algebraic calculations show their relation to the Laplacian
2 2 2 2
2 2
: 4 4
z z z zx y
∂ ∂ ∂ ∂
∆ = + = =
∂ ∂ ∂ ∂∂ ∂
. (12)
Further, we apply the theory of the wellknown integral operators
1 ( ) 1 ( )
( ) : ( ) , ( ) : ( )
d m w d m w
Tg z g w T g z g w
z w z w
= =
π − π −∫ ∫
£ £
defined for integrable functions with a compact support K and studied in detail.
The following theorem on the Newtonian potential strengthens Theorem 2 from [8]. It is
important to obtain solutions of a higher regularity to the Poisson equations (8), as well as to
the corresponding semilinear equations.
Here and later on, given a domain D in £ , a function :g D → ¡ is assumed to be exten
ded onto £ by zero outside of D .
Theorem 1. Let D be a bounded domain in £ . Suppose that 1( )g L D∈ . Then 1,
loc ( )q
gN W∈ £
for all [1, 2)q ∈ , and there exist the generalized derivatives by Sobolev
2
gN
z z
∂
∂ ∂
and
2
gN
z z
∂
∂ ∂
and
2 2
4 4 . .g g
g
N N
N g a e in
z z z z
∂ ∂
= ∆ = =
∂ ∂ ∂ ∂
£ . (13)
Moreover, loc( )s
gN L∈ £ for all [1, )s ∈ ∞ . More precisely,
1 ln | | [1, ) ,g s sN g s⋅ ξ ∀ ∈ ∞P P P P P P� (14)
where g sNP P is in rD for all (0, )r ∈ ∞ , and ln | | sξP P is in R r+D if RD ⊆D .
If ( )pg L D∈ for some (1, 2]p ∈ , then 2,
loc ( )p
gN W∈ £ and
1,
loc ( ) (1, ) , 2 /(2 ) 2 .gN W q where q p pγ∈ ∀γ ∈ = − >£ (15)
In addition, the collection { }gN is locally β Hölder equicontinuous in £ for all (0,1 2/ )qβ ∈ − , and
the collection { }gN ′ of its first partial derivatives is strictly compact in ( )L Dγ for all (1, )qγ ∈ , if the
collection { }g is bounded in ( )pL D .
6 ISSN 10256415. Dopov. Nac. akad. nauk Ukr. 2020. № 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Finally, if ( )pg L D∈ for some > 2p , then 1,
loc ( )gN C α∈ £ with ( 2)/p pα = − . Furthermore, the col
lection { }gN ′ is locally Hölderequicontinuous in £ with the given α, if { }g is bounded in ( )pL D .
Proof. Note that gN is the convolution gψ ∗ , where ( ) ln | |ψ ξ = ξ . Hence, (14) follows
(e.g., by Corollary 4.5.2 in [9]). Moreover,
g
g
z z
∂ψ ∗ ∂ψ
= ∗
∂ ∂
and
g
g
z z
∂ψ ∗ ∂ψ
= ∗
∂ ∂
(see, e.g., (4.2.5)
in [9]). By elementary calculations,
1 1 1 1
ln | | , ln | | .
2 2
z w z w
z z w z z w
∂ ∂
β − = − =
∂ − ∂ −
(16)
Consequently,
( ) ( )1 1
( ) , ( ) .
4 4
g gN z N z
Tg z T g z
z z
∂ ∂
= =
∂ ∂
(17)
Thus, the rest conclusions for 1( )g L D∈ follow by Theorems 1.13—1.14 in [5].
Next, if ( )pg L D∈ with (1, 2]p ∈ , then 1,
loc ( )gN W γ∈ £ for all (1, )qγ ∈ , where 2 /(2 ) 2q p p= − >
2 /(2 ) 2q p p= − > , by Theorem 1.27, (1.27) in [5] and, moreover, 2,
loc ( )p
gN W∈ £ by Theorems 1.36—1.37
in [5]. In addition, a collection { }gN is locally β H lderequicontinuous in £ for all (0,1 2/ )qβ ∈ −
(see, e.g., Lemma 2.7 in [3]), and the collection { }gN ′ of its first partial derivatives is strictly
compact in ( )L Dγ for all (1, )qγ ∈ , if the collection { }g is bounded by the norm in ( )pL D
(see, e.g., Theorem 1.4.3 in [10] and Theorem 1.27 in [5]).
The last item of Theorem 1 was derived in Theorem 2 from [8].
Remark 1. Note, generally speaking, that 2,1
locgN W∉ and gN C∉ , if 1( )g L∈ £ (see, for in
stance, Example 7.5 in [11], and the example in Remark 2 from [8]).
Corollary 1. Let D be a subdomain of D , :g D → ¡ be in 1( )L D and in loc( )pL D for so me
1p > . Then 2,
loc ( )p
gN W D∈ and satisfies (13) a.e. in D . Moreover, 1,
loc ( )q
gN W D∈ for 2q > ,
and gN is locally Höldercontinuous in D . Furthermore, 1,
loc ( )gN C Dα∈ with ( 2)/p pα = − , if
loc( )pg L D∈ for 2p > .
In addition, the collection { }gN is locally β Hölderequicontinuous in D for all (0,1 2/ )β ∈ −
(0,1 2/ )qβ ∈ − , and the collection { }gN ′ of its first partial derivatives is strictly compact in loc( )L Dγ
for all (1, )qγ ∈ , if a collection { }g is bounded in 1( )L D and in loc( )pL D for some (1, 2]p ∈ , where
q is defined in (15).
Finally, the collection { }gN ′ is locally α Hölderequicontinuous in D with the given α , if a
collection { }g is bounded in 1( )L D and in loc( )pL D for 2p > .
Proof. Given 0z D∈ and 00 dist( , )R z D< < ∂ ,
1 2g g gN N N= + with 2 1:g g g= − and 1 :g g= ⋅χ,
where χ is the characteristic function of the disk 0( )R zD . The first summand satisfies all de
sired properties by Theorem 1, and the second one is a harmonic function in 0( )R zD (see, e.g.,
Theorem 3.1.2 in [12]). Thus, the first part follows. Under the proof of the rest part, the same
decomposition is applied. However, in the case we need the following 2 explicit estimates for
the second summand in a smaller disk 0( )r zD , (0, )r R∈ ,
2 2
12 2
2 1 2 12 2
1 1
1
| ( ) ( ) | | |
2 ( )
g g
g g
z z
z zN N g
N z N z dz dz z z
z z R r
∂ ∂
− + −
∂ ∂ π −∫ ∫
P P
� �
7ISSN 10256415. Допов. Нац. акад. наук Укр. 2020. № 1
On the quasilinear Poisson equations in the complex plane
and, since the function 2Tg is analytic in 0( )r zD and the function 2 2T g Tg= (for the realva
lued function 2g ) is antianalytic in 0( )r zD , similarly,
2 2
2
12
2 1 2 12
1
1 1
| ( ) ( ) | | | .
4 4 ( )
g g
z
z
gTg
N z N z dz z z
z R r
∂
− −′ ′
∂ π −∫
P P
� �
Here, we denote, by
2gN ′ , any of the first partial derivatives of
2gN (see (11)):
, , ,i z x iy
x z z y z z
∂ ∂ ∂ ∂ ∂ ∂ = + = ⋅ − = + ∂ ∂ ∂ ∂ ∂ ∂
take relation (16) into account, and calculate the given integrals over the segment 1 2 0[ , ] ( )rz z D z⊂
of the straight line going through 1z , 2 0( )rz z∈D .
3. On the solvability of quasilinear Poisson equations. In this section, we study the sol
vability problem for quasilinear Poisson equations of the form ( ).U hf U∆ = The wellknown
Le ray—Schauder approach allows us to reduce the problem to the study of the corresponding
linear Poisson equation from the previous section.
Theorem 2. Let :h →£ ¡ be a function in the class ( )pL £ for 1p > with compact support.
Suppose that a function :f →¡ ¡ is continuous and
( )
0 .lim
t
f t
t→∞
= (18)
Then there is a continuous function :U →£ ¡ in the class 2,
loc ( )pW £ such that
( ) ( ) ( ( )) . .U z h z f U z a e∆ = (19)
and gU N= , where :g →£ ¡ is a function in pL whose support is in the support of h, and the up
per bound of pgP P depends only on phP P and on the function f . Moreover, 1,
loc ( )qU W∈ £ for so me
2q > , and U is locally Höldercontinuous. Furthermore, 1,
loc ( )U C α∈ £ with ( 2)/p pα = − , if > 2p .
In particular, 1,
loc ( )U C α∈ £ for all (0,1)α = , if h is bounded in Theorem 2.
Proof. If 0ph =P P or 0cf =P P , then any constant function U in £ gives the desired solution
of (19). Thus, we may assume that 0ph ≠P P and 0cf ≠P P . Set *
| |
( ) | ( ) |max
t s
f s f t=
�
, : [0, )s +∈ = ∞¡ .
Then the function :f + +
∗ →¡ ¡ is continuous and nondecreasing. Moreover, *( ) / 0f s s → as
s → ∞ by (18).
By Theorem 1 in [8], we obtain a family of operators ( ; ) : ( ) ( )p p
h hF g L Lτ →£ £ , ( )p
hL £ is the
subspace of ( )pL £ with supports in the support of h ,
( ; ) : ( ) [0,1]gF g h f Nτ = τ ⋅ ∀τ ∈ (20)
which satisfies all groups of hypotheses H1H3 of Theorem 1 in [13]. Indeed:
H1. First of all, ( ; ) : ( )p
hF g Lτ £ for all [0,1]τ ∈ and ( )p
hg L∈ £ , because, by Theorem 1 in [8],
the function ( )gf N is continuous and
*( ; ) ( ) [0,1] ,p p pF g h f M gτ < ∞ ∀τ ∈P P P P P P�
8 ISSN 10256415. Dopov. Nac. akad. nauk Ukr. 2020. № 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
where M is the constant from estimate (19) in [8]. Thus, by Theorem 1 from [8] in combination
with the Arzela—Ascoli theorem (see, e.g., Theorem IV.6.7 in [14]), the operators ( ; )F g τ are
completely continuous for every [0,1]τ ∈ and even uniformly continuous in the parameter [0,1]τ ∈ .
H2. The index of the operator ( ; 0)F g is obviously equal to 1 .
H3. By Theorem 1 in [8], we have, for solutions of the equations ( ; )g F g= τ :
*( )p p pg h f M gP P P P P P� ,
i.e.,
*( ) 1p
p p
f M g
M g M h
P P
P P P P
� . (21)
Hence, pgP P is bounded in the parameter [0,1]τ ∈ by condition (18).
Thus, by Theorem 1 in [13], there is a function ( )p
hg L D∈ with ( ;1)F g g= , and, by Theo
rem 1, the function : gU N= gives the desired solution of (19).
Theorem 3. Let D be a domain in D , :h D → ¡ be in
1
loc( ) ( )
p
L D L D∩ , 1p > , and let :f →¡ ¡
satisfy the hypothesis of Theorem 2. Then there is a weak solution :U D → ¡ of the quasilinear
Poisson equation (19) in the class 1,
loc( ) ( )qC D W D∩ for some 2q > which is locally Höldercon
tinuous in D .
Proof. Let kD be an expanding sequence of domains in £ with kD D⊂ , 1, 2,k = … , exha us
ting D , i.e.
1
k
k
D D
∞
=
=∪ . Let us extend h by zero outside of D . Set k kh h= χ , where kχ is a cha
racteristic function of kD in £, and k gk
U N= , where kg corresponds to kh by Theorem 2. Now,
by Corollary 1, the family of functions { }gk
N is H lderequicontinuous on each mD , 1, 2,m = … .
Moreover, by Theorems 1 and 2, as well as by the H lder integral inequality, we have
1
1
1 1 1 1
1 1
ln | | ( ) ln | |
( ) ( ) ( )
p
g k pk L L L
N g h
−
ε +ε +ε
⋅ ξ π Ψ ⋅ ξP P P P P P P P P P� �
D D D
(22)
for small enough 0ε > such that 1Dε ⊂D , where the function : + +Ψ →R R depends only on f . Hence,
for each 1, 2,k = …, there is a point kz ε∈D with
1
2
1
1
| ( ) | : ( ) ln | | /
( )
p
p
g k pk L
N z c h
−
+ ε
= π Ψ ⋅ ξ πεP P P P�
D
,
because, by (22), the mean integral value of | ( ) |g kk
N z over the disk εD cannot be greater than
the given number c . Combining the latter fact with H lderequicontinuity of the sequence
kgN , 1, 2,k = … , on each mD , 1, 2,m = … , we obtain also its boundedness. Thus, by the Arzela—
Ascoli theorem (see, e.g., Theorem IV.6.7 in [14]), the family of functions { }gk
N is compact on
each mD , 1, 2,m = … .
Without loss of generality, we may assume that (1, 2]p ∈ . Then, by Corollary 1, the Newtonian
potential { }gk
N , 1, 2,m = … , is in the class 1,
loc
qW for some 2q > , and the family { }gk
N ′ is also
compact on each mD , 1, 2,m = … by the norm of qL . Consequently, the sequence { }gk
N is com
pact on each mD , 1, 2,m = … by any norm ⋅P P of 1, qW , too (see, e.g., Theorem 2.5.1 in [15]).
Next, let us apply the socalled diagonal process. Namely, let (1)
kU , 1, 2,k = … , be a subse
quence of { }gk
N that converges uniformly to a function 1:U D → R by the norm ⋅P P on the
9ISSN 10256415. Допов. Нац. акад. наук Укр. 2020. № 1
On the quasilinear Poisson equations in the complex plane
domain 1D . Of course, we may assume that (1) 1/CkU U k− <P P , as well as (1) 1/kU U k− <P P for
all 1, 2,k = … . Similarly, a subsequence (2)
kU of (1)
kU with respect to the domain 2D is defined.
Let us continue the process by induction and, finally, consider the diagonal subsequence
( ): m
m mU U= , 1, 2,m = … of the sequence gk
N .
It is clear by the construction that |m DU converges to a function :U D → ¡ locally uni
formly and also in 1,
loc ( )qW D , 2q > . Thus, 1,
loc( ) ( )qU C D W D∈ ∩ , and, consequently, U is local ly
H l dercontinuous in D . Moreover, U is a weak solution of Eq. (19) in the domain D . Indeed,
by (10) and the definition of generalized derivatives, we have that mU satisfy the relations
0( ), ( ) ( ) ( ) ( ( )) ( ) ( ) 0 ( )m m m
D D
U z z dm z h z f U z z dm z C D∞〈∇ ∇ψ 〉 + ψ = ∀ψ ∈∫ ∫ .
Passing to the limit as m → ∞ , we obtain the desired conclusion.
This work was partially supported by grants of Ministry of Education and Science of Uk raine,
project number is 0119U100421.
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Received 20.10.2019
10 ISSN 10256415. Dopov. Nac. akad. nauk Ukr. 2020. № 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
В.Я. Гутлянський 1,
О.В. Нєсмєлова 1, 2, В.І. Рязанов 1, 3
1 Інститут прикладної математики і механіки НАН України, Слов’янськ
2 Донбаський державний педагогічний університет, Слов’янськ
3 Черкаський національний університет ім. Богдана Хмельницького
Email: vgutlyanskii@gmail.com, staro@ukr.net, vl.ryazanov1@gmail.com
ПРО КВАЗІЛІНІЙНІ РІВНЯННЯ ПУАССОНА
В КОМПЛЕКСНІЙ ПЛОЩИНІ
Вивчено існування і регулярність розв’язків для лінійних рівнянь Пуассона виду ( ) ( )U z g z∆ = в обме
жених областях D комплексної площини £ із зарядами g в класах 1
loc( ) ( )pL D L D∩ , 1p > . З викорис
танням підходу Лере—Шаудера доведено існування неперервних за Гельдером розв’язків U в класі
2,
loc ( )pW D для квазілінійних рівнянь Пуассона виду ( ) ( ) ( ( ))U z h z f U z∆ = ⋅ з h із того самого класу, що і
g , та неперервними функціями :f →¡ ¡ такими, що ( ) / 0f t t → при t → ∞ . Отримані результати мо
жуть бути застосовані до різномантіних задач математичної фізики.
Ключові слова: теорія потенціалу, квазілінійні рівняння Пуассона, напівлінійні рівняння, анізотропні та
неоднорідні середовища, квазіконформні відображення.
В.Я. Гутлянский 1,
О.В. Несмелова 1, 2, В.И. Рязанов 1, 3
1 Институт прикладной математики и механики НАН Украины, Славянск
2 Донбасский государственный педагогический университет, Славянск
3 Черкасский национальный университет им. Богдана Хмельницкого
Email: vgutlyanskii@gmail.com, staro@ukr.net, vl.ryazanov1@gmail.com
О КВАЗИЛИНЕЙНЫХ УРАВНЕНИЯХ ПУАССОНА
В КОМПЛЕКСНОЙ ПЛОСКОСТИ
Изучено существование и регулярность решений для линейних уравнений Пуассона вида ( ) ( )U z g z∆ =
в ограниченных областях D комплексной плоскости £ с зарядами g в классах 1
loc( ) ( )pL D L D∩ , 1p > .
С применением подхода Лере—Шаудера доказано существование непрерывных по Гёльдеру решений U в
классе 2,
loc ( )pW D для квазилинейных уравнений Пуассона вида ( ) ( ) ( ( ))U z h z f U z∆ = ⋅ с h из того же
класса, что и g , и непрерывными функциями :f →¡ ¡ такими, что ( ) / 0f t t → при t → ∞ . Получен
ные результаты могут быть применены к различным задачам математической физики.
Ключевые слова: теория потенциала, квазилинейные уравнения Пуассона, полулинейные уравнения, ани
зотропные и неоднородные среды, квазиконформные отображения.
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