On the regularity of solutions of quasilinear Poisson equations
We study the Dirichlet problem for quasilinear partial differential equations of the form Δu(z) = h(z)f(u(z)) in the unit disk D ⊂ C with continuous boundary data. Here, the function h : D→R belongs to the class L^p(D), p > 1, and the continuous function f : R→R is assumed to have the nondecrea...
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| Цитувати: | On the regularity of solutions of quasilinear Poisson equations / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 10. — С. 9-17. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1445182018-12-28T01:22:52Z On the regularity of solutions of quasilinear Poisson equations Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. Математика We study the Dirichlet problem for quasilinear partial differential equations of the form Δu(z) = h(z)f(u(z)) in the unit disk D ⊂ C with continuous boundary data. Here, the function h : D→R belongs to the class L^p(D), p > 1, and the continuous function f : R→R is assumed to have the nondecreasing |f| of |t| and such that f(t) / t →0 as t →∞. We prove the existence of a continuous solution u of the problem in the Sobolev class W^2,p loc (D). Moreover, we show that if p > 2 , then u∈ C^1,α loc (D) with α = (p − 2)/p. Вивчається задача Діріхле для квазілінійних диференціальних рівнянь у частинних похідних виду Δu(z) = h(z)f(u(z)) в одиничному колі D ⊂ C з неперервними граничними умовами. Тут функція h :D→R належить класу Lp(D), p > 1, і неперервна функція f : R→R припускається такою, що її | f | як функція від | t | є неспадною і такою, що f (t) / t →0 при t →∞ . Доводиться існування неперервного розв’язку u даної проблеми в класі Соболєва W^2,p loc (D). Більш того, показано, що якщо p > 2 , то C^1,α loc (D) з α = (p − 2)/p. Изучается задача Дирихле для квазилинейных дифференциальных уравнений в частных производных вида Δu(z) = h(z)f(u(z)) в единичном круге D ⊂ C с непрерывными граничными условиями. Здесь функция h :D→R принадлежит классу Lp(D), p > 1, и непрерывная функция f : R→R предполагается такой, что ее | f | как функция от | t | является неубывающей и такой, что f (t) / t →0 при t →∞. Доказывается существование непрерывного решения u рассматриваемой проблемы в классе Соболева W^2,p loc (D). Более того, показано, что если p > 2 , то C^1,α loc (D) с α = (p − 2)/p. 2018 Article On the regularity of solutions of quasilinear Poisson equations / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 10. — С. 9-17. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2018.10.009 http://dspace.nbuv.gov.ua/handle/123456789/144518 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Математика Математика |
| spellingShingle |
Математика Математика Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. On the regularity of solutions of quasilinear Poisson equations Доповіді НАН України |
| description |
We study the Dirichlet problem for quasilinear partial differential equations of the form Δu(z) = h(z)f(u(z)) in the
unit disk D ⊂ C with continuous boundary data. Here, the function h : D→R belongs to the class L^p(D), p > 1, and
the continuous function f : R→R is assumed to have the nondecreasing |f| of |t| and such that f(t) / t →0 as
t →∞. We prove the existence of a continuous solution u of the problem in the Sobolev class W^2,p loc (D). Moreover,
we show that if p > 2 , then u∈ C^1,α loc (D) with α = (p − 2)/p. |
| format |
Article |
| author |
Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_facet |
Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_sort |
Gutlyanskiĭ, V.Ya. |
| title |
On the regularity of solutions of quasilinear Poisson equations |
| title_short |
On the regularity of solutions of quasilinear Poisson equations |
| title_full |
On the regularity of solutions of quasilinear Poisson equations |
| title_fullStr |
On the regularity of solutions of quasilinear Poisson equations |
| title_full_unstemmed |
On the regularity of solutions of quasilinear Poisson equations |
| title_sort |
on the regularity of solutions of quasilinear poisson equations |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2018 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/144518 |
| citation_txt |
On the regularity of solutions of quasilinear Poisson equations / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 10. — С. 9-17. — Бібліогр.: 15 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT gutlyanskiivya ontheregularityofsolutionsofquasilinearpoissonequations AT nesmelovaov ontheregularityofsolutionsofquasilinearpoissonequations AT ryazanovvi ontheregularityofsolutionsofquasilinearpoissonequations |
| first_indexed |
2025-07-10T19:33:18Z |
| last_indexed |
2025-07-10T19:33:18Z |
| _version_ |
1837289716095909888 |
| fulltext |
9ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 10
© V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, 2018
1. Introduction. We study the existence of regular solutions to the Dirichlet problem for the qua-
silinear Poisson equation
( ) ( ) ( ( ))u z h z f u zΔ = (1)
in the unit disk { :| | 1}z z= <D of the complex plane with continuous boundary values. In ge-
neral, we assume that the function :h →D is in the class ( )pL D , 1,p > and the continuous
function :f → is either bounded or has the non-decreasing | |f of | |t and such that
( )
0,lim
t
f t
t→∞
= (2)
without any assumptions on the sign and zeros of the right-hand side in (1). We analyze how the
degree of regularity of solutions depends on the degree of integrability of the multiplier h.
On the one hand, the interest in this subject is well known both from a purely theoretical
point of view, due to its deep relations to linear and nonlinear harmonic analysis, and because of
numerous applications of equations of this type in various areas of physics, differential geometry,
logistic problems, etc. In particular, in the excellent book by M. Marcus and L. Veron [1], the
reader can find a comprehensive analysis of the Dirichlet problem for the semilinear equation
( ) ( , ( ))u z f z u zΔ = (3)
doi: https://doi.org/10.15407/dopovidi2018.10.009
UDC 517.5
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com
On the regularity of solutions
of quasilinear Poisson equations
Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ
We study the Dirichlet problem for quasilinear partial differential equations of the form ( ) ( ) ( ( ))u z h z f u zΔ = in the
unit disk ⊂D with continuous boundary data. Here, the function :h →D belongs to the class ( )pL D , 1,p > and
the continuous function :f → is assumed to have the nondecreasing | |f of | |t and such that ( ) / 0f t t → as
t →∞ . We prove the existence of a continuous solution u of the problem in the Sobolev class 2,
loc ( )pW D . Moreover,
we show that if 2p > , then 1,
loc ( )u C α∈ D with ( 2)/p pα = − .
Keywords: quasilinear Poisson equation, potential theory, logarithmic and Newtonian potentials, Dirichlet problem,
Sobolev classes, quasiconformal mappings.
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 10
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
in smooth (C 2) domains D in n, 3n� , with boundary data in 1L . Here, ( , )t f t→ ⋅ is a continuous
mapping from to a weighted Lebesgue space 1( , )L D ρ , and ( , )z f z→ ⋅ is a non-decreasing
function for every z D∈ , ( , 0) 0f z ≡ , such that
( , )
,lim
t
f z t
t→∞
= ∞ (4)
uniformly with respect to the parameter z in compact subsets of D .
On the other hand, Eqs. (1) naturally arise under the study of some semilinear equations in
the divergent form. Indeed, we have established [2] that, in arbitrary simply connected domains
D ⊂ , solutions of the semilinear equations
div[ ( ) ( )] ( ( ))A z U z f U z∇ = (5)
with suitable matrix functions ( )A z can be represented as the composition U u= ω , where ω is
a quasiconformal mapping of D onto D associated with A , and u is a solution of Eq. (1) with
.h J= Here, J stands for the Jacobian of the mapping 1−ω . Hence, the results on regular solu-
tions for Eqs. (1) presented in this paper and the comprehensively developed theory of quasicon-
formal mappings in the plane, see, e.g., [3-5], are good basiс tools for the further study of Eqs. (5).
The latter opens up a new approach to the study of a number of semi-linear equations of mathe-
matical physics in anisotropic and inhomogeneous media.
In Section 2, we give a necessary background for the Poisson equation ( ) ( )u z g zΔ = due to
the theory of the Newtonian potential and to the theory of singular integrals in . First, we recall
that, correspondingly to the key fact of the potential theory, see Proposition 1, the Newtonian
potential
1
( ) : (ln | |) ( ) ( )
2gN z z w g w dm w= −
π ∫ (6)
of arbitrary integrable densities g of charge with compact support satisfies the Poisson equation
in a distributional sense, see Corollary 1. Moreover, gN is continuous for ( )pg L∈ , and, further-
more, the Newtonian operator : ( ) ( )pN L C→ is completely continuous for 1p > , see Theorem
1. The example in Remark 2 shows that gN for 1( )g L∈ can be not continuous and even not in
loc( )L∞ . Theorem 2 describes additional regularity properties of gN depending on a degree of
integrability of g . Finally, resulting Corollary 2 states the existence, representation, and regu-
larity of solutions to the Dirichlet problem for the Poisson equation with arbitrary continuous
boundary data.
Section 3 contains the main result of the paper. It is well known that solutions to the qua-
silinear Poisson equation (1) in a unit disk D for arbitrary continuous boundary data belong to
the Sobolev space 1,
loc ( )qW D for some > 2q , and U is locally Hölder continuous in D , see, e.g., [6].
If, in addition, ϕ is Hölder continuous, then U is Hölder continuous in D . In Theorem 3, we
prove the existence of solutions in the Sobolev class 2,
loc ( )pW D , if the multiplier :h →D is in the
class ( )pL D , >1p . Moreover, we show that if 2p > , then 1,
loc ( )U C α∈ D with ( 2)/p pα = − . The
proof of Theorem 3 is realized through reducing the problem to the case of the linear Poisson
equation by the Leray—Schauder approach.
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 10
On the regularity of solutions of quasilinear Poisson equations
2. Potentials and the Poisson equation. Let D be the unit disk in the complex plane .
For z and w∈D with z w≠ , let
2
2
1 1 | |
( , ) : log and ( , ) :
|1 |
it
it
zw z
G z w P z e
z w ze−
− −= =
− −
(7)
be the Green function and Poisson kernel in D . If ( )Cϕ ∈ ∂D and ( )g C∈ D , then a solution to the
Poisson equation
( ) ( )f z g zΔ = (8)
satisfying the boundary condition |f ∂ = ϕD is given by the formula
( ) ( ) ( )gf z z zϕ= −P G , (9)
where
2
0
1
( ) ( , ) ( ) , ( ) ( , ) ( ) ( ),
2
it it
gz P z e e dt z G z w g w dm w
π
−
ϕ = ϕ =
π ∫ ∫
D
P G (10)
see, e.g., [7], p. 118-120. Here, ( )m w denotes the Lebesgue measure in .
In this section, we give the representation of solutions of the Poisson equation in the form
of the Newtonian (normalized antilogarithmic) potential that is more convenient for our re-
search. On this basis, we prove the existence and representation theorem for solutions of the
Dirichlet problem to the Poisson equation under the corresponding conditions of integrability
of sources g .
Correspondingly to 3.1.1 in [8], given a finite Borel measure ν on with compact support,
its potential is the function : [ , )pν → −∞ ∞C defined by
( ) ln | | ( ).p z z w d wν = − ν∫ (11)
Remark 1. Note that the function pν is subharmonic by Theorem 3.1.2 in [8] and, con-
sequently, it is locally integrable on by Theorem 2.5.1 in [8]. Moreover, pν is harmonic out-
side the support of ν .
This definition can be extended to finite charges ν with compact support (named also signed
measures), i. e., to real-valued sigma-additive functions on Borel sets in , because + −ν = ν − ν ,
where +ν and −ν are Borel measures by the well-known Jordan decomposition.
The key fact is the following statement, see, e.g., Theorem 3.7.4 in [8].
Proposition 1. Let ν be a finite charge with compact support in . Then
2pνΔ = πν (12)
in the distributional sense, i. e.,
0( ) ( ) ( ) 2 ( ) ( ) ( ).p z z dm z z d z C∞
ν Δψ = π ψ ν ∀ψ ∈∫ ∫ (13)
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 10
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
As usual, 0 ( )C∞ denotes the class of all infinitely differentiable functions :ψ → with
compact support in ,
2 2
2 2x y
∂ ∂Δ = +
∂ ∂
is the Laplace operator, and ( )dm z corresponds to the
Le besgue measure in .
Corollary 1. In particular, if, for every Borel set B in ,
( ) : ( ) ( ),
B
B g z dm zν = ∫ (14)
where :g → is an integrable function with compact support, then
,gN gΔ = (15)
where
1
( ) : (ln | |) ( ) ( )
2gN z z w g w dm w= −
π ∫ (16)
in the distributional sense, i. e.,
0( ) ( ) ( ) ( ) ( ) ( ) ( ).gN z z d m z z g z dm z C∞Δψ = ψ ∀ψ ∈∫ ∫ (17)
Here, the function g is called a density of charge ν and the function gN is said to be the
Newtonian potential of g .
The next statement on continuity in the mean of functions :ψ → in ( )qL , [1, )q∈ ∞ ,
with respect to shifts is useful for the study of the Newtonian potential, see, e.g., Theorem 1.4.3 in [9].
Lemma 1. Let ( )qLψ ∈ , [1, )q∈ ∞ , have a compact support. Then
0
| ( ) ( ) | ( ) 0.lim q
z
z z z dm z
Δ →
ψ + Δ −ψ =∫ (18)
Theorem 1. Let :g → be in ( )pL , 1p > , with compact support. Then gN is continuous.
A collection { }gN is equicontinuous on compacta, if the collection { }g is bounded by the norm in
( )pL with supports in a fixed disk K. Moreover, under these conditions, on each compact set in ,
.g C pN M g⋅� (19)
Proof. By the Hölder inequality with
1 1
1
q p
+ = , we have
1
| ( ) ( ) | |ln | | ln | || ( )
2
q
p q
g g
K
g
N z N z w w dm w
⎡ ⎤
− ζ ⋅ − − ζ − =⎢ ⎥
π ⎢ ⎥⎣ ⎦
∫�
1
| ( ) ( ) | ( )
2
q
p qg
z dmζ ζ
⎡ ⎤
= ⋅ ψ ξ + Δ −ψ ξ ξ⎢ ⎥
π ⎢ ⎥⎣ ⎦
∫ ,
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 10
On the regularity of solutions of quasilinear Poisson equations
where wξ = ζ − , z zΔ = −ζ, ( ) := ( )ln | |Kζ +ζψ ξ χ ξ ξ . Thus, the first conclusion follows from
Lemma 1, because locln | | ( )qLξ ∈ for all [1, )q∈ ∞ .
The second conclusion follows by continuity of the integral on the right-hand side in the
above estimate with respect to the parameter ζ ∈ . Indeed,
1
*
| ln | || ( )
q
q
q dmζ ζ
Δ
⎧ ⎫⎪ ⎪ψ −ψ = ξ ξ⎨ ⎬
⎪ ⎪⎩ ⎭
∫ ,
where Δ denotes the symmetric difference of the disks K +ζ and *K +ζ . Thus, the statement
follows from the absolute continuity of the indefinite integral.
The third conclusion similarly follows through the direct estimate
1 1
| ( ) | |ln | || ( ) | ( )| ( ) .
2 2
q q
p pq q
g
K
g g
N w dm w dmζ
⎡ ⎤ ⎡ ⎤
ζ ζ − = ψ ξ ξ⎢ ⎥ ⎢ ⎥
π π⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
∫ ∫�
Remark 2. It is easy to verify that the function
( ) : (| |), , ( ) 0, \ ,g z z z g z z= ω ∈ ≡ ∈D D
where
2( ) 1/ (1 ln ) , (0,1], (1, 2), (0) ,t t t tαω = − ∈ α ∈ ω = ∞
is in 1( )L , and its potential gN is not continuous. Furthermore, locgN L∞∉ .
The following theorem on the Newtonian potentials is important to obtain solutions of
the Dirichlet problem to the Poisson equation of a higher regularity.
Theorem 2. Let :g → have compact support. If 1( )g L∈ , then loc
r
gN L∈ for all [1, )r ∈ ∞ ,
1,
loc
q
gN W∈ for all [1, 2)q∈ , moreover, 2,1
locgN W∈ ,
2 2
4 4 . .g g
g
N N
N g a e
z z z z
∂ ∂
= Δ = =
∂ ∂ ∂ ∂
(20)
If ( )pg L∈ C , 1p > , then 2,
loc
p
gN W∈ , gN gΔ = a. e. and, moreover, 1,
loc
q
gN W∈ for 2q > , consequently,
gN is locally Hölder continuous. If ( )pg L∈ , 2p > , then 1,
locgN C α∈ , where ( 2)/p pα = − .
In this connection, recall the definition of formal complex derivatives:
1 1
: , : , .
2 2
i i z x iy
z x y x yz
⎧ ⎫ ⎧ ⎫∂ ∂ ∂ ∂ ∂ ∂= − = + = +⎨ ⎬ ⎨ ⎬∂ ∂ ∂ ∂ ∂∂⎩ ⎭ ⎩ ⎭
The elementary algebraic calculations show that the Laplacian
2 2 2 2
2 2
: 4 4
z z z zx y
∂ ∂ ∂ ∂Δ = + = =
∂ ∂ ∂ ∂∂ ∂
.
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 10
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Proof. Note that gN is the convolution gψ∗ , where ( ) ln | |ψ ζ = ζ , and, hence, loc
r
gN L∈
for all [1, )r ∈ ∞ , see, e. g., Corollary 4.5.2 in [10]. Moreover, as is well known,
g
g
z z
∂ψ∗ ∂ψ= ∗
∂ ∂
and
g
g
z z
∂ψ∗ ∂ψ= ∗
∂ ∂
, see, e. g., (4.2.5) in [10]. In addition by elementary calculations, we get
1 1 1 1
ln | | , ln | | .
2 2
z w z w
z z w z z w
∂ ∂− = − =
∂ − ∂ −
Consequently,
( ) ( )1 1
( ), ( ),
4 4
g gN z N z
Tg z T g z
z z
∂ ∂
= =
∂ ∂
where Tg and T g are the well-known integral operators
1 ( ) 1 ( )
( ) : ( ) , ( ) : ( ) .
dm w d m w
Tg z g w T g z g w
z w z w
= =
π − π −∫ ∫
Thus, all the rest conclusions for 1( )g L∈ follow from Theorems 1.13-1.14 in [11]. If ( )pg L∈ ,
1p > , then 1,
loc
q
gN W∈ , 2q > , by Theorem 1.27, (6.27) in [11]. Consequently, gN is locally Hölder
continuous, see, e. g., Theorem 8.22 in [12], and 2,
loc
p
gN W∈ by Theorems 1.36-1.37 in [11]. If
( )pg L∈ , 2p > , then 1,
locgN C α∈ with
2p
p
−α = by Theorem 1.19 in [11].
By Theorem 2 and the known Poisson formula, see, e. g., I.D.2 in [13], we come to the fol-
lowing consequence on the existence, regularity, and representation of solutions for the Di-
richlet problem to the Poisson equation in the unit disk D , where we assume the charge density
g to be extended by zero outside D .
Corollary 2. Let :ϕ ∂ →D be a continuous function and let :g →D belong to the class
( )pL D , 1p > . Then the function *:= g N g
U N ϕ− +P P , * : |g gN N ∂= D , is continuous in D with
|U ∂ = ϕD , belongs to the class 2,
loc ( )pW D , and U gΔ = a.e. in D . Moreover, 1,
loc ( )qU W∈ D for some
2q > and U is locally Hölder continuous. If, in addition, ϕ is Hölder continuous, then U is Hölder
conti nuous in D . If ( )pg L∈ D , 2p > , then 1,
loc ( )U C α∈ D , where ( 2)/p pα = − .
Here, the Hölder continuity of U for Hölder continuous ϕ follows from the corresponding
result for integrals of the Cauchy type over the unit circle, see, e. g., Theorem 1.10 in [11], be-
cause the Poisson kernel has the representation ( , ) Re
it
it
it
e z
P z e
e z
+=
−
.
3. The case of quasilinear Poisson equations. The case is reduced to the Poisson equation
by the Leray—Schauder approach.
Theorem 3. Let :ϕ ∂ →D be a continuous function, let :h →D be a function in the class
( )pL D , 1p > , and let :f → be a continuous function with the nondecreasing function | |f of
| |t such that
( )
0.lim
t
f t
t→+∞
= (21)
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 10
On the regularity of solutions of quasilinear Poisson equations
Then there is a continuous function :U →D with |U ∂ = ϕD , such that 2,
loc ( )pU W∈ D and
( ) ( ) ( ( ))U z h z f U zΔ = for a. e. z ∈D . (22)
Moreover, 1,
loc ( )qU W∈ D for some 2q > , and U is locally Hölder continuous in D . If, in ad di-
tion, ϕ is Hölder continuous, then U is Hölder continuous in D . Furthermore, if 2p > , then
1,
loc ( )U C α∈ D where ( 2)/p pα = − . In particular, 1,
loc ( )U C α∈ D for all (0,1)α ∈ if ( )h L∞∈ D .
Proof. If 0ph = or 0cf = , then the Poisson integral ϕP gives the desired solution of the
Dirichlet problem for Eq. (22), see, e. g., I.D.2 in [13]. Hence ,we may assume further that 0ph ≠
and 0cf ≠ .
By Theorem 1 and the maximum principle for harmonic functions, we obtain the family of
operators ( ; ) : ( ) ( )p pF g L Lτ →D D , [0,1]τ ∈ :
*
*( ; ) : ( ), : | , [0,1],g g gN g
F g h f N N Nϕ ∂τ = τ ⋅ − + = ∀ τ ∈DP P (23)
which satisfies all hypotheses H1-H3 of Theorem 1 in [14]:
H1. First of all, ( ; ) ( )pF g Lτ ∈ D for all [0,1]τ ∈ and ( )pg L∈ D , because by Theorem 1,
*( )g N g
f N ϕ− +P P is a continuous function and, moreover, by (19),
( ; ) | (2 ) | [0,1].p p p CF g h f M gτ + ϕ < ∞ ∀τ ∈�
Thus, by Theorem 1 in combination with the Arzela—Ascoli theorem, see, e. g., Theorem
IV.6.7 in [15], the operators ( ; )F g τ are completely continuous for each [0,1]τ ∈ and even uni-
formly continuous with respect to the parameter [0,1]τ ∈ .
H2. The index of the operator ( ; 0)F g is obviously equal to 1 .
H3. By inequality (19) and the maximum principle for harmonic functions, we have the esti-
mate for solutions pg L∈ of the equations ( ; )g F g= τ :
| (2 ) | | (3 ) |p p p C p pg h f M g h f M g+ ϕ� �
whenever /p Cg M≥ ϕ , i.e. then it should be
| (3 ) | 1
3 3
p
p p
f M g
M g M h
� , (24)
and, hence, pg should be bounded in view of condition (21).
Thus, by Theorem 1 in [14], there is a function ( )pg L∈ D such that ( ;1)g F g= . Consequently,
by Corollaries 2, the function *: g N g
U N ϕ= − +P P gives the desired solution of the Dirichlet prob-
lem to the quasilinear Poisson equation (22).
Remark 3. As is clear from the proof, Theorem 3 remains valid, if f is an arbitrary continuous
bounded function. These results can be extended to arbitrary smooth domains and applied to the
study of some physical problems.
16 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 10
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
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equations in the plane. Ukr. Mat. Visn., 14, No. 2, pp. 161-191.
3. Bojarski, B., Gutlyanskii, V., Martio, O. & Ryazanov, V. (2013). Infinitesimal geometry of quasiconformal and
bi-lipschitz mappings in the plane. EMS Tracts in Mathematics (Vol. 19). Zürich: European Mathematical
Society.
4. Gutlyanskii, V., Ryazanov, V., Srebro, U. &Yakubov, E. (2012). The Beltrami equation: A geometric approach.
Developments in Mathematics (Vol. 26). New York etc.: Springer.
5. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2009). Moduli in modern mapping theory. New York:
Springer.
6. Ladyzhenskaya, O. A. & Ural’tseva, N. N. (1968). Linear and quasilinear elliptic equations. New York-London:
Academic Press.
7. Hörmander, L. (1994). Notions of convexity. Progress in Mathematics (Vol. 127). Boston: Birkhäuser.
8. Ransford, T. (1995). Potential theory in the complex plane. London Mathematical Society Student Texts
(Vol. 28). Cambridge: Cambridge Univ. Press.
9. Sobolev, S. L. (1991). Some applications of functional analysis in mathematical physics. Math. Mon.
(Vol. 90). Providence, RI: AMS.
10. Hörmander, L. (1983). The analysis of linear partial differential operators. Vol. I. Distribution theory and
Fourier analysis. Grundlehren der Mathematischen Wissenschaften. (Vol. 256). Berlin: Springer-Verlag.
11. Vekua, I. N. (1962). Generalized analytic functions, London-Paris-Frankfurt: Pergamon Press, Addison-
Wesley, Reading, Mass.
12. Gilbarg, D. &Trudinger, N. (1983). Elliptic partial differential equations of second order. Grundlehren
der Mathematischen Wissenschaften. (Vol. 224). Berlin: Springer.
13. Koosis, P. (1998). Introduction to pH spaces. Cambridge Tracts in Mathematics (Vol. 115). Cambridge:
Cambridge Univ. Press.
14. Leray, J. & Schauder, Ju. (1934). Topologie et equations fonctionnelles. Ann. Sci. Ecole Norm., Sup. 51, No. 3,
pp. 45-78; (1946). Topology and functional equations, Uspehi Matem. Nauk (N.S.). 1, No. 3-4, pp.71-95.
15. Dunford, N. & Schwartz, J. T. (1958). Linear operators. I. General theory. Pure and Applied Mathematics
(Vol. 7). New York, London: Interscience Publishers.
Received 04.07.2018
В.Я. Гутлянський,
О.В. Нєсмєлова, В.І. Рязанов
Інститут прикладної математики і механіки НАН України, Слов’янськ
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com
ПРО РЕГУЛЯРНІСТЬ РОЗВ’ЯЗКІВ
КВАЗІЛІНІЙНИХ РІВНЯНЬ ПУАССОНА
Вивчається задача Діріхле для квазілінійних диференціальних рівнянь у частинних похідних виду
( ) ( ) ( ( ))u z h z f u zΔ = в одиничному колі ⊂D з неперервними граничними умовами. Тут функція :h →D
належить класу ( )pL D , 1,p > і неперервна функція :f → припускається такою, що її | |f як функція
від | |t є неспадною і такою, що ( ) / 0f t t → при t →∞ . Доводиться існування неперервного розв’язку
u даної проблеми в класі Соболєва 2,
loc ( )pW D . Більш того, показано, що якщо 2p > , то 1,
loc ( )u C α∈ D з
( 2)/p pα = − .
Ключові слова: квазілінійне рівняння Пуассона, теорія потенціалу, логарифмічний та ньютонів потен-
ціали, задачі Діріхле, класи Соболєва, квазіконформні відображення.
17ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 10
On the regularity of solutions of quasilinear Poisson equations
В.Я. Гутлянский,
О.В. Несмелова, В.И. Рязанов
Институт прикладной математики и механики НАН Украины, Славянск
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com
О РЕГУЛЯРНОСТИ РЕШЕНИЙ
КВАЗИЛИНЕЙНЫХ УРАВНЕНИЙ ПУАССОНА
Изучается задача Дирихле для квазилинейных дифференциальных уравнений в частных производных
вида ( ) ( ) ( ( ))u z h z f u zΔ = в единичном круге ⊂D с непрерывными граничными условиями. Здесь функ-
ция :h →D принадлежит классу ( )pL D , 1,p > и непрерывная функция :f → предполагается такой,
что ее | |f как функция от | |t является неубывающей и такой, что ( ) / 0f t t → при t →∞ . Доказывается
существование непрерывного решения u рассматриваемой проблемы в классе Соболева 2,
loc ( )pW D . Более
того, показано, что если 2p > , то 1,
loc ( )u C α∈ D с ( 2)/p pα = − .
Ключевые слова: квазилинейное уравнение Пуассона, теория потенциала, логарифмический и ньютонов
потенциалы, задача Дирихле, классы Соболева, квазиконформные отображения.
|