On semilinear equations in the complex plane
We study the Dirichlet problem for the semilinear partial differential equations div (A∇u) = f (u) in simply connected domains D of the complex plane C with continuous boundary data. We prove the existence of the weak solutions u in the class C∩Wloc¹’²(D), if a Jordan domain D satisfies the quasihyp...
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irk-123456789-1599522019-10-20T01:25:45Z On semilinear equations in the complex plane Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. Математика We study the Dirichlet problem for the semilinear partial differential equations div (A∇u) = f (u) in simply connected domains D of the complex plane C with continuous boundary data. We prove the existence of the weak solutions u in the class C∩Wloc¹’²(D), if a Jordan domain D satisfies the quasihyperbolic boundary condition by Gehring—Martio. An example of such a domain that fails to satisfy the standard (A)-condition by Ladyzhenskaya—Ural'tseva and the known outer cone condition is given. Some applications of the results to various processes of diffusion and absorption in anisotropic and inhomogeneous media are presented. Досліджено задачу Діріхле для напівлінійних рівнянь в частинних похідних div (A∇u) = f (u) в однозв’язних областях D комплексної площини C з неперервними граничними умовами. Доведено існування слабких розв’язків u у класі C∩Wloc¹’²(D), якщо Жорданова область D задовольняє квазігіперболічну граничну умову Герінга—Мартіо. Наведено приклад такої області, яка не задовольняє стандартну (А)- умову Ладиженської–Уральцевої та відому умову зовнішнього конуса. Також наведено деякі застосування отриманих результатів до різних процесів дифузії та поглинання в анізотропних і неоднорідних середовищах. Исследована задача Дирихле для полулинейных уравнений в частных производных div (A∇u) = f (u) в односвязных областях D комплексной плоскости C с непрерывными граничными условиями. Доказано существование слабых решений u в классе C∩Wloc¹’²(D), если Жорданова область удовлетворяет квазигиперболическому граничному условию Геринга—Мартио. Приведен пример такой области, которая не удовлетворяет стандартному (А)-условию Ладыженской—Уральцевой и известному условию внешнего конуса. Также приведены некоторые применения полученных результатов к различным процессам диффузии и поглощения в анизотропных и неоднородных средах. 2019 Article On semilinear equations in the complex plane / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2019. — № 7. — С. 9-16. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2019.07.009 http://dspace.nbuv.gov.ua/handle/123456789/159952 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Математика Математика |
| spellingShingle |
Математика Математика Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. On semilinear equations in the complex plane Доповіді НАН України |
| description |
We study the Dirichlet problem for the semilinear partial differential equations div (A∇u) = f (u) in simply connected domains D of the complex plane C with continuous boundary data. We prove the existence of the weak solutions u in the class C∩Wloc¹’²(D), if a Jordan domain D satisfies the quasihyperbolic boundary condition by Gehring—Martio. An example of such a domain that fails to satisfy the standard (A)-condition by Ladyzhenskaya—Ural'tseva and the known outer cone condition is given. Some applications of the results to various processes of diffusion and absorption in anisotropic and inhomogeneous media are presented. |
| format |
Article |
| author |
Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
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Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_sort |
Gutlyanskiĭ, V.Ya. |
| title |
On semilinear equations in the complex plane |
| title_short |
On semilinear equations in the complex plane |
| title_full |
On semilinear equations in the complex plane |
| title_fullStr |
On semilinear equations in the complex plane |
| title_full_unstemmed |
On semilinear equations in the complex plane |
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on semilinear equations in the complex plane |
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Видавничий дім "Академперіодика" НАН України |
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2019 |
| topic_facet |
Математика |
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http://dspace.nbuv.gov.ua/handle/123456789/159952 |
| citation_txt |
On semilinear equations in the complex plane / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2019. — № 7. — С. 9-16. — Бібліогр.: 15 назв. — англ. |
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Доповіді НАН України |
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2025-07-14T12:32:07Z |
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9ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 7
1. Introduction. In our last paper [1], we have established the existence of regular solutions of the
Dirichlet problem for the quasilinear Poisson equation
( ) ( ) ( ( ))U z h z f U zΔ = ⋅ (1)
in the unit disk { : | | 1}z z= ∈ <D with continuous boundary values. We assumed that :h →D
is in ( )pL D , 1,p > and :f → is continuous and ( ) / 0f t t → as t →∞ .
This result and the theory of quasiconformal mappings (see, e.g., [2]), give a base for the study
of the semilinear equations
div [ ( ) ( )] ( ( ))A z u z f u z∇ = (2)
describing many physical phenomena in anisotropic and inhomogeneous media.
Given a simply connected domain D in the complex plane , denote, by 2 2( )KM D× , the class
of all 2 2× symmetric matrix functions ( ) { ( )}jkA z a z= with measurable real-valued entries and
det ( ) =1,A z satisfying the uniform ellipticity condition
2 21
| | ( ) , | | . .A z K a e in D
K
ξ 〈 ξ ξ〉 ξ� � (3)
© V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, 2019
https://doi.org/10.15407/dopovidi2019.07.009
UDC 517.5
V.Ya. Gutlyanskiĭ1, O.V. Nesmelova 1,3, V.I. Ryazanov 1,2
1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk
2 Bogdan Khmelnytsky National University of Cherkasy
3 Donbas State Pedagogical University, Slov’yansk
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com
On semilinear equations in the complex plane
Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ
We study the Dirichlet problem for the semilinear partial differential equations div ( ) ( )A u f u∇ = in simply connected
domains D of the complex plane with continuous boundary data. We prove the existence of the weak solutions u in
the class 1, 2
loc ( )C W D∩ , if a Jordan domain D satisfies the quasihyperbolic boundary condition by Gehring—Martio.
An example of such a domain that fails to satisfy the standard (A)-condition by Ladyzhenskaya—Ural'tseva
and the known outer cone condition is given. Some applications of the results to various processes of diffusion and
absorption in anisotropic and inhomogeneous media are presented.
Keywords: Dirichlet problem, semilinear elliptic equations, conformal and quasiconformal mappings, anisotropic
and inhomogeneous media.
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 7
V.Ya. Gutlyanski, O.V. Nesmelova, V.I. Ryazanov
for every 2ξ ∈ , where 1 K < ∞� .
Equations (2) are closely relevant to the so-called Beltrami equations. Let : Dμ → be a
measurable function with | ( ) | 1zμ < a.e. The equation
( )z zzω = μ ⋅ω , (4)
where ( ) / 2z x yiω = ω + ω , ( ) / 2z x yiω = ω − ω , z x iy= + , xω and yω are partial derivatives of
the function ω with respect to x and y , is said to be a Beltrami equation. Equation (4) is said
to be nondegenerate, if || || 1∞μ < . Homeomorphic solutions of the nondegenerate equations (4)
with the first generalized derivatives by Sobolev are called quasiconformal mappings (see, e.g., [2]).
We say that a quasiconformal mapping ω satisfying (4) is agreed with 2 2( )KA M D×∈ , if
22 11 12( ) ( ) 2 ( )
( )
det( ( ))
a z a z ia z
z
I A z
− −
μ =
+
, (5)
where I is the unit 2 2× matrix. Condition (3) is now written as
1
| ( ) | . . .
1
K
z a e in D
K
−μ
+
� (6)
Vice versa, given a measurable function : Dμ → , satisfying (6), one can invert the algebraic
system (5) to obtain the matrix function 2 2( )KA M D×∈ :
2
2 2
2
2 2
|1 | 2Im
1 | | 1 | |
( ) .
2Im |1 |
1 | | 1 | |
A z
⎛ ⎞−μ − μ
⎜ ⎟− μ − μ⎜ ⎟= ⎜ ⎟
− μ + μ⎜ ⎟
⎜ ⎟− μ − μ⎝ ⎠
(7)
Note that, by the known existence theorem for the Beltrami equations (see, e.g., Theorem
V.1.3 in [2]), any 2 2( )KA M D×∈ with condition (3) in a simply connected domain D generates
a quasiconfomal mapping : Dω →D through Eq. (4) with μ given by (5), where D is the unit
disk in .
2. Some definitions and preliminary remarks. Following [3], under a weak solution of
Eq. (2), we understand a function 1, 2
loc ( )u C W∈ ∩ Ω such that, for all 1, 2
0 ( )C W Dη∈ ∩ ,
( ) ( ), ( ) ( ) ( ( )) ( ) ( ) 0.
D D
A z u z z dm z f u z z dm z〈 ∇ ∇η 〉 + η =∫ ∫ (8)
A fundamental role in the study of the posed problem will play the following factorization
theorem (see, e.g., [4], Theorem 1, or [3], Theorem 4.1). A function :u D→ is a weak solution of
(2) in the class 1, 2
loc ( )C W D∩ , iff u U= ω , where : Dω →D is a quasiconformal mapping agreed
with A , and U is a weak solution in the class 1, 2
loc ( )C W∩ D of the quasilinear Poisson equation
( ) ( ) ( ( )), ,U w J w f U w wΔ = ⋅ ∈D (9)
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 7
On semilinear equations in the complex plane
J denotes the Jacobian of the inverse quasiconformal mapping 1 : D−ω →D .
The regularity properties of solutions of Eq. (9) strongly depends on the degree of integrabi-
lity of ( )J w . Note that the mapping * 1: −ω = ω is extended to a quasiconformal mapping of
on to itself, if D∂ is the so-called quasicircle (see, e.g., Theorem II.8.3 in [2]). The well-known
Bojarski result (see, e.g., [5]) says that the generalized derivatives of a quasiconformal map-
ping in the plane are locally integrable with some power 2q > . Note also that the Jac obian
* 2 * 2( ) | | | |w wJ w = ω − ω . Consequently, in this case, ( )pJ L∈ D for some 1p > .
Recall that the image of the unit disk D under a quasiconformal mapping of onto itself
is called a quasidisk and its boundary is called a quasicircle or a quasiconformal curve. Recall
also that a Jordan curve is a continuous one-to-one image of the unit circle in . As known, such
a smooth ( 1C ) or Lipschitz curve is a quasiconformal curve and, at the same time, quasicon for-
mal curves can be even locally non-rectifiable, as it follows from the well-known Van Koch
snowflake example (see, e.g., point II.8.10 in [2]).
By Theorem 4.7 in [6], cf. also Theorem 1 and Corollary in [7], the Jacobian of a quasicon-
formal homeomorphism : D∗ω →D is in ( )pL D , 1p > , iff D satisfies the quasihyperbolic boun-
dary condition by Gehring—Martio (see [8]), i.e.
0
0
( , )
( , ) ln
( , )D
d z D
k z z a b z D
d z D
∂
⋅ + ∀ ∈
∂
� (10)
for some constants a and b and a fixed point 0z D∈ , where 0( , )Dk z z is the quasihyperbolic dis-
tance between the points z and 0z in the domain D ,
0( , ) : .inf
( , )D
ds
k z z
d Dγ γ
=
ζ ∂∫ (11)
Here, ( , )d Dζ ∂ denotes the Euclidean distance from a point Dζ ∈ to the boundary of D and
the infimum is taken over all rectifiable curves γ joining the points z and 0z in D .
Recall that a domain D in n , 2n� , is called satisfying (A)-condition, if
0 0mes ( , ) mes ( , ) ,D B B D∩ ζ ρ Θ ⋅ ζ ρ ∀ζ ∈∂ ρ ρ� � , (12)
for some 0Θ and 0 (0,1)ρ ∈ (see 1.1.3 in [9]). Recall also that a domain D in n , 2n� , is said
to satisfy the outer cone condition, if there is a cone that makes possible to be touched by its top to
every boundary point of D from the completion of D after its suitable rotation and shift. It is
clear that the outer cone condition implies (A)-condition.
Remark 1. Note that the quasidisks D satisfy (A)-condition. Indeed, the quasidisks are the
so-called QED − domains by Gehring–Martio (see, Theorem 2.22 in [10]), and the latter satisfy
the condition
mes ( , ) mes ( , ) , diaD B B D D∗∩ ζ ρ Θ ⋅ ζ ρ ∀ζ ∈∂ ρ� � (13)
for some (0,1)∗Θ ∈ (see, Lemma 2.13 in [10]), and the quasidisks (as domains with quasihyper-
bolic boundary) have boundaries of the zero Lebesgue measure (see, e.g., Theorem 2.4 in [6]).
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 7
V.Ya. Gutlyanski, O.V. Nesmelova, V.I. Ryazanov
Thus, it remains to note that, by definition, the completions of quasidisks D in the the extended
complex plane : { }= ∪ ∞ are also quasidisks up to the inversion with respect to a circle in D .
Probably, the first example of a simply connected plane domain D with the quasihyperbolic
boundary condition, which is not a quasidisk, was constructed in [7], Theorem 2. However, this
domain satisfieds (A)-condition. Probably, one of the simplest examples of a domain D with
the quasihyperbolic boundary condition and without (A)-condition is the union of 3 open disks
with the radius 1 centered at the points 0 and 1 i± . It is clear that the domain has zero interior
angle at its boundary point 1 and, consequently, by Remark 1, it is not a quasidisk.
3. Dirichlet problem for semilinear equations.
Theorem 1. Let D be a Jordan domain in satisfying the quasihyperbolic boundary condi-
tion, 2 2( )KA M D×∈ , let : Dϕ ∂ → be a continuous function, and let :f → be a continuous
function such that
( )
0.lim
t
f t
t→∞
= (14)
Then there is a weak solution :u D→ of Eq. (2), which is locally Hölder-continuous in D and con-
tinuous in D with | Du ∂ = ϕ. If, in addition, ϕ is Hölder-continuous, then u is Hölder-continuous in D.
Proof. By Theorem 1 in [4] (see, also Theorem 4.1 in [3]), if u is a weak solution of (2), then
u U= ω , where ω is a quasiconformal mapping of D onto the unit disk D agreed with A , and
U is a weak solution of Eq. (9) with h J= , where J stands for the Jacobian of 1−ω . It is also easy
to see that if U is a weak solution of (9) with h J= , then u U= ω is a weak solution of (2). This
allows us to reduce the Dirichlet problem for Eq. (2) with a continuous boundary function ϕ in
the simply connected Jordan domain D to the Dirichlet problem for Eq. (9) in the unit disk D
with the continuous boundary function 1−ψ = ϕ ω . Indeed, ω is extended to a homeomorphism
of D onto D (see, e.g., Theorem I.8.2 in [2]). Thus, the function ψ is well defined and really is
continuous on the unit circle.
It is well-known that the quasiconformal mapping ω is locally Hölder-continuous in D
(see Theorem 3.5 in [5]). Taking into account that D is a Jordan domain in satisfying a qua-
sihyperbolic boundary condition, we can show that both mappings ω and 1−ω are Hölder-con-
tinuous in D and D , correspondingly. Indeed, Hω = Ω , where Ω is a conformal (Riemann)
mapping of D onto D , and H is a quasiconformal mapping of D onto itself. The mappings Ω and
1−Ω are Hölder-continuous in D and in D , correspondingly, by Theorem 1 and its corollary in
[7]. Next, by the reflection principle, H can be extended to a quasiconformal mapping of onto
itself (see, e.g., I.8.4 in [2]), and, consequently, H and 1H − are also Hölder-continuous in D (see
again Theorem 3.5 in [5]). Thus, the Hölder continuity of ω and 1−ω in closed domains follows
immediately.
Finally, it is easy to see that if ϕ is Hölder-continuous, then ψ is also so, and all the con-
clusions of Theorem 1 follow from Theorem 3 in [1].
Remark 2. In Theorem 3 of [1], we assumed additionally that | |f is nondecreasing with
respect to t . However, setting *
| |
( ) | ( ) |max
t s
f s f t=
�
, : [0, )s +∈ = ∞ , we see that the function
* :f + +→ is continuous and nondecreasing. Moreover, *( ) / 0f s s→ as s→∞ by (14). Hence,
all estimates in the proof of Theorem 3 in [1] remain valid without this additional condition after
the change *f f→ .
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 7
On semilinear equations in the complex plane
Corollary 1. In particular, under the hypotheses of Theorem 1 on D, ϕ, and f, there is a weak
solution U of the quasilinear Poisson equation
( ) ( ( )) . .U z f U z for a e z DΔ = ∈ (15)
which is locally Hölder-continuous in D and continuous in D with U | D∂ = ϕ . If, in addition, ϕ is
Hölder-continuous, then U is Hölder-continuous in D .
4. Some applied corollaries. 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, differen-
tial geometry, logistic problems, etc. (see, e.g., [11], [12], and the references therein). In parti-
cular, the excellent book by M. Marcus and L. Veron [12] contains a comprehensive analysis of
the Dirichlet problem for the semilinear equation
( ) ( , ( ))u z f z u zΔ = (16)
in smooth ( 2C ) domains D in n , 3n� , with boundary data in 1L . Here, ( , )t f t→ ⋅ is a con-
tinuous mapping from to a weighted Lebesgue space 1( , )L D ρ , and ( , )z f z→ ⋅ is a nonde-
creasing function for every z D∈ , ( , 0) 0f z ≡ , with
( , )
.lim
t
f z t
t→∞
= ∞ (17)
uniformly with respect to the parameter z in compact subsets of D .
The mathematical modeling of some reaction-diffusion problems leads to the study of the cor-
responding Dirichlet problem for Eq. (1) with specified right-hand side. Following [13], a nonlin-
ear system can be obtained for the density u and the temperature T of a reactant. Upon elimi-
nating T , the system can be reduced to a scalar problem for the concentration
( ),u f uΔ = λ ⋅ (18)
where λ stands for a positive constant.
It turns out that the reactant density u may be zero in a closed interior region 0D called a
dead core. If, for instance, in Eq. (18), ( ) qf u u= , 0q > , a particularization of the results in Chapter
1 of [11] shows that a dead core may exist, if and only if 0 1q< < and λ is large enough. See also
the corresponding examples of dead cores in [3]. We have, by Theorem 1, the following:
Theorem 2. Let D be a Jordan domain in satisfying the quasihyperbolic boundary condition,
2 2( )KA M D×∈ , : Dϕ ∂ → be a continuous function. Then there is a weak solution :u D→ of the
semilinear equation
div [ ( ) ( )] ( ), 0 1qA z u z u z q∇ = < < (19)
which is locally Hölder-continuous in D and continuous in D with | Du ∂ = ϕ . If, in addition, ϕ is
Hölder-continuous, then u is Hölder-continuous in D .
We have also the following consequence of Corollary 1.
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 7
V.Ya. Gutlyanski, O.V. Nesmelova, V.I. Ryazanov
Corollary 2. Let D be a smooth Jordan domain in , and let : Dϕ ∂ → be a continuous fun-
ction. Then there is a weak solution U of the quasilinear Poisson equation
( ) = ( ) , 0 < <1,qU z U z qΔ (20)
which is continuous in D with U | D∂ = ϕ and 1,
loc ( )U C Dα∩ for all (0,1)α ∈ . If, in addition, ϕ is
Hölder-continuous with some order (0,1)β ∈ , then U is also Hölder-continuous in D with the
same order.
Recall also that certain mathematical models of a heated plasma lead to nonlinear equations
of the type (18). Indeed, it is known that some of them have the form ( ) ( )u f uΔψ = with
(0)ψ = +∞′ and ( ) 0uψ >′ , if 0u ≠ , as, for instance, 1( ) | |qu u u−ψ = under 0 1q< < (see, e.g., [14]
and [11, p. 4]). With the replacement of the function ( ) | | signqU u u u= ψ = ⋅ , we have that
| | signQu U U= ⋅ , 1/Q q= , and, with the choice
2
( ) | | signqf u u u= ⋅ , we come to the equation
| | sign ( )qU U U UΔ = ⋅ = ψ .
Corollary 3. Let D be a smooth Jordan domain in , and let : Dϕ ∂ → be a continuous func-
tion. Then there is a weak solution U of the quasilinear Poisson equation
1( ) | ( ) | ( ), 0 1,qU z U z U z q−Δ = < < (21)
which is continuous in D with U | D∂ = ϕ and 1,
loc ( )U C Dα∩ for all (0,1)α ∈ . If, in addition, ϕ is
Hölder-continuous with some order (0,1)β ∈ , then U is also Hölder-continuous in D with the
same order.
In the combustion theory, the following model equation
( , ) 1
, 0, ,uu z t
u e t z D
t
∂ = ⋅Δ + ∈
∂ δ
� (22)
occupies a special place (see, e.g., [15] and the references therein). Here, 0u� is the temperature
of the medium, and δ is a certain positive parameter. We restrict ourselves by stationary solutions
of (22) and generalizations in anisotropic and inhomogeneous media, although our approach ma-
kes it possible to consider the parabolic case as well (see [3]). Namely, by Theorem 1, we have:
Theorem 3. Let D be a Jordan domain in satisfying the quasihyperbolic boundary condition,
2 2( )KA M D×∈ , and let : Dϕ ∂ → be a continuous function. Then there is a weak solution :u D→
of the semilinear equation
( )div [ ( ) ( )] U zA z U z e−∇ = δ ⋅ , (23)
which is locally Hölder-continuous in D and continuous in D with | Du ∂ = ϕ . If, in addition, ϕ is
Hölder-continuous, then u is Hölder-continuous in D .
Finally, we obtain the following consequence of Corollary 1.
Corollary 4. Let D be a smooth Jordan domain in , and let : Dϕ ∂ → be a continuous
function. Then there is a weak solution U of the quasilinear Poisson equation
( )( ) U zU z eΔ = δ⋅ (24)
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 7
On semilinear equations in the complex plane
which is continuous in D with U | D∂ = ϕ and 1,
loc ( )U C Dα∩ for all (0,1)α ∈ . If, in addition, ϕ
is Hölder-continuous with some order (0,1)β ∈ , then U is also Hölder-continuous in D with
the same order.
This work was partially supported by grants of Ministry of Education and Science of Ukraine,
project number is 0119U100421.
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2018.02.012
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Received 26.03.2019
В.Я. Гутлянський 1,
О.В. Нєсмєлова 1, 3, В.І. Рязанов 1, 2
1 Інститут прикладної математики і механіки НАН України, Слов’янськ
2 Черкаський національний університет ім. Богдана Хмельницького
3 Донбаський державний педагогічний університет, Слов’янськ
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com
ПРО НАПІВЛІНІЙНІ РІВНЯННЯ В КОМПЛЕКСНІЙ ПЛОЩИНІ
Досліджено задачу Діріхле для напівлінійних рівнянь в частинних похідних div ( ) ( )A u f u∇ = в одно-
зв’язних областях D комплексної площини з неперервними граничними умовами. Доведено існування
слабких розв’язків u у класі 1, 2
loc ( )C W D∩ , якщо Жорданова область D задовольняє квазігіперболічну гра-
16 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 7
V.Ya. Gutlyanski, O.V. Nesmelova, V.I. Ryazanov
ничну умову Герінга—Мартіо. Наведено приклад такої області, яка не задовольняє стандартну (А)-умову
Ладиженської–Уральцевої та відому умову зовнішнього конуса. Також наведено деякі застосування отри-
маних резуль т а тів до різних процесів дифузії та поглинання в анізотропних і неоднорідних середовищах.
Ключові слова: задача Діріхле, напівлінійні еліптичні рівняння, конформні та квазіконформні відображен-
ня, анізотропні та неоднорідні середовища.
В.Я. Гутлянский 1,
О.В. Несмелова 1,3, В.И. Рязанов 1,2
1 Институт прикладной математики и механики НАН Украины, Славянск
2 Черкасский национальный университет им. Богдана Хмельницкого
3 Донбасский государственный педагогический университет, Славянск
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com
О ПОЛУЛИНЕЙНЫХ УРАВНЕНИЯХ НА КОМПЛЕКСНОЙ ПЛОСКОСТИ
Исследована задача Дирихле для полулинейных уравнений в частных производных div ( ) ( )A u f u∇ = в
односвязных областях D комплексной плоскости с непрерывными граничными условиями. Доказано
существование слабых решений u в классе 1, 2
loc ( )C W D∩ , если Жорданова область удовлетворяет квази-
гиперболическому граничному условию Геринга—Мартио. Приведен пример такой области, которая не
удовлетворяет стандартному (А)-условию Ладыженской—Уральцевой и известному условию внешнего
конуса. Также приведены некоторые применения полученных результатов к различным процессам диф-
фузии и поглощения в анизотропных и неоднородных средах.
Ключевые слова: задача Дирихле, полулинейные эллиптические уравнения, конформные и квазиконформ-
ные отражения, анизотропные и неоднородные среды.
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