General elliptic boundary-value problems in Hörmander—Roitberg spaces
We study semilinear partial differential equations in the plane, the linear part of which is written in a divergence form. The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation can be represented as a composition of a weak solution of...
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| Cite this: | General elliptic boundary-value problems in Hörmander—Roitberg spaces / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 2. — С. 12-18. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1326352018-04-25T03:03:02Z General elliptic boundary-value problems in Hörmander—Roitberg spaces Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. Математика We study semilinear partial differential equations in the plane, the linear part of which is written in a divergence form. The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation can be represented as a composition of a weak solution of the corresponding isotropic equation in a canonical domain and a quasiconformal mapping agreed with a matrix-valued measurable coefficient appearing in the divergence part of the equation. The latter makes it possible, in particular, to remove the regularity restrictions on the boundary in the study of boundary-value problems for such semilinear equations. Вивчено напівлінійні диференціальні рівняння в частинних похідних на площині, лінійна частина яких подана в дивергентній формі. Основний результат сформульований у вигляді теореми факторизації. Ця теорема стверджує, що будь-який слабкий розв'язок такого рівняння можна подати у вигляді композиції слабкого розв’язку відповідного ізотропного рівняння в канонічній області і квазіконформного відображення, узгодженого з матричнозначним вимірюваним коефіцієнтом, який входить до дивергентної частини вихідного рівняння. Свобода у виборі канонічної області дозволяє, зокрема, зняти деякі обмеження на регулярність границі при дослідженні крайових задач для таких напівлінійних рівнянь. Изучены полулинейные дифференциальные уравнения в частных производных на плоскости, линейная часть которых представлена в дивергентной форме. Основной результат сформулирован в виде теоремы факторизации. Эта теорема утверждает, что любое слабое решение такого уравнения представимо в виде композиции слабого решения соответствующего изотропного уравнения в канонической области и квазиконформного отображения, согласованного с матричнозначным измеримым коэффициентом, входящим в дивергентную часть исходного уравнения. Свобода в выборе канонической области позволяет, в частности, снять некоторые ограничения на регулярность границы при исследовании краевых задач для таких полулинейных уравнений. 2018 Article General elliptic boundary-value problems in Hörmander—Roitberg spaces / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 2. — С. 12-18. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2018.02.012 http://dspace.nbuv.gov.ua/handle/123456789/132635 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Математика Математика |
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Математика Математика Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. General elliptic boundary-value problems in Hörmander—Roitberg spaces Доповіді НАН України |
| description |
We study semilinear partial differential equations in the plane, the linear part of which is written in a divergence form.
The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation
can be represented as a composition of a weak solution of the corresponding isotropic equation in a canonical domain
and a quasiconformal mapping agreed with a matrix-valued measurable coefficient appearing in the divergence
part of the equation. The latter makes it possible, in particular, to remove the regularity restrictions on the boundary
in the study of boundary-value problems for such semilinear equations. |
| format |
Article |
| author |
Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_facet |
Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_sort |
Gutlyanskii, V.Ya. |
| title |
General elliptic boundary-value problems in Hörmander—Roitberg spaces |
| title_short |
General elliptic boundary-value problems in Hörmander—Roitberg spaces |
| title_full |
General elliptic boundary-value problems in Hörmander—Roitberg spaces |
| title_fullStr |
General elliptic boundary-value problems in Hörmander—Roitberg spaces |
| title_full_unstemmed |
General elliptic boundary-value problems in Hörmander—Roitberg spaces |
| title_sort |
general elliptic boundary-value problems in hörmander—roitberg spaces |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2018 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/132635 |
| citation_txt |
General elliptic boundary-value problems in Hörmander—Roitberg spaces / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2018. — № 2. — С. 12-18. — Бібліогр.: 15 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT gutlyanskiivya generalellipticboundaryvalueproblemsinhormanderroitbergspaces AT nesmelovaov generalellipticboundaryvalueproblemsinhormanderroitbergspaces AT ryazanovvi generalellipticboundaryvalueproblemsinhormanderroitbergspaces |
| first_indexed |
2025-07-09T17:49:13Z |
| last_indexed |
2025-07-09T17:49:13Z |
| _version_ |
1837192555611029504 |
| fulltext |
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 2
The main goal of this paper is to point out one application of quasiconformal mappings to the
study of some nonlinear partial differential equations in the plane.
Let Ω be a domain in the complex plane C. It is well known that the Beltrami equation
( ) ,z zz zω = μ ω ∈Ω , (1)
where
1
( )
2z x yiω = ω − ω ,
1
( )
2z x yiω = ω + ω , z x iy= + , is turned out to be instrumental in the
study of Riemann surfaces, Teichmüller spaces, Kleinian groups, meromorphic functions, low di-
mensional topology, holomorphic motion, complex dynamics, Clifford analysis, and control theory.
As known, a K-quasiconformal mapping :ω Ω→C, 1K� , is just a homeomorphic 1, 2
loc ( )W Ω
solution to the Beltrami equation when the measurable coefficient μ satisfies the strong ellip-
ticity condition | ( ) | ( 1) / ( 1)z K Kμ − +� almost everywhere in Ω . In particular, if 0μ = in a
domain Ω⊂C, then the Beltrami equation reduces to the Cauchy—Riemann equation and a
so lution ω is analytic in Ω, see, e.g., [1, 2], see also [3], and the references therein.
We will deal with semilinear partial differential equations
div [ ( ) ] ( )A z u f u∇ = , (2)
© V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, 2018
doi: https://doi.org/10.15407/dopovidi2018.02.012
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
Semilinear equations in the plane
with measurable data
Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ
We study semilinear partial differential equations in the plane, the linear part of which is written in a divergence form.
The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation
can be represented as a composition of a weak solution of the corresponding isotropic equation in a canonical domain
and a quasiconformal mapping agreed with a matrix-valued measurable coefficient appearing in the divergence
part of the equation. The latter makes it possible, in particular, to remove the regularity restrictions on the boundary
in the study of boundary-value problems for such semilinear equations.
Keywords: semilinear elliptic equations, quasiconformal mappings, Beltrami equation.
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 2
Semilinear equations in the plane with measurable data
linear part of which contains the elliptic operator in the divergence form, where the matrix func-
tion ( )A z is in the class 2 2( )M × Ω of 2 2× symmetric matrix functions with measurable en-
tries ( )jka z , , 1, 2j k = , the determinant 1, and the uniform ellipticity condition
2 21
| | ( ) , | | a.e. inA z K
K
ξ 〈 ξ ξ〉 ξ Ω� � (3)
for every ξ ∈C, where 1 K < ∞� . For the case of smooth ( )A z , see [4, 5].
We prove the following Factorization Theorem: Every weak solution 1, 2
loc ( )u C W∈ ∩ Ω of
the semilinear equation (2) with arbitrary continuous ( )f u can be represented as u T= ω ,
where : Gω Ω→ ⊆ is a quasiconformal mapping agreed with the matrix function A , and
1, 2
loc ( )T C W G∈ ∩ is a weak solution of the semilinear Poisson equation
( ) in ,T mf T GΔ = (4)
where ( )m w , w G∈ , is the Jacobian of the inverse mapping 1( )w−ω . In particular, we obtain here
the semilinear Gauss—Bieberbach—Rademacher equation with the weight ( )m w for the case
( ) uf u e= .
1. Some definitions and preliminary remarks. Given 2 2( )A M ×∈ Ω , let us first consider
the second-order elliptic homogeneous equation
indiv( ( ) ) 0 a.e.A z u∇ = Ω . (5)
A function u is called a weak solution to the equation if
0( ) , 0 ( )A z u C∞
Ω
〈 ∇ ∇ϕ〉 = ∀ϕ ∈ Ω∫ . (6)
This is meaningful at least for 1,1
loc ( )u W∈ Ω , where 1,
loc ( )pW Ω stands for the well-known Sobolev
space. Here, we will assume a little more regularity, namely that 1, 2
loc ( )u C W∈ ∩ Ω .
Let 2 2( )A M ×∈ Ω and 1, 2
loc ( )u C W∈ ∩ Ω be a weak solution to (5). Then there exists
1, 2
loc ( )v C W∈ ∩ Ω called the stream function of ,u such that
0 1
a.e. in , where
1 0
v H A u H
−⎛ ⎞
∇ = ∇ Ω = ⎜ ⎟⎝ ⎠
. (7)
Setting ( ) ( ) ( )z u z iv zω = + we see that ω satisfies the Beltrami equation
in( ) ( ) ( ) a.e.z zz z zω = μ ω Ω , (8)
where the complex dilatation ( )zμ is given by
22 11 12( ) ( ) 2 ( )
( )
det( ( ))
a z a z ia z
z
I A z
− −
μ =
+
, (9)
see, e.g., Theorem 16.1.6 in [6]. The condition of ellipticity (3) now is written as
in1
| ( ) | a.e.
1
K
z
K
−μ Ω
+
� . (10)
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 2
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Thus, given any 2 2( )A M ×∈ Ω , one produces by (9) the complex dilatation ( )zμ for which,
in turn, by the Measurable Riemann mapping theorem, see, e.g., Theorem V.B.3 in [1] and Theo-
rem V.1.3 in [2], the Beltrami equation (8) generates, as its solution, a quasiconfomal homeo-
morphism ω . We say that the matrix function A generates the corresponding quasiconformal
mapping ω , or that A and ω are agreed.
Note also the useful fact that the quasiconformal mappings ω admit a change of variables in
integrals, because homeomorphisms of the class 1, 2
locW are absolute continuous with respect to the
area measure, see, e.g., Theorem III.6.1, Lemmas III.2.1 and III.3.3 in [2].
We complete this section with the following very important result on the composition opera-
tors in Sobolev spaces, see, e.g., [7—9].
Proposition 1. Let :ω Ω→C be a quasiconformal homeomorphism and let :Gϕ →C belong to
the class 1, 2
locW (Ω). Then the composition function 1, 2
loc ( )W Gϕ ω ∈ .
The study of the superposition operators on Sobolev spaces stems from the classical article
[10], see also, e.g., [11—13] for the detailed history and bibliography.
2. The basic identity. It is well known that every positive definite quadratic form
2 2 2( , ) 2 ( , ) ( , )ds a x y dx b x y dxdy c x y dy= + + , (11)
defined in a plane domain Ω , can be reduced, by means of a suitable quasiconformal change of
variables, to the canonical form
in2 2 2( ), 0 a.e.ds du dv= Λ + Λ = Ω, (12)
provided that 2
0 0ac b− Δ >� , 0a > , a.e. in Ω , see, e.g., [14, pp. 10-12]. This key result can be
extended to every linear divergent operator of the form div[ ( ) ( )]A z u z∇ , z x iy= + , with an
arbitrary matrix function 2 2( )A M ×∈ Ω .
Namely, we have already seen by direct computation that if the function T and the entries
of A are sufficiently smooth, then
div [ ( ) ( ( ( )))] ( ) ( ( )),A z T z J z T z zω∇ ω = Δ ω ∈Ω , (13)
see [4, 5]. Here, ( )wJ z stands for the Jacobian of the mapping ( )w z , i.e., ( ) det ( )J z D zω ω= ,
where ( )wD z is the Jacobian matrix of the mapping ω at the point z ∈Ω . Equality (14) below
can be viewed as a weak counterpart to equality (13).
Proposition 2. Let Ω be a domain in C, 2 2( )A M ×∈ Ω and :ω Ω→G be a quasiconformal
mapping agreed with A. Then the equality
1( ) ( ( ( ))), ( ) ( ) ( ( )), ( ) ( )z zA z T z z dm D z T z z J z dm−
ω ω
Ω Ω
〈 ∇ ω ∇ϕ 〉 = 〈 ∇ ω ∇ϕ 〉∫ ∫ (14)
holds for every 1, 2
loc ( )T W G∈ and for all 1, 2
0 ( )Wϕ ∈ Ω .
Proof. Assuming that 1, 2
loc ( )T W G∈ and that :ω Ω→G is a quasiconformal mapping agreed
with ( )A z , we see, by Proposition 1, that 1, 2
loc:u T W= ω ∈ (Ω). Since
( ) ( ) ( ( ))tu z D z T zω∇ = ∇ ω , (15)
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 2
Semilinear equations in the plane with measurable data
where ( )tD zω stands for the transpose matrix to ( )D zω and ω satisfies the Beltrami equation
(8), that can be written in the matrix form as
1( ) ( ) ( ) ( )tA z D z D z J z−
ω ω ω= , (16)
we arrive successively at the required equality (14):
1
( ) ( ( ( ))), ( ) ( ) ( ) ( ( )), ( )
( ) ( ( )), ( ) ( ) .
t
z z
z
A z T z z dm A z D z T z z dm
D z T z z J z dm
ω
Ω Ω
−
ω ω
Ω
〈 ∇ ω ∇ϕ 〉 = 〈 ∇ ω ∇ϕ 〉 =
= 〈 ∇ ω ∇ϕ 〉
∫ ∫
∫
(17)
3. The main result. Let Ω be a bounded domain in C and let f : R → R be a continuous func-
tion. In this section, we study a model semilinear equation
div [ ( ) ( )] ( ( )),A z u z f u z z∇ = ∈Ω , (18)
as well as its Laplace counterp3art:
( ) ( ) ( ( )), ( )T w J w f T w w GΔ = ∈ = ω Ω , (19)
where : Gω Ω→ is a quasiconformal mapping agreed with ( )A z and ( )J w stands for the Ja-
co bian of the inverse mapping
1 :G−ω →Ω.
We say that a function 1, 2
loc ( )u C W∈ ∩ Ω is a weak solution to Eq. (18) if
1, 2
0( ) ( ), ( ) ( ( )) ( ) 0 ( )z zA z u z z dm f u z z dm C W
Ω Ω
〈 ∇ ∇ϕ 〉 + ϕ = ∀ϕ ∈ ∩ Ω∫ ∫ . (20)
We also say that a function 1, 2
loc ( )T C W G∈ ∩ is a weak solution to Eq. (19) if
1, 2
0( ), ( ) ( ) ( ( )) ( ) 0 ( )w w
G G
T w w dm J w f T w w dm C W G〈∇ ∇ψ 〉 + ψ = ∀ψ ∈ ∩∫ ∫ . (21)
Since ( )J w is the Jacobian of the mapping 1( )w−ω it is easy to verify, by performing the chan-
ge of a variable by the formula ( )w z= ω that the second integral in (21) is well-defined. Here,
we again made use of the fact from Proposition 1 that the composed mapping ( ) ( ( ))u z T z= ω
is in 1, 2
loc ( )C W∩ Ω if 1, 2
loc ( )T C W G∈ ∩ and ω is quasiconformal.
Theorem 1. Let Ω be a domain in C, 2 2( )A M ×∈ Ω and let f : R → R be a continuous function.
Then every weak solution u of the semilinear equation
div [ ( ) ( )] ( ( )),A z u z f u z z∇ = ∈Ω , (22)
can be represented as the composition
( ) ( ( ))u z T z= ω , (23)
where : Gω Ω→ is a quasiconformal mapping agreed with A and T is a weak solution to the equation
( ) ( ) ( ( )),T w J w f T w w GΔ = ∈ . (24)
16 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 2
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
Corollary 1. If ( ) 0f u � , then the function T in Theorem 1 is subharmonic.
Proof. Let u be a weak solution of the semilinear equation (22) and 1T u −= ω . Then
1, 2
loc ( )T C W G∈ ∩ by Proposition 1, and we have that
( ) ( ( ( ))), ( ) ( ( ( ))) ( ) 0z zA z T z z dm f T z z dm
Ω Ω
〈 ∇ ω ∇ϕ 〉 + ω ϕ =∫ ∫ (25)
for all 1, 2
0 ( )C Wϕ ∈ ∩ Ω . Next, by Proposition 2,
1( ) ( ( ( ))), ( ) ( ) ( ( )), ( ) ( )z zA z T z z dm D z T z z J z dm−
ω ω
Ω Ω
〈 ∇ ω ∇ϕ 〉 = 〈 ∇ ω ∇ϕ 〉∫ ∫ , (26)
аnd, therefore,
1( ) ( ) ( ( )), ( ) ( ( ( ))) ( ) 0z zJ z D z T z z dm f T z z dm−
ω ω
Ω Ω
〈 ∇ ω ∇ϕ 〉 + ω ϕ =∫ ∫ (27)
for all 1, 2
0 ( )C Wϕ ∈ ∩ Ω .
Given an arbitrary function 1, 2
0( ) ( )w C W Gψ ∈ ∩ , we can set ( ) ( ( ))z zϕ = ψ ω in (25) and
(26), because such 1, 2
0 ( )C Wϕ ∈ ∩ Ω again by Proposition 1. Performing the change of a variable
in (27) by the formula 1( )z w−= ω , we obtain
1 1 1 1( ( )) ( ( )) ( ), ( ( )) ( ) ( )
( ) ( ( )) ( ) 0.
t
w
G
w
G
J w D w T w D w w J w dm
J w f T w w dm
− − − −
ω ω ω〈 ω ω ∇ ω ∇ψ 〉 +
+ ψ =
∫
∫
Since, by elementary algebraic arguments,
1 1 1 1
1
( ( )) ( ( )) ( ), ( ( )) ( )
( ( )) ( ), ( ) ,
tJ w D w T w D w w
J w T w w
− − − −
ω ω ω
−
ω
〈 ω ω ∇ ω ∇ψ 〉 =
= ω 〈∇ ∇ψ 〉
and
1( ( )) 1/ ( )J w J w−
ω ω = ,
we see that the identity
( ), ( ) ( ) ( ( )) ( ) 0w w
G G
T w w dm J w f T w w dm〈∇ ∇ψ 〉 + ψ =∫ ∫ (28)
holds for all 1, 2
0( ) ( )w C W Gψ ∈ ∩ . Thus, T is a weak solution to Eq. (24).
Remark 1. Inversely, since the arguments given above are invertible, we see that if T is a weak
solution to Eq. (24), then the function ( ) ( ( ))u z T z= ω is a weak solution to Eq. (22). Note also
that, among the quasiconformal mappings : Gω Ω→ , there is a variety of the so-called volume-
preserving maps, for which ( ) 1J z ≡ , z ∈Ω . If A generates such ω , then T is a weak solution of
17ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 2
Semilinear equations in the plane with measurable data
the quasilinear Poisson equation
( )T f T in GΔ = . (29)
4. The final remarks. By the Measurable Riemann mapping theorem, see, e.g., Theorem
V.B.3 in [1] and Theorem V.1.3 in [2], given ( )zμ , z ∈Ω, agreed with the matrix function
2 2( )A M ×∈ Ω , there exists a quasiconformal mapping : Gω Ω→ with the complex dilatation μ.
Here, if Ω is finitely connected, then G can be chosen as a circular domain whose boundary
con sists of circles or points, see, e.g., Theorem V.6.2 in [15]. If Ω is simply connected with a non-
degenerate boundary, then we may assume that G is the unit disk D in . The latter makes it
possible to remove the restrictions on the regularity of the boundary in the study of boundary-
value problems for Eq. (24).
The corresponding factorization theorems can be established for other similar semilinear
equations in the anisotropic case such as the nonlinear heat equation like
div [ ( ) ( )] ( )tu A z u z f u− ∇ = (30)
(the same equation describes the Brownian motion, diffusion models of the population dyna-
mics, and many other phenomena), the nonlinear Schrödinger equation, and the nonlinear wave
equation
div [ ( ) ( )] ( )ttu A z u z f u− ∇ = . (31)
Namely, one can show that every weak solution in a suitable sense for semilinear equations of such
type can be factorized as the composition of a weak solution to the corresponding isotropic equa-
tion and a quasiconformal mapping agreed with the matrix function ( )A z as above.
REFERENCES
1. Ahlfors, L. V. (1966). Lectures on quasiconformal mappings. Princeton, N.J.: Van Nostrand. Reprinted by
Wadsworth Ink. Belmont, 1987.
2. Lehto, O. & Virtanen, K. I. (1973). Quasiconformal mappings in the plane. 2nd ed. Berlin, Heidelberg,
New York: Springer.
3. Gutlyanskii, V., Ryazanov, V., Srebro, U. & Yakubov, E. (2012). The Beltrami Equation: A Geometric Ap-
proach. Developments in Mathematics. Vol. 26. New York: Springer.
4. Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2016). On a model semilinear elliptic equation in the plane.
Ukr. Mat. Visn., 13, No. 1, pp. 91-105; J. Math. Sci., 2017, 220, No. 5, pp. 603-614.
5. Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2017). Semilinear equations in a plane and qua-
siconformal mappings. Dopov. Nac. akad. nauk Ukr., No. 1, pp. 10-16. doi: https://doi.org/10.15407/
dopovidi2017.01.010
6. Astala, K., Iwaniec, T. & Martin, G. (2009). Elliptic partial differential equations and quasiconformal map-
pings in the plane. Princeton Mathematical Series. Vol. 48. Princeton, N.J.: Princeton Univ. Press.
7. Gol’dshtein, V. & Ukhlov, A. (2010). About homeomorphisms that induce composition operators on Sobo-
lev spaces. Complex Var. Elliptic Equ., 55, No. 8-10, pp. 833-845.
8. Ukhlov, A. (1993). Mappings that generate embeddings of Sobolev spaces. Sibirsk. Mat. Zh., 34, No. 1,
pp. 185-192 (in Russian); Siberian Math. J., 34, No.1, pp. 165-171.
9. Vodopyanov, S. K. & Ukhlov, A. (1998). Sobolev spaces and ( , )P Q -quasiconformal mappings of Carnot
groups. Sib. Mat. Zh., 39, No. 4, pp. 665-682 (in Russian); Math. J., 39, No. 4, pp. 665-682.
10. Sobolev, S. L. (1941). On some transformation groups of an n-dimensional space. Dokl. AN SSSR, 32, No. 6,
pp. 380-382 (in Russian).
18 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 2
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov
11. Gol’dshtein, V., Gurov, L. & Romanov, A. (1995). Homeomorphisms that induce monomorphisms of Sobolev
spaces. Israel J. Math., 91, No. 1-3, pp. 31-60.
12. Vodop’yanov, S. K. (2012). On the regularity of mappings inverse to the Sobolev mapping. Mat. Sb., 203,
No. 10, pp. 3-32 (in Russian); Sb. Math., 203, No. 9-10, pp. 1383-1410.
13. Vodop’yanov, S. K. & Evseev, N. A. (2014). Isomorphisms of Sobolev spaces on Carnot groups and quasi-iso-
metric mappings. Sibirsk. Mat. Zh., 55, No. 5, pp. 1001-1039 (in Russian); Math. J., 55, No. 5, pp. 817-848.
14. 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. Zurich: European Mathematical
Society.
15. Goluzin, G.M. (1969). Geometric Theory of functions of a complex variable. Providence, Rhode Island: Amer.
Math. Soc.
Received 20.10.2017
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
ПОЛУЛИНЕЙНЫЕ УРАВНЕНИЯ НА ПЛОСКОСТИ
С ИЗМЕРИМЫМИ ДАННЫМИ
Изучены полулинейные дифференциальные уравнения в частных производных на плоскости, линейная
часть которых представлена в дивергентной форме. Основной результат сформулирован в виде теоремы
факторизации. Эта теорема утверждает, что любое слабое решение такого уравнения представимо в виде
композиции слабого решения соответствующего изотропного уравнения в канонической области и квази-
конформного отображения, согласованного с матричнозначным измеримым коэффициентом, входящим в
дивергентную часть исходного уравнения. Свобода в выборе канонической области позволяет, в частно-
сти, снять некоторые ограничения на регулярность границы при исследовании краевых задач для таких
полулинейных уравнений.
Ключевые слова: полулинейные эллиптические уравнения, квазиконформные отображения, уравнение
Бельтрами.
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
НАПІВЛІНІЙНІ РІВНЯННЯ НА ПЛОЩИНІ
З ВИМІРНИМИ ДАНИМИ
Вивчено напівлінійні диференціальні рівняння в частинних похідних на площині, лінійна частина яких
подана в дивергентній формі. Основний результат сформульований у вигляді теореми факторизації.
Ця теорема стверджує, що будь-який слабкий розв’язок такого рівняння можна подати у вигляді компо-
зиції слабкого розв’язку відповідного ізотропного рівняння в канонічній області і квазіконформного ві-
дображення, узгодженого з матричнозначним вимірюваним коефіцієнтом, який входить до дивергент-
ної частини вихідного рівняння. Свобода у виборі канонічної області дозволяє, зокрема, зняти деякі об-
меження на регулярність границі при дослідженні крайових задач для таких напівлінійних рівнянь.
Ключові слова: напівлінійні еліптичні рівняння, квазіконформні відображення, рівняння Бельтрамі.
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