Finite Lipschitz mappings on Finsler manifolds
We consider ring Q-homeomorphisms with respect to p-modulus on Finsler manifolds, n - 1 < p < n, and establish sufficient conditions for these mappings to be finitely Lipschitzian. Рассматриваются кольцевые Q-гомеоморфизмы относительно p-модуля на финслеровых многообразиях, n - 1 < p < n...
Saved in:
| Published in: | Труды Института прикладной математики и механики |
|---|---|
| Date: | 2016 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2016
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/140851 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Finite Lipschitz mappings on Finsler manifolds / O.S. Afanas’eva // Труды Института прикладной математики и механики НАН Украины. — Слов’янськ: ІПММ НАН України, 2016. — Т. 30. — С. 13-20. — Бібліогр.: 28 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-140851 |
|---|---|
| record_format |
dspace |
| spelling |
Afanas’eva, O.S. 2018-07-17T10:56:55Z 2018-07-17T10:56:55Z 2016 Finite Lipschitz mappings on Finsler manifolds / O.S. Afanas’eva // Труды Института прикладной математики и механики НАН Украины. — Слов’янськ: ІПММ НАН України, 2016. — Т. 30. — С. 13-20. — Бібліогр.: 28 назв. — англ. 1683-4720 https://nasplib.isofts.kiev.ua/handle/123456789/140851 517.5 We consider ring Q-homeomorphisms with respect to p-modulus on Finsler manifolds, n - 1 < p < n, and establish sufficient conditions for these mappings to be finitely Lipschitzian. Рассматриваются кольцевые Q-гомеоморфизмы относительно p-модуля на финслеровых многообразиях, n - 1 < p < n, устанавливаются достаточные условия конечной липшицевости этих отображений. Розглядаються кiльцевi Q-гомеоморфiзми вiдносно p-модуля на фiнслерових многовидах, n-1 < p < n, та встановлюються достатнi умови кiнцевої лiпшицевостi таких вiдображень. en Інститут прикладної математики і механіки НАН України Труды Института прикладной математики и механики Finite Lipschitz mappings on Finsler manifolds Конечно липшицевы отображения на финслеровых многообразиях Кiнцево лiпшицевi вiдображення на фiнслерових многовидах Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Finite Lipschitz mappings on Finsler manifolds |
| spellingShingle |
Finite Lipschitz mappings on Finsler manifolds Afanas’eva, O.S. |
| title_short |
Finite Lipschitz mappings on Finsler manifolds |
| title_full |
Finite Lipschitz mappings on Finsler manifolds |
| title_fullStr |
Finite Lipschitz mappings on Finsler manifolds |
| title_full_unstemmed |
Finite Lipschitz mappings on Finsler manifolds |
| title_sort |
finite lipschitz mappings on finsler manifolds |
| author |
Afanas’eva, O.S. |
| author_facet |
Afanas’eva, O.S. |
| publishDate |
2016 |
| language |
English |
| container_title |
Труды Института прикладной математики и механики |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| title_alt |
Конечно липшицевы отображения на финслеровых многообразиях Кiнцево лiпшицевi вiдображення на фiнслерових многовидах |
| description |
We consider ring Q-homeomorphisms with respect to p-modulus on Finsler manifolds, n - 1 < p < n, and establish sufficient conditions for these mappings to be finitely Lipschitzian.
Рассматриваются кольцевые Q-гомеоморфизмы относительно p-модуля на финслеровых многообразиях, n - 1 < p < n, устанавливаются достаточные условия конечной липшицевости этих отображений.
Розглядаються кiльцевi Q-гомеоморфiзми вiдносно p-модуля на фiнслерових многовидах, n-1 < p < n, та встановлюються достатнi умови кiнцевої лiпшицевостi таких вiдображень.
|
| issn |
1683-4720 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/140851 |
| citation_txt |
Finite Lipschitz mappings on Finsler manifolds / O.S. Afanas’eva // Труды Института прикладной математики и механики НАН Украины. — Слов’янськ: ІПММ НАН України, 2016. — Т. 30. — С. 13-20. — Бібліогр.: 28 назв. — англ. |
| work_keys_str_mv |
AT afanasevaos finitelipschitzmappingsonfinslermanifolds AT afanasevaos konečnolipšicevyotobraženiânafinslerovyhmnogoobraziâh AT afanasevaos kincevolipšicevividobražennânafinslerovihmnogovidah |
| first_indexed |
2025-11-26T23:44:01Z |
| last_indexed |
2025-11-26T23:44:01Z |
| _version_ |
1850783147620302848 |
| fulltext |
ISSN 1683-4720 Труды ИПММ НАН Украины. 2016. Том 30
UDK 517.5
c⃝2016. O. S. Afanas’eva
FINITE LIPSCHITZ MAPPINGS ON FINSLER MANIFOLDS
We consider ring Q-homeomorphisms with respect to p-modulus on Finsler manifolds, n − 1 <
p < n, and establish sufficient conditions for these mappings to be finitely Lipschitzian.
Key words: Finsler manifolds, ring Q-homeomorphisms, p-modulus, finite Lipschitz mappings.
1. Introduction.
In this article we continue our study of mappings on Finsler manifolds (Mn,Φ)
started in [1]. For historical remarks and needed definitions, we refer to [1]. The main
tools involve the method of moduli applied to ring Q-homeomorphisms and the method
of p-capacities recently developed for Finsler manifolds. For the latter see [2]–[4].
Recall that a mapping f : D → D′ between Finsler manifolds (Mn,Φ) and (Mn
∗ ,Φ∗),
n ≥ 2, is called Lipschitz if there is a finite constant C > 0 such that the inequality
d∗Φ(f(x), f(y)) ≤ C · dΦ(x, y) holds for all x, y ∈ Mn, cf. [5]. We say that a continuous
mapping f : D → D′ is finitely Lipschitzian on the domain D if
L(x, f) = lim sup
y→x
d∗Φ(f(x), f(y))
dΦ(x, y)
<∞
for all x ∈ D, cf. [6].
The main result of the paper is the following statement.
Theorem 1. Let D and D′ be domains in (Mn, Φ̃) and (Mn
∗ , Φ̃∗), n ≥ 2, respectively.
Assume that Q : D → [0, ∞] is a locally integrable function such that
lim sup
ε→0
1
σ
Φ̃
(B(x0, ε))
∫
B(x0,ε)
Q(x) dσ
Φ̃
(x) <∞ (1)
and f : D → D′ is a ring Q-homeomorphism with respect to a p-modulus at any x0 ∈ D,
n− 1 < p < n. Then f is finitely Lipschitzian on D.
The similar results for homeomorphisms and mappings with branching were earlier
obtained in Rn, n ≥ 2, see [7]. The Lipschitzian continuity for mappings in Rn, n ≥ 2,
with a uniformly bounded function Q has been established by Gehring [8]. The same
condition for Riemannian manifolds was proved in [9].
2. Definitions and preliminary results.
Recall some needed definitions. By domain in a topological space T we mean an open
linearly connected set. The domain D is called locally connected at a point x0 ∈ ∂D, if
for any neighborhood U of x0 there is a neighborhood V ⊆ U of x0 such that V ∩D is
connected, cf. [10, c. 232]. Similarly, we say that a domain D is locally linearly connected
13
O. S. Afanas’eva
at a point x0 ∈ ∂D, if for any neighborhood U of x0 there exists a neighborhood V ⊆ U
of x0 such that V ∩D is linearly connected. Recall that the n-dimensional topological
manifold Mn is a Hausdorff topological space with a countable base such that every
point has a neighborhood homeomorphic to Rn. The manifold of the class Cr with
r ≥ 1 is called smooth.
Let further D denote a domain in the Finsler space (Mn,Φ), n ≥ 2, and TMn =
∪ TxMn be a tangent bundle of (Mn,Φ) for all x ∈ Mn. By a Finsler manifold (Mn,Φ),
n ≥ 2, we mean a smooth manifold of class C∞ with defined Finsler structure Φ(x, ξ),
where Φ(x, ξ) : TMn → R+ is a function satisfying the following conditions:
1) Φ ∈ C∞(TMn \ {0});
2) Φ(x, aξ) = aΦ(x, ξ) holds for all a > 0 and Φ(x, ξ) > 0 holds for ξ ̸= 0;
3) the n× n Hessian matrix gij(x, ξ) = 1
2
∂2Φ2(x,ξ)
∂ξi∂ξj
is positive defined at every point
of TMn \ {0}, cf. [4].
By the geodesic distance dΦ(x, y) we mean the infimum of lengths of piecewise-
smooth curves joining x and y in (Mn,Φ), n ≥ 2. It is well-known that for such metric
only two axioms of metric spaces hold, namely identity and triangle inequality axioms.
Therefore, the Finsler manifold provides a quasimetric space for which symmetry axiom
fails (see, e.g. [11]).
Remark 1. Consider a Finsler structure of the type
Φ̃(x, ξ) =
1
2
(Φ(x, ξ) + Φ(x,−ξ)). (2)
In this case we obtain a Finsler manifold (Mn, Φ̃) with symmetrized (reversible) function
Φ̃. Clearly, if Φ̃ is reversible, then the induced distance function d
Φ̃
is reversible, i.e.,
d
Φ̃
(x, y) = d
Φ̃
(y, x), for all pairs of points x, y ∈ Mn. It is also known that the reversible
Finsler metric coincides with the Riemannian one, see, e.g., [11]. Therefore, in our
further discussion we can rely on the results of [12].
Let γ : [a, b] → Mn be a piecewise-smooth curve and x(t) be its parametrization. An
element of length in (Mn, Φ̃), n ≥ 2, we define as a differential of path for infinitesimal
measured part of a curve γ ∈ D by ds2
Φ̃
=
n∑
i,j=1
gij(x, ξ)dηidηj ; see, e.g. [13]. So, the
distance ds
Φ̃
in the Finsler space, as in the case of a Riemannian space, is determined
by a metric tensor. Using the quadratic form ds
Φ̃
, we determine the length of γ ⊂ D
by s
Φ̃
(γ) =
∫
γ
ds
Φ̃
=
t2∫
t1
Φ̃(x, dx)dt, see, e.g. [11]. The invariance of this integral requires
the restrictions 2)-3), given above, on the Lagrangian Φ̃(x, dx).
In the Finsler geometry there are various definitions for the volume: by Holmes-
Thompson, Loewner, Busemann and others. In this paper we agree with the volume
definition by Busemann (Busemann-Hausdorff). Following [14], an element of volume
on the Finsler manifold is defined by dσΦ(x) =
|Bn|
|Bnx |
dx1...dxn, where |Bn| denotes the
Euclidean volume of the unit n-ball, whereas |Bn
x | is the Euclidean volume of the set
14
Finite Lipschitz mappings on Finsler manifolds
Bn
x =
{
(ξ1, ..., ξn) ∈ Rn : Φ
(
x,
n∑
1
(ξi, ei(x))
)
< 1
}
with an arbitrary basis {ei(x)}ni=1
in Rn depending on x. It is known that the volume in the Finsler space coincides with
its Hausdorff measure induced by metric dΦ(x, y), if Φ(x, ξ) is an invertible function,
see, e.g. [14]. In view of Remark 1, we have dσ
Φ̃
(x) =
√
det gij(x, ξ) dx
1...dxn, cf. [15].
Let Γ be a family of curves in a domain D. By the family of curves Γ we mean a
fixed set of curves γ, and for arbitrary mapping f : Mn → Mn
∗ , f(Γ) := {f ◦ γ|γ ∈ Γ}.
The p-modulus of the family Γ, p ∈ (1,∞), is defined by
Mp(Γ) = inf
∫
Mn
ρp(x) dσ
Φ̃
(x) , (3)
where the infimum is taken over all nonnegative Borel functions ρ such that the
condition
∫
γ
ρΦ̃(x, dx) =
∫
γ
ρds
Φ̃
≥ 1 holds for any curve γ ∈ Γ. The functions ρ,
satisfying this condition, are called admissible for Γ, cf. [4].
The quantity (3) can be interpreted as an outer measure in the space of curves.
For sets A,B and C from (Mn, Φ̃), n ≥ 2, by ∆(A,B;C) we denote a set of all
curves γ : [a, b] → Mn, which join A and B in C, i.e. γ(a) ∈ A, γ(b) ∈ B and γ(t) ∈ C
for all t ∈ (a, b).
Remark 2. One can apply the following well-known facts: Proposition 1 and
Remark 1 in [12] (due to Remark 1), and thus assume that the geodesic spheres S(x0, r),
geodesic balls B(x0, r) and geodesic rings A = A(x0, r1, r2) lie in a normal neighborhood
of a point x0.
Let D and D′ be domains in (Mn, Φ̃) and (Mn
∗ , Φ̃∗), n ≥ 2, respectively, and
Q : Mn → (0,∞) be a measurable function, p ∈ (1,∞), x0 ∈ D. We say that a
homeomorphism f : D → D′ is a ring Q-homeomorphism with respect to a p-modulus
at the point x0 if the inequality
Mp
(
∆(f(Sε), f(Sε0);D
′)
)
≤
∫
A
Q(x) · ηp
(
d
Φ̃
(x, x0)
)
dσ
Φ̃
(x) (4)
holds for every geodesic ring A = A(x0, ε, ε0), 0 < ε < ε0 < d0 = dist(x0, ∂D),
and for every measurable function η : (ε, ε0) → [0,∞], such that
ε0∫
ε
η(r)dr ≥ 1. Here
Sε = S(x0, ε), Sε0 = S(x0, ε0). We also say that f is a ring Q-homeomorphism with
respect to a p-modulus in the domain D if f is a ring Q-homeomorphism at every point
x0 ∈ D.
Let us recall that the idea to introduce the ring Q-homeomorphisms goes back to
Gehring’s ring definition of quasiconformality in Rn, n = 3, see [16]. These homeo-
morphisms first appeared in the plane for study of the Beltrami equations (see, e.g.
[17]), and later in Rn, n ≥ 2, cf. [18]. Further, the notion of ring homeomorphisms was
extended to boundary points of domains in the plane [19] and then in the space [20]. It
15
O. S. Afanas’eva
is well known that the theory of boundary behavior is one of the difficult and interesting
parts of the mapping theory; see the monographs [19, 6] and references therein. Note
also that the ring Q-homeomorphisms have rich applications in the theory of boundary
behavior of Sobolev and Orlic–Sobolev classes of mappings on Riemannian manifolds;
see [21]. The notion of ring Q-homeomorphisms at boundary points with respect to
p-modulus for p = 2 was introduced and applied for study the Beltrami equations
with a degenerate condition of strong ellipticity in [22]. Later a criterium for arbitrary
homeomorphisms to be ring Q-homeomorphisms with respect to p-modulus, p ̸= n, at
interior points of domains in the n-dimensional Euclidean space Rn was established in
[23].
3. p-capacities and Finsler manifolds.
By a condenser we mean a pair E = (A,G), where A ⊂ Mn is open and G ⊂ Mn is
a non-empty compact set contained in A. We shall say that E is a ringlike condenser if
B = A\G is a geodesic ring, i.e. B is a domain whose complement D\B has exactly two
components. We shall say that E is a bounded condenser if A is bounded. A condenser
E = (A,G) lies in a domain D if A ⊂ D.
Each condenser has p-capacity (where p ≥ 1) defined by the equality
capp E = capp (A,G) = inf
u
∫
A\G
|∇u|p dσ
Φ̃
(x), (5)
where the infimum is taken over all Lipschitz functions u with compact support in A.
In the local coordinates, the gradient at a point x ∈ Mn is defined by (∇f)i = gij ∂f
∂xj
,
1 ≤ i ≤ n, where the matrix gij is the inverse matrix of the matrix gij ; see [24].
Recall that in Rn, n ≥ 2, for 1 < p < n,
capp E ≥ nΩ
p
n
n
(
n− p
p− 1
)p−1
[m(G)]
n−p
n , (6)
see, e.g. (8.9) in [25]. Finally, for n− 1 < p ≤ n in Rn, the following lower bound
(
capp E
)n−1 ≥ γ
d(G)p
m(A)1−n+p
, (7)
where d(G) is the diameter of the compact set G and γ is a positive constant depending
only on n and p (see Proposition 6 in [26]) holds.
4. Proof of Theorem 1.
It suffices to show the following. Let Q : D → [0,∞] be a locally integrable function
and f : D → D′ be a ring Q-homeomorphism with respect to a p-modulus (n − 1 <
p < n) at an arbitrary point x0 ∈ D satisfying
Q0 = lim sup
ε→0
1
σ
Φ̃
(B(x0, ε))
∫
B(x0,ε)
Q(x) dσ
Φ̃
(x) <∞.
16
Finite Lipschitz mappings on Finsler manifolds
We show that
L(x0, f) = lim sup
x→x0
d∗
Φ̃
(f(x0), f(x))
d
Φ̃
(x0, x)
≤ λn,pQ
1
n−p
0 ,
where λn,p is a positive constant depending only on n and p.
Consider a geodesic ring A = A(x0, ε1, ε2) ⊂ D with 0 < ε1 < ε2 such that
A(x0, ε1, ε2) lies in a normal neighborhood at x0 (see Remark 2). Of course, if f : D →
D′ is open and E = (A,G) is a condenser in D, then f(E) = (f(A), f(G)) is also
condenser in D′, see Lemma A.1 in [6] and [27]. Then
(
f(B (x0, ε2)), f(B (x0, ε1))
)
is
the ringlike condenser in D′, in view of Remark 1. Follow the Theorem 2 in [4] we have
capp (f(B(x0, ε2)), f(B(x0, ε1))) =Mp(△(∂f(B(x0, ε2)), ∂f(B(x0, ε1); f(A))).
This equality is invariant with respect to change of the local coordinates. Since f is a
homeomorphism, then
△ (∂f(B (x0, ε2)), ∂f(B (x0, ε1)); f(A)) = f((△ (∂B(x0, ε2)), ∂B(x0, ε1);A)) .
Letting
η(t) =
{ 1
ε2−ε1 , t ∈ (ε1, ε2),
0, t ∈ R \ (ε1, ε2),
and applying the definition of ring Q-homeomorphisms with respect to p-module, we
obtain
capp (f(B(x0, ε2)), f(B(x0, ε1))) ≤
1
(ε2 − ε1)p
∫
A(x0,ε1,ε2)
Q(x) dσ
Φ̃
(x). (8)
Choose ε1 = 2ε and ε2 = 4ε, then
capp (f(B(x0, 4ε)), f(B(x0, 2ε))) ≤
1
(2ε)p
∫
B(x0,4ε)
Q(x) dσ
Φ̃
(x). (9)
Due to Remark 1 (see also proposition 5.11 (d) [28]), inequality (6) holds in sufficiently
small neighborhoods of the point x0 with respect to the normal coordinates, i.e.
capp (f(B(x0, 4ε)), f(B(x0, 2ε))) ≥ Cn,p
[
σ
Φ̃
(fB(x0, 2ε))
]n−p
n , (10)
where Cn,p is a positive constant depending only on n and p. Combining (9) and (10)
and taking into account the local n-regularity of measures (see Lemma 2.1 in [1]), we
obtain
σ
Φ̃
(f(B(x0, 2ε)))
σ
Φ̃
(B(x0, 2ε))
≤ cn,p
[
1
σ
Φ̃
(B(x0, 4ε))
∫
B(x0,4ε)
Q(x) dσ
Φ̃
(x)
] n
n−p
, (11)
where cn,p is a positive constant depending only on n and p.
17
O. S. Afanas’eva
Now choosing in (8), ε1 = ε and ε2 = 2ε, we have
capp (f(B(x0, 2ε)), f(B(x0, ε))) ≤
1
εp
∫
B(x0,2ε)
Q(x) dσ
Φ̃
(x) . (12)
Arguing similar to above, one gets from (7) the following lower bound
(
capp (f(B(x0, 2ε)), f(B(x0, ε)))
)n−1
≥ C̃n,p
dp
Φ̃
(f(B(x0, ε)))
σ1−n+p
Φ̃
(f(B(x0, 2ε)))
, (13)
where C̃n,p is a positive constant that depends only on n and p. Combining (12) and
(13) and taking again into account the Lemma 2.1 in [1], we obtain
d∗
Φ̃
(f(B(x0, ε)))
ε
≤ γn,p
(
σ
Φ̃
(f(B(x0, 2ε)))
σ
Φ̃
(B(x0, 2ε))
) 1−n+p
p
×
×
1
σ
Φ̃
(B(x0, 2ε))
∫
B(x0,2ε)
Q(x)dσ
Φ̃
(x)
n−1
p
, (14)
where γn,p is a positive constant depending only on n and p. The estimates (11) and
(14) imply
d∗
Φ̃
(f(B(x0, ε)))
ε
≤ λn,p
(
1
σ
Φ̃
(B(x0, 4ε))
∫
B(x0,4ε)
Q(x) dσ
Φ̃
(x)
)n(1−n+p)
p(n−p)
×
×
[
1
σ
Φ̃
(B(x0, 2ε))
∫
B(x0,2ε)
Q(x) dσ
Φ̃
(x)
]n−1
p
.
Letting ε→ 0, we obtain the desired estimate
L(x0, f) = lim sup
x→x0
d∗
Φ̃
(f(x0), f(x))
d
Φ̃
(x0, x)
≤ lim sup
ε→0
d∗
Φ̃
(f(B(x0, ε)))
ε
≤ λn,pQ
1
n−p
0
with a positive constant λn,p depending on n and p.
Since x0 was chosen arbitrary, the proof of Theorem 1 is completed.
Corollary 1. Let D and D′ be domains in (Mn, Φ̃) and (Mn
∗ , Φ̃∗), n ≥ 2, respectively,
and f : D → D′ be a ring Q-homeomorphism with respect to a p-modulus, n−1 < p < n.
Assume that Q(x) is bounded almost everywhere (a.e.) in D by a positive constant K.
Then f is locally Lipschitzian and, moreover,
L(x0, f) ≤ λn,pK
1
n−p ,
18
Finite Lipschitz mappings on Finsler manifolds
where λn,p is a constant depending only on n and p.
Remark 4. Condition (1) in Theorem 1 is sufficient. However, it cannot be omitted.
Here we refer to an example of homeomorphism in Rn [7] which does not satisfy (1)
and fails to be finitely Lipschizian.
Remark 5. Finitely Lipschitz mappings possess the property of the absolute con-
tinuity on surfaces of any dimension (see, e.g. [6]).
1. Afanas’eva E. S. The boundary behavior of Q-homeomorphisms on the Finsler spaces // Ukr.
Mat. Vis. – 2015. – V. 12, no. 3. – P. 311–325; transl. in J. Math. Sci. – 2016. – V. 214, no. 2. –
P. 161–171.
2. Bidabad B., Hedayatian S. Capacity on Finsler Spaces // Iranian journal of science and technology
transaction A-science – 2008. – V. 32, N A1. – P. 17–24.
3. Borcea V. T., Neagu A. p-modulus and p-capacity in a Finsler space // Math. Report – 2000. –
52. – P. 431–439.
4. Dymchenko Yu. V. Equality of the capacity and modulus of a condenser in Finsler spaces // Mat.
Zametki. – 2009. – V. 85, no. 4. – P. 594–602; transl. in Math. Notes. – 2009. – V. 85, no. 3–4. –
P. 566–573.
5. Garrido M. I., Jaramillo J. A., Rangel Y. C. Smooth Approximation of Lipschitz Functions on
Finsler Manifolds // Journal of Function Spaces and Applications V. 2013. – 2013. – 10 pp.
6. Martio O., Ryazanov V., Srebro U., Yakubov E. Moduli in Modern Mapping Theory. — Springer,
New York, 2009.
7. Salimov R. On Finitely Lipschitz space // Sib. Electr. Math. Rep. – 2011. – V. 8. – P. 284–295.
8. Gehring F. W. Lipschitz mappings and the p-capacity of ring in n-space // Advances in the theory
of Riemann surfaces. – Proc. Conf. Stony Brook, N.Y., 1969. – P. 175–193; Ann. of Math. Studies.
– 1971. – V. 66.
9. Nakai M. Existence of quasiisometric mappings and royden compactifications // Ann. Acad. Sci.
Fenn., Ser. AI, Math. – 2000. – V. 25, no. 1. – P. 239–260.
10. Kuratowski K. Topology. Vol. II. – Academic Press, New York-London, 1968.
11. Bao D., Chern S., Shen Z. An Introduction to Riemann-Finsler Geometry. – Graduate Texts in
Mathematics, 200. Springer-Verlag, New York, 2000.
12. Afanas’eva E. S. Boundary behavior of ring Q-homeomorphisms on Riemannian manifolds //
Ukr. Math. J. – 2011. – V. 63, no. 10, P. 1–15; transl. in J. Math. Sci. – 2012. – V. 63, no. 10. –
P. 1479–1493.
13. Rutz S. F., Paiva F. M. Gravity in Finsler spaces // Finslerian geometries. – Edmonton, AB,
1998. – P. 223–244; Fund. Theories Phys., 109, Kluwer Acad. Publ., Dordrecht, 2000.
14. Shen Z. Lectures on Finsler geometry. – World Scientific Publishing Co., Singapore, 2001.
15. Rund H. The differential geometry of Finsler spaces. – Die Grundlehren der Mathematischen
Wissenschaften, Bd. 101 Springer-Verlag, Berlin-Gцttingen-Heidelberg, 1959.
16. Gehring F. W. Rings and quasiconformal mappings in space // Trans. Amer. Math. Soc. – 1962.
– V. 103. – P. 353–393.
17. Ryazanov V., Srebro U., Yakubov E. On ring solutions of Beltrami equations // J. Anal. Math. –
2005. – V. 96. – P. 117–150.
18. Ryazanov V., Sevost’yanov E. Equicontinuous classes of ring Q-homeomorphisms // Sibirsk. Mat.
Zh. – 2007. – V. 48, no. 6. – P. 1361–1376; transl. in Siberian Math. J. – 2007. – V. 48, no. 6. –
P. 1093–1105.
19. Gutlyanskii V., Ryazanov V., Srebro U. and Yakubov E. The Beltrami equation. A geometric
approach. – Developments in Mathematics, 26. Springer, New York, 2012.
20. Golberg A. Differential properties of (a,Q)-homeomorphisms // Further progress in analysis. –
World Sci. Publ., Hackensack, NJ, 2009. – P. 218–228.
21. Afanas’eva E. S., Ryazanov V. I. and Salimov R. R. On mappings in Orlicz-Sobolev classes on
19
O. S. Afanas’eva
Riemannian manifolds // Ukr. Mat. Visn. – 2011. – V. 8, no. 3. – P. 319–342, 461; transl. in J.
Math. Sci. – 2012. – V. 181, no. 1. – P. 1–17.
22. Ryazanov V., Srebro U. and Yakubov E. On strong solutions of the Beltrami equations // Complex
Var. Elliptic Equ. – 2010. – V. 55, no. 1–3. – P. 219–236.
23. SalimovR. R. Estimation of the measure of the image of the ball // Sibirsk. Mat. Zh. – 2012. –
V. 53, no. 4. – P. 920–930; transl. in Sib. Math. J. – 2012. – V. 53, no. 4. – P. 739–747.
24. Grigor’yan A. Heat Kernel and Analysis on Manifolds // AMS/IP Studies in Advanced Mathe-
matics 47. – Amer. Math. Soc., Providence, RI, 2009.
25. Maz’ya V. Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces
// Contemp. Math. – 2003. – V. 338. – P. 307–340.
26. Kruglikov V. I. Capacity of condensers and spatial mappings quasiconformal in the mean //
Math. USSR Sb. – 1987. – V. 58, no. 1. – P. 185–205.
27. Martio O., Rickman S., Väisälä J. Definitions for quasiregular mappings // Ann. Acad. Sci.
Fenn. Ser. A1. Math. – 1969. – V. 448, no. 40. – P. 1–40.
28. Lee J. M. Riemannian Manifolds: An Introduction to Curvature. – New York, Springer, 1997. –
224 pp.
Е. С. Афанасьева
Конечно липшицевы отображения на финслеровых многообразиях.
Рассматриваются кольцевые Q-гомеоморфизмы относительно p-модуля на финслеровых много-
образиях, n − 1 < p < n, устанавливаются достаточные условия конечной липшицевости этих
отображений.
Ключевые слова: Финслеровы многообразия, кольцевые Q-гомеоморфизмы, p-модули, конечно
липшицевы отображения.
О. С. Афанасьєва
Кiнцево лiпшицевi вiдображення на фiнслерових многовидах.
Розглядаються кiльцевi Q-гомеоморфiзми вiдносно p-модуля на фiнслерових многовидах, n−1 <
p < n, та встановлюються достатнi умови кiнцевої лiпшицевостi таких вiдображень.
Ключовi слова: Фiнслеровi многовиди, кiльцевi Q-гомеоморфiзми, p-модулi, кiнцево лiпшицевi
вiдображення.
Ин-т прикл. математики и механики НАН Украины, Славянск
es.afanasjeva@yandex.ru
Received 25.10.16
20
|