Local behavior of mappings defined by their distributional derivatives
We study asymptotic behavior of spatial mappings at the origin. These mappings belong to the Orlic-Sobolev class and are defined by their Jacobimatrices. The theorem provides a result of a Schwarz Lemma type. The sharpness of conditions is illustrated by an example.
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Інститут математики НАН України
2015
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552691962478592 |
|---|---|
| author | Golberg, A. Salimov, R. Golberg, A. Salimov, R. |
| author_facet | Golberg, A. Salimov, R. Golberg, A. Salimov, R. |
| author_institution_txt_mv | [
{
"author": "A. Golberg",
"institution": null
},
{
"author": "R. Salimov",
"institution": null
}
] |
| author_sort | Golberg, A. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-23T12:10:05Z |
| description | We study asymptotic behavior of spatial mappings at the origin. These mappings belong to the Orlic-Sobolev class and are defined by their Jacobimatrices. The theorem provides a result of a Schwarz Lemma type. The sharpness of conditions is illustrated by an example. |
| first_indexed | 2026-08-04T01:03:28Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2015, том 12, № 3, 110–120
УДК 517.5
A. Golberg 1, R. Salimov 2
1 (Holon institute of Technology, Holon, Israel)
2 (Institute of Mathematics of NAS of Ukraine, Kyiv)
1 golberga@hit.ac.il, 2 ruslan623@yandex.ru
Local behavior of mappings defined by
their distributional derivatives
We study asymptotic behavior of spatial mappings at the origin. These
mappings belong to the Orlic-Sobolev class and are defined by their Jacobi
matrices. The theorem provides a result of a Schwarz Lemma type. The
sharpness of conditions is illustrated by an example.
1. Introductory remarks.
1.1. Let D be a domain in Rn, n ≥ 2, f = (f1, . . . , fn) : D −→ Rn be a
W 1,1
loc -mapping, and f ′(x) = (∂fi/∂xj) denote its Jacobi matrix at x ∈ D.
Denote by Mn the space of n× n matrices.
In this paper, we study the asymptotic behavior of local solutions
y = f(x) of the matrix differential equation
y′(x) = M(x) (1)
with given M(x) ∈ Mn (determined almost everywhere in D).
We shall use the both Hilbert-Schmidt and operator norms
∥M(x)∥ =
√√√√ n∑
i=1
n∑
j=1
m2
i,j(x), |M(x)| = sup
|h|=1
|M(x) · h|
of matrices. Each matrix M(x) determines the p-outer dilatation coefficient
(p ≥ 1) of solution f(x) by
Kp (M(x)) =
|M(x)|p
det(M(x)) , if detM(x) ̸= 0 ,
1 , if M(x) = 0 ,
∞ , otherwise .
(2)
c⃝ Institute of Mathematics, 2015
Local behavior ... 111
We consider homeomorphisms f ∈ W 1,1
loc (D,Rn) having finite
distortion, i.e. such that |M(x)|n/detM(x) < ∞ almost everywhere in
D. This means that f ′(x) = 0 for almost all points of the set Z = {x ∈
D : det f ′(x) = 0}; see [8], [6].
1.2. Consider for a given convex increasing function
φ : [0,∞) −→ [0,∞), φ(0) = 0, the corresponding Orlic space Lφ of
functions g : D −→ R, satisfying∫
D
φ
(
|g(x)|
λ
)
dm(x) < ∞
for some λ > 0 (cf., e.g. [10]), where m denotes the n-dimensional Lebesgue
measure in Rn.
The Orlic-Sobolev class W 1,φ
loc (D) is a collection of locally integrable
functions g in D with first distributional derivatives, whose gradient ∇g
belongs to the Orlic class locally in D. Note that W 1,φ
loc ⊂ W 1,1
loc and
g ∈ W 1,p
loc when φ(t) = tp, p ≥ 1.
For a locally integrable vector-function
f = (f1(x1, . . . , xn), . . . , fm(x1, . . . , xn))
with fi ∈ W 1,1
loc , we put∫
D
φ (∥∇f(x)∥) dm(x) < ∞ ,
where ∥∇f(x)∥ =
√
m∑
i=1
n∑
j=1
(
∂fi
∂xj
)2
, and f ∈ W 1,φ
loc . We shall also use the
notation W 1,φ
loc for more general functions φ, than for the Orlic classes,
assuming the convexity of φ with a normalization φ(0) = 0.
1.3. We say that a matrix function M : D −→ Mn (n ≥ 3) has the
Fφ-property if:
1) M(x) = 0 for almost all points of the set Z = {x ∈ D :
detM(x) = 0};
2) the function φ : [0,∞) −→ [0,∞) is monotone increasing and
satisfies a Calderon type condition
∞∫
1
[
t
φ(t)
] 1
n−2
dt < ∞ (cf. [1]);
112 A. Golberg, R. Salimov
3)
∫
D
φ(∥M(x)∥) dm(x) < ∞ .
In fact, the Fφ-property can be extended to the planar case (n = 2).
In this case it suffices to require from φ only the monotonicity property
(since all mappings from W 1,1
loc (R2) are differentiable almost everywhere).
2. Moduli of surface families and capacities of
condensers.
2.1. Let S be a k-dimensional surface in Rn, which means that
S : Ds −→ Rn is a continuous image of a open domain Ds ⊂ Rk :=
:= Rk ∪ {∞}. We denote by
N(S, y) = cardS−1(y) = card{x ∈ Ds : S(x) = y}
the multiplicity function of the surface S at the point y ∈ Rn. It is known
that the multiplicity function is semicontinuous from below and therefore
it is measurable with respect to the Hausdorff measure Hk (cf. [12]).
For a given Borel function ρ : Rn −→ [0,∞], the integral of ρ over S is
defined by ∫
S
ρ dA =
∫
Rn
ρ(y)N(S, y) dHky.
Let Sk be a family of k-dimensional surfaces S in Rn, 1 ≤ k ≤ n − 1
(curves for k = 1). The p-module of Sk is defined as
Mp(Sk) = inf
∫
Rn
ρp(x) dm(x), p ≥ k,
where the infimum is taken over all Borel measurable functions ρ ≥ 0 and
such that ∫
S
ρk dA ≥ 1
for every S ∈ Sk. We call each such ρ an admissible function for Sk ( ρ ∈
∈ admSk ).
Following [9], a metric ρ is said to be extensively admissible for Sk
(ρ ∈ extpadmSk) with respect to p-module if ρ ∈ adm (Sk\S̃k) such that
Mp(S̃k) = 0.
Accordingly, we say that a property P holds for almost every k-
dimensional surface, if P holds for all surfaces except a family of zero
α-module.
Local behavior ... 113
2.2. We also use especially another tool which is important in Potential
Theory and Mathematical Analysis.
Following in general [11], a pair E = (A,C), where A ⊂ Rn is an open
set and C ⊂ A is a nonempty compactum, is called the condenser. We
say that the condenser E is the ring condenser, if R = A \ C is a ring
domain, i.e. its complement consists of two components. The condenser
E is bounded, if A is bounded. We also say that a condenser E = (A,C)
lies in a domain G when A ⊂ G. Obviously, for an open and continuous
mapping f : G −→ Rn and for any condenser E = (A,C) ⊂ G, the pair
(f(A), f(C)) is a condenser in f(G). In this case we shall use the notation
f(E) = (f(A), f(C)).
Let E = (A,C) be a condenser. Denote by C0(A) the set of all continuous
functions u : A −→ R1 with compact support in A. Consider the set
W0(E) = W0(A,C) of all nonnegative functions u : A −→ R1 such that
1) u ∈ C0(A), 2) u(x) ≥ 1 for x ∈ C and 3) u belongs ACL. Put
capp E = capp (A,C) = inf
u∈W0(E)
∫
A
|∇u|p dx, p ≥ 1,
where, as usual
|∇u| =
(
n∑
i=1
(∂iu)
2
)1/2
.
This quantity is called p-capacity of condenser E .
It was proven in [7] that for p > 1
capp E = Mp(∆(∂A, ∂C;A \ C)), (3)
where ∆(∂A, ∂C;A\C)) denotes the set of all continuous curves which join
the boundaries ∂A and ∂C in A\C (cf. [2,14]). For general properties of p-
capacities and their relation to the mapping theory, we refer, for instance,
to [5] and [13]. In particular, for 1 < p < n,
capp E ≥ nΩ
p
n
n
(
n− p
p− 1
)p−1
[m(C)]
n−p
n , (4)
where Ωn denotes the volume of the unit ball in Rn, and mC is the n-
dimensional Lebesgue measure of C.
114 A. Golberg, R. Salimov
3. Lower Q-homeomorphisms and auxiliary results.
We shall use the notations
B(x0, r) = {x ∈ Rn : |x− x0| < r} , Br = B(0, r), Bn = B(0, 1),
S(x0, r) = {x ∈ Rn : |x− x0| = r} , Sr = S(0, r).
Given a Lebesgue measurable function Q : G → [0,∞], we define for
the measurable sets E ⊂ Rn set
−
∫
E
Q(x) dm(x) =
1
m(E)
∫
E
Q(x) dm(x) .
Suppose that D and D′ are two domains in Rn, n ≥ 2, x0 ∈ D, and
Q : D −→ (0,∞) is a Lebesgue measurable function. A homeomorphism
f : D −→ D′ is called lower Q-homeomorphism with respect to p-module
at x0, if the following bound holds
Mp (f(ΣR)) ≥ inf
ρ∈extpadmΣR
∫
R
ρp(x)
Q(x)
dm(x)
for each ring
R = R(x0, ε1, ε2) = {x ∈ Rn : ε1 < |x− x0| < ε2} , 0 < ε1 6 ε2 < d0,
where d0 = dist(x0, ∂D) , and ΣR denotes the family of spheres S(x0, r),
r ∈ (ε1, ε2) .
The following statement is given in [3] and provides the necessary and
sufficient condition for homeomorphisms to be lower Q-homeomorphisms.
Lemma 1. Let D be a domain in Rn, n ≥ 2, x0 ∈ D and
Q : D −→ (0,∞) be a measurable function. A homeomorphism
f : D −→ Rn is lower Q-homeomorphism at x0 with respect to p-module,
p > n− 1, if and only if the following inequality holds
Mp(f(ΣR)) ≥
r2∫
r1
dr( ∫
S(x0,r)
Q
n−1
p−n+1 (x)dA
) p−n+1
n−1
, ∀ 0 < ε1 < ε2 < d0,
Local behavior ... 115
where d0 = dist(x0, ∂D) , ΣR is the family of spheres S(x0, r) = {x ∈ Rn :
|x− x0| = r} with r ∈ (ε1, ε2).
In the following lemma we establish the necessary condition
characterizing the lower Q-homeomorphisms and obtain an upper bound
for α-module of the family of curves, α = p/(p− n+ 1).
Lemma 2. Let D be a domain in Rn, n ≥ 2, x0 ∈ D, and
Q : D −→ (0,∞) be measurable function. Suppose that f : D −→ Rn is
a lower Q-homeomorphism at x0 with respect to p-module with p > n− 1.
Then
M p
p−n+1
(Γ∗) ≤
r2∫
r1
dr( ∫
S(x0,r)
Q
n−1
p−n+1 (x)dA
) p−n+1
n−1
− n−1
p−n+1
,
where Γ∗ = ∆(f(S1), f(S2), f(D)) is the family of all curves connection
Sj = S(x0, rj), j = 1, 2, in f(D).
Proof. Pick arbitrary spheres Si = S(x0, ri), i = 1, 2, such that
0 < r1 < r2 < d(x0, ∂D). Then due to the relations of moduli and
capacities by Hesse [7] and Ziemer [15], we have
M p
p−n+1
(f (∆(S1, S2, D))) ≤ 1
M
n−1
p−n+1
p (f (ΣR))
, (5)
because f (ΣR) ⊂ Σ(f(S1), f(S2), f(D)) . Here ΣR denotes a collection
of all spheres centered at x0, which lie between S1 and S2;
Σ (f(S1), f(S2), f(D)) is the family of all (n − 1)-dimensional surfaces in
f(D), that separate f(S1) and f(S2). Now the assertion of the lemma
follows from the inequality (5) and Lemma 1.
The following statement is crucial in the proof of the main result and
follows from Theorem 5 of [4] obtained for open discrete mappings.
Lemma 3. Let D and D′ be domains in Rn, n ≥ 3. Assume that
M : D −→ Mn possesses Fφ-property and f : D −→ D′ is a homemorphic
solution of the equation (1). Then f : D −→ D′ is lower Q-homeomorphism
with respect to p-module with Q(x) = Kp (M(x)) and p > n− 1.
116 A. Golberg, R. Salimov
4. Main result.
Our main result is the following theorem which implies an upper
estimate for stretching of mappings reconstructed by the Jacobi matrix
at the origin.
Theorem. Let a matrix M : D −→ Mn be defined in the unit ball
Bn and possess Fφ-property. Assume that f : Bn −→ Bn, n ≥ 3, is a
homeomorphic solution of the equation (1) in Bn normalized by f(0) = 0.
If for p ∈ (n,+∞)
kp = lim inf
ε→0
(
−
∫
Bε
Kα
p (M(x)) dm(x)
) 1
α
< ∞ , α =
n− 1
p− n+ 1
, (6)
then
lim inf
x→0
|f(x)|
|x|
≤ ν0 · k
1
p−n
p < ∞ , (7)
where ν0 is a positive constant depending only on n and p.
Proof. Consider a spherical ring R = R(0, ε, 2ε) with 0 < ε < 1
2 . Then
E =
(
B2ε, Bε
)
and f(E) =
(
f(B2ε), f(Bε)
)
are the ring condensers in Bn.
Consider the curve family Γ∗
ε = ∆(f(Sε), f(S2ε), f(R)). Then from (3),
capq f(E) = Mq (Γ
∗
ε) ,
with q = p/(p− n+ 1).
By Lemmas 2 and 3, one gets
capq f(E) ≤
2ε∫
ε
dr(∫
Sr
Kβ
p (M(x)) dA
)1/β
−β
, (8)
where β = (n− 1)/(p− n+ 1).
Noting that
ε =
2ε∫
ε
∫
Sr
Kβ
p (M(x)) dA
1/q
dr(∫
Sr
Kβ
p (M(x)) dA
)1/q
Local behavior ... 117
and applying the Fubini theorem and the Hölder inequality with the
exponents q = p
p−n+1 , q′ = p
n−1 , one obtains
2ε∫
ε
dr(∫
Sr
Kβ
p (M(x)) dA
)1/β
−β
≤ 1
εq
∫
R
Kβ
p (M(x)) dm(x) . (9)
Combining (9) and (8) yields
capq f(E) ≤ 1
εq
∫
R
Kβ
p (M(x)) dm(x) , (10)
where q = p
p−n+1 .
On the other hand, due to the inequality (4), we have
capq f(E) ≥ c1 [m(f(Bε))]
n−q
n , (11)
with the same q = p
p−n+1 and a positive constant c1 depending only on n
and p.
Comparing (10) and (11), one obtains the following upper estimate
m(f(Bε))
Ωnεn
≤ c2
(
−
∫
B2ε
Kβ
p (M(x)) dm(x)
) n
n−q
. (12)
Here c2 > 0 also depends only n and p.
Denote lf (ε) = min
|x|=ε
|f(x)|. Since f(0) = 0, Ωn l
n
f (ε) ≤ m(f(Bε)) or
equivalently,
lf (ε) ≤
(
m(f(Bε))
Ωn
) 1
n
,
one gets
lim inf
x→0
|f(x)|
|x|
= lim inf
ε→0
lf (ε)
ε
≤ lim inf
ε→0
(
m(f(Bε))
Ωnεn
) 1
n
,
and, combining with (12),
lim inf
x→0
|f(x)|
|x|
≤ c0 lim inf
ε→0
(
−
∫
B2ε
Kβ
p (M(x)) dm(x)
) 1
n−q
= c0 k
p−n+1
(n−1)(p−n)
p
118 A. Golberg, R. Salimov
with q = p
p−n+1 and a positive constant c0 which depends only on n and
p. The proof is completed.
Now we show that the sufficient condition (6) of the theorem can not
be dropped. The following example shows that in the case of infinite lower
limit in (6), the limit in (7) controlling the asymptotic behavior can be also
infinite. Indeed, consider the homeomorphic automorphism of the unit ball
Bn defined by
f(x) =
x
|x|
1 + (p− n)
1∫
|x|
dt
tp−n+1 ln
p−n+1
n−1 ( et )
− 1
p−n
for any |x| < 1, x ̸= 0 and f(0) = 0 for any fixed p in the interval (n,∞) .
Because of the radial symmetry of the mapping, one can rewrite it via
f(x) =
x
|x|
φ(|x|), x ̸= 0, and f(0) = 0,
with
φ(|x|) =
1 + (p− n)
1∫
|x|
dt
tp−n+1 ln
p−n+1
n−1 ( et )
− 1
p−n
.
Note that φ(|x|) → 0 as x → 0, and φ(|x|) → 1 as |x| → 1. In addition, we
can restrict ourselves to the case when x = (r, 0, 0, . . . , 0), r ∈ (0, 1). Then
the Jacobi matrix of f is of the form
f ′(x) =
φ′(r) 0 0 . . . 0
0 φ(r)
r 0 . . . 0
0 0 φ(r)
r . . . 0
...
...
...
. . .
...
0 0 0 . . . φ(r)
r
,
at any x ∈ Bn, x ̸= 0. A direct computation yields(
φ(r)
r
)p−n+1
= φ′(r) ln
p−n+1
n−1
(e
r
)
,
Local behavior ... 119
and since for 0 < r < 1, φ(r)/r > φ′(r), the p-outer dilatation coefficient
Kp(M(x)) defined by (2) assumes the form
Kp(M(x)) =
|M(x)|p
detM(x)
=
(
φ(r)
r
)p
(
φ(r)
r
)n−1
φ′(r)
= ln
p−n+1
n−1 (
e
|x|
) .
For this mapping,
lim inf
ε→0
−
∫
B(0,ε)
(Kp(M(x)))
n−1
p−n+1 dm(x) = ∞ , (13)
hence the condition (6) does not hold.
On the other hand, L’Hospital’s rule yields |f(x)|
|x| → ∞ as x → 0.
References
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|
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English Ukrainian |
| last_indexed | 2026-08-04T01:03:28Z |
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| spelling | oai:trim.imath.kiev.ua:article-1712018-01-23T12:10:05Z Local behavior of mappings defined by their distributional derivatives Golberg, A. Salimov, R. Golberg, A. Salimov, R. We study asymptotic behavior of spatial mappings at the origin. These mappings belong to the Orlic-Sobolev class and are defined by their Jacobimatrices. The theorem provides a result of a Schwarz Lemma type. The sharpness of conditions is illustrated by an example. Інститут математики НАН України 2015-04-23 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/171 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 3 (2015): Analysis and Applications; 110–120 Сборник Трудов Института математики НАН Украины; Том 12 № 3 (2015): Анализ и приложения; 110–120 Збірник Праць Інституту математики НАН України; Том 12 № 3 (2015): Аналіз та застосування; 110–120 3083-7529 1815-2910 en uk https://trim.imath.kiev.ua/index.php/trim/article/view/171/139 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Golberg, A. Salimov, R. Golberg, A. Salimov, R. Local behavior of mappings defined by their distributional derivatives |
| title | Local behavior of mappings defined by their distributional derivatives |
| title_full | Local behavior of mappings defined by their distributional derivatives |
| title_fullStr | Local behavior of mappings defined by their distributional derivatives |
| title_full_unstemmed | Local behavior of mappings defined by their distributional derivatives |
| title_short | Local behavior of mappings defined by their distributional derivatives |
| title_sort | local behavior of mappings defined by their distributional derivatives |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/171 |
| work_keys_str_mv | AT golberga localbehaviorofmappingsdefinedbytheirdistributionalderivatives AT salimovr localbehaviorofmappingsdefinedbytheirdistributionalderivatives AT golberga localbehaviorofmappingsdefinedbytheirdistributionalderivatives AT salimovr localbehaviorofmappingsdefinedbytheirdistributionalderivatives |