General Beltrami equations and BMO
We study the Beltrami equations ∂f = μ(z)∂f + ν(z)∂f under the assumption that the coefficients μ, ν satisfy the inequality |μ| + |ν| < 1 almost everywhere. Sufficient conditions for the existence of homeomorphic ACL solutions to the Beltrami equations are given in terms of the bounded mean oscil...
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1243432017-09-24T03:03:11Z General Beltrami equations and BMO Bojarski, B.V. Gutlyanskii, V.V. Ryazanov, V.I. We study the Beltrami equations ∂f = μ(z)∂f + ν(z)∂f under the assumption that the coefficients μ, ν satisfy the inequality |μ| + |ν| < 1 almost everywhere. Sufficient conditions for the existence of homeomorphic ACL solutions to the Beltrami equations are given in terms of the bounded mean oscillation by John and Nirenberg. 2008 Article General Beltrami equations and BMO / B.V. Bojarski, V.V. Gutlyanskii, V.I. Ryazanov // Український математичний вісник. — 2008. — Т. 5, № 3. — С. 305-326. — Бібліогр.: 38 назв. — англ. 1810-3200 2000 MSC. 30C65, 30C75. http://dspace.nbuv.gov.ua/handle/123456789/124343 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We study the Beltrami equations ∂f = μ(z)∂f + ν(z)∂f under the assumption that the coefficients μ, ν satisfy the inequality |μ| + |ν| < 1 almost everywhere. Sufficient conditions for the existence of homeomorphic ACL solutions to the Beltrami equations are given in terms of the bounded mean oscillation by John and Nirenberg. |
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Bojarski, B.V. Gutlyanskii, V.V. Ryazanov, V.I. |
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Bojarski, B.V. Gutlyanskii, V.V. Ryazanov, V.I. General Beltrami equations and BMO Український математичний вісник |
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Bojarski, B.V. Gutlyanskii, V.V. Ryazanov, V.I. |
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Bojarski, B.V. |
| title |
General Beltrami equations and BMO |
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General Beltrami equations and BMO |
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General Beltrami equations and BMO |
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General Beltrami equations and BMO |
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General Beltrami equations and BMO |
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general beltrami equations and bmo |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/124343 |
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General Beltrami equations and BMO / B.V. Bojarski, V.V. Gutlyanskii, V.I. Ryazanov // Український математичний вісник. — 2008. — Т. 5, № 3. — С. 305-326. — Бібліогр.: 38 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT bojarskibv generalbeltramiequationsandbmo AT gutlyanskiivv generalbeltramiequationsandbmo AT ryazanovvi generalbeltramiequationsandbmo |
| first_indexed |
2025-07-09T01:17:38Z |
| last_indexed |
2025-07-09T01:17:38Z |
| _version_ |
1837130172683255808 |
| fulltext |
Український математичний вiсник
Том 5 (2008), № 3, 305 – 326
General Beltrami equations and BMO
Bogdan V. Bojarski, Vladimir V. Gutlyanskĭı,
Vladimir I. Ryazanov
Abstract. We study the Beltrami equations ∂f = µ(z)∂f + ν(z)∂f
under the assumption that the coefficients µ, ν satisfy the inequality
|µ| + |ν| < 1 almost everywhere. Sufficient conditions for the existence
of homeomorphic ACL solutions to the Beltrami equations are given in
terms of the bounded mean oscillation by John and Nirenberg.
2000 MSC. 30C65, 30C75.
Key words and phrases. Degenerate Beltrami Equations, quasicon-
formal mappings, bounded mean oscillation.
1. Introduction
Let D be a domain in the complex plane C. We study the Beltrami
equation
∂f = µ(z)∂f + ν(z)∂f a.e. in D (1.1)
where ∂f = (fx + ify)/2 and ∂f = (fx − ify)/2, z = x + iy, and µ
and ν are measurable functions in D with |µ(z)| + |ν(z)| < 1 almost
everywhere in D. Equation (1.1) arises, in particular, in the study of
conformal mappings between two domains equipped with different mea-
surable Riemannian structures, see [22]. Equation (1.1) and second order
PDE’s of divergent form are also closely related. For instance, given a do-
main D, let σ be the class of symmetric matrices with measurable entries,
satisfying
1
K(z)
|h|2 ≤ 〈σ(z)h, h〉 ≤ K(z)|h|2, h ∈ C.
Assume that u ∈ W 1,2
loc (D) is a week solution of the equation
div[σ(z)∇u(z)] = 0.
Received 25.09.2008
ISSN 1810 – 3200. c© Iнститут математики НАН України
306 General Beltrami equations...
Consider the mapping f = u + iv where ∇v(z) = Jf (z)σ(z)∇u(z) and
Jf (z) stands for the Jacobian determinant of f . It is easily to verify that
f satisfies the Beltrami equation (1.1). On the other hand, the above
second order partial differential equation naturally appears in a number
of problems of mathematical physics, see, e.g., [3].
In the case ν(z) ≡ 0 in (1.1) we recognize the classical Beltrami equa-
tion, which generates the quasiconformal mappings in the plane. Given
an arbitrary measurable coefficient µ(z) with ‖µ‖∞<1 in D ⊆ C, the well-
known measurable Riemann mapping theorem for the Beltrami equation
∂f = µ(z)∂f a.e. in D ⊆ C, (1.2)
see, e.g., [1, Chapter 5], [24, Chapter 5], guarantees the existence of a
homeomorphic solution f ∈ W 1,2
loc (D) to the equation (1.2), which maps
D onto an arbitrary conformally equivalent domain G. Moreover, the
mapping f can be represented in the form f = F◦ω, where F stands for an
arbitrary conformal mapping of D onto G and ω is a quasiconformal self-
mapping of D with complex dilatation µ(z) a.e. in D. The corresponding
measurable Riemann mapping theorem for the general Beltrami equation
(1.1) was given in [5] and [6], Theorem 5.1 and Theorem 6.8, see also [38,
Chapter 3, §17]. Bellow we give its statement in the form, convenient for
our application.
Theorem 1.1. Let B be the unit disk and G be a simply connected do-
main in C. If µ and ν are measurable functions in B with |µ(z)|+|ν(z)| ≤
q < 1 a.e. in B, then there exists a quasiconformal mapping f : B → G,
satisfying the equation (1.1). The mapping f has the representation
f = F ◦ ω where F stands for a conformal mapping of B onto G and
the quasiconformal self-mapping ω of B can be normalized by ω(0) = 0,
ω(1) = 1. If F (ω) ≡ ω, then the normalized solution is unique.
Notice, that if f is a W 1,2
loc solution to (1.1), then f is also a solution
to (1.2) where µ is replaced by µ̃ = µ + ν∂f/∂f if ∂f 6= 0 and µ̃ = 0 if
∂f = 0.
The case, when the assumption of strong ellipticity condition |µ(z)|+
|ν(z)| ≤ q < 1 is replaced by the assumption |µ(z)| + |ν(z)| < 1 a.e., the
similar existence and uniqueness problem was not studied so far. Let us
consider a couple of illustrative examples. Define the following Beltrami
coefficients
µ(z) =
( |z|2 − 1
3|z|2 + 1
+ |z|
)
· z
z
, ν(z) = −|z| · z
z
,
in the punctured unit disk 0 < |z| < 1. Since |µ(z)| → 1 as z → 0, we see
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 307
that the equation fz − µ(z) fz − ν(z)fz = 0 degenerates near the origin.
It is easy to verify that the radial stretching
f(z) = (1 + |z|2) z
|z| , 0 < |z| < 1.
satisfies the above equation and is a homeomorphic mapping of the punc-
tured unit disk onto the annulus 1 < |w| < 2. Thus, we observe the effect
of cavitation. For the second example we choose
µ(z) =
i
2
z
z
, ν(z) =
i
2
z
z
e2i log |z|2 .
In this case |µ(z)| + |ν(z)| = 1 holds for every z ∈ C. In other words,
we deal with “global” degeneration. However, the corresponding globally
degenerate general Beltrami equation (1.1) admits the spiral mapping
f(z) = zei log |z|2
as a quasiconformal solution. The above observation shows, that in order
to obtain existence or uniqueness results, some extra constraints must be
imposed on µ and ν.
In this paper we give sufficient conditions for the existence of a home-
omorphic ACL solution to the Beltrami equation (1.1), assuming that
the degeneration of µ and ν is is controlled by a BMO function. More
precisely we assume that the maximal dilatation function
Kµ,ν(z) =
1 + |µ(z)| + |ν(z)|
1 − |µ(z)| − |ν(z)| (1.3)
is dominated by a function Q(z) ∈ BMO, where BMO stands for the
class of functions with bounded mean oscillation in D, see [21].
Recall that, by John and Nirenberg in [21], a real-valued function u
in a domain D in C is said to be of bounded mean oscillation in D, u ∈
BMO(D), if u ∈ L1
loc(D), and
‖u‖∗ := sup
B
1
|B|
∫
B
|u(z) − uB| dx dy < ∞ (1.4)
where the supremum is taken over all discs B in D and
308 General Beltrami equations...
uB =
1
|B|
∫
B
u(z) dx dy.
We also write u ∈ BMO if D = C. If u ∈ BMO and c is a constant, then
u+ c ∈ BMO and ‖u‖∗ = ‖u+ c‖∗. The space of BMO functions modulo
constants with the norm given by (1.4) is a Banach space. Note that
L∞ ⊂ BMO ⊂ Lp
loc for all p ∈ [1,∞), see e.g. [21, 30]. Fefferman and
Stein [13] showed that BMO can be characterized as the dual space of
the Hardy space H1. The space BMO has become an important concept
in harmonic analysis, partial differential equations and related areas.
The case, when ν = 0 and the degeneration of µ is expressed in terms
of |µ(z)|, has recently been extensively studied, see, e.g., [7–10,16,19,20,
23,25,27,32,33,36], and the references therein.
In this article, unless otherwise stated, by a solution to the Beltrami
equation (1.1) in D we mean a sense-preserving homeomorphic mapping
f : D → C in the Sobolev space W 1,1
loc (D), whose partial derivatives
satisfy (1.1) a.e. in D.
Theorem 1.2. Let µ, ν be measurable functions in D ⊂ C, such that
|µ| + |ν| < 1 a.e. in D and
1 + |µ(z)| + |ν(z)|
1 − |µ(z)| − |ν(z)| ≤ Q(z) (1.5)
a.e. in D for some function Q(z) ∈ BMO(C). Then the Beltrami equa-
tion (1.1) has a homeomorphic solution f : D → C which belongs to the
space W 1,s
loc (D) for all s ∈ [1, 2). Moreover, this solution admits a homeo-
morphic extension to C such that f is conformal in C\D and f(∞) = ∞.
For the extended mapping f−1 ∈ W 1,2
loc , and for every compact set E ⊂ C
there are positive constants C, C ′, a and b such that
C exp
(
− a
|z′ − z′′|2
)
≤ |f(z′) − f(z′′)| ≤ C ′
∣
∣
∣
∣
log
1
|z′ − z′′|
∣
∣
∣
∣
−b
(1.6)
for every pair of points z′, z′′ ∈ E provided that |z′ − z′′| is sufficiently
small.
Remark 1.1. Note that C ′ is an absolute constant, b depends only on
E and Q.
Remark 1.2. Prototypes of Theorem 1.2 when ν(z) ≡ 0 can be found in
the pioneering papers on the degenerate Beltrami equation [27] and [10],
see also [32] and [36].
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 309
In [2] it was shown that a necessary and sufficient condition for a
measurable function K(z) ≥ 1 to be majorized in D ⊂ C by a function
Q ∈ BMO is that
∫∫
D
eηK(z) dx dy
1 + |z|3 < ∞ (1.7)
for some positive number η. Thus, the inequality (1.7) can be viewed as
a test for Kµ,ν(z) to satisfy the hypothesis of Theorem 1.2.
2. Auxiliary lemmas
For the proof of Theorem 1.2 we need the following lemmas.
Lemma 2.1. Let fn : D → C be a sequence of homeomorphic ACL
solutions to the equation (1.1) converging locally uniformly in D to a
homeomorphic limit function f . If
Kµn,νn(z) ≤ Q(z) ∈ Lp
loc(D) (2.1)
a.e. in D for some p > 1, then the limit function f belongs to W 1,s
loc
where s = 2p/(1+ p) and ∂fn and ∂fn converge weakly in Ls
loc(D) to the
corresponding generalized derivatives of f .
Proof. First, let us show that the partial derivatives of the sequence fn
are bounded by the norm in Ls over every disk B with B ⊂ D. Indeed,
|∂fn| ≤ |∂fn| ≤ |∂fn| + |∂fn| ≤ Q1/2(z) · J1/2
n (z)
a.e. in B and by the Hölder inequality and Lemma 3.3 of Chapter III
in [24]
‖∂fn‖s ≤ ‖Q‖1/2
p · |fn(B)|1/2
where s = 2p/(1 + p), Jn is the Jacobian of fn and ‖ · ‖p denotes the
Lp−norm in B.
By the uniform convergence of fn to f in B, for some λ > 1 and large
n, |fn(B)| ≤ |f(λB)| and, consequently,
‖∂fn‖s ≤ ‖Q‖1/2
p · |f(λB)|1/2.
Hence fn ∈ W 1,s
loc , see e.g. Theorem 2.7.1 and Theorem 2.7.2 in [26].
On the other hand, by the known criterion of the weak compactness
in the space Ls, s ∈ (1,∞), see [12, Corollary IV.8.4], ∂fn → ∂f and
∂fn → ∂f weakly in Ls
loc for such s. Thus, f belongs to W 1,s
loc where
s = 2p/(1 + p).
310 General Beltrami equations...
Lemma 2.2. Under assumptions of Lemma 2.1, if µn(z) → µ(z) and
νn(z) → ν(z) a.e. in D, then the limit function f is a W 1,s
loc solution to
the equation (1.1) with s = 2p/(1 + p).
Proof. We set ζ(z) = ∂f(z)−µ(z) ∂f(z)−ν(z) ∂f(z) and, assuming that
µn(z) → µ(z) and νn(z) → ν(z) a.e. in D, we will show that ζ(z) = 0 a.e.
in D. Indeed, for every disk B with B ⊂ D, by the triangle inequality
∣
∣
∣
∣
∣
∫
B
ζ(z) dx dy
∣
∣
∣
∣
∣
≤ I1(n) + I2(n) + I3(n)
where
I1(n) =
∣
∣
∣
∣
∣
∫
B
(
∂f(z) − ∂fn(z)
)
dx dy
∣
∣
∣
∣
∣
,
I2(n) =
∣
∣
∣
∣
∣
∫
B
(µ(z) ∂f(z) − µn(z) ∂fn(z)) dx dy
∣
∣
∣
∣
∣
,
I3(n) =
∣
∣
∣
∣
∣
∫
B
(
ν(z) ∂f(z) − νn(z) ∂fn(z)
)
dx dy
∣
∣
∣
∣
∣
.
By Lemma 2.1, ∂fn and ∂fn converge weakly in Ls
loc(D) to the corre-
sponding generalized derivatives of f . Hence, by the result on the repre-
sentation of linear continuous functionals in Lp, p ∈ [1,∞), in terms of
functions in Lq, 1/p + 1/q = 1, see [12, IV.8.1 and IV.8.5], we see that
I1(n) → 0 as n → ∞. Note that I2(n) ≤ I ′2(n) + I ′′2 (n), where
I ′2(n) =
∣
∣
∣
∣
∣
∫
B
µ(z)(∂f(z) − ∂fn(z)) dx dy
∣
∣
∣
∣
∣
and
I ′′2 (n) =
∣
∣
∣
∣
∣
∫
B
(µ(z) − µn(z))∂fn(z) dx dy
∣
∣
∣
∣
∣
,
and we see that I ′2(n) → 0 as n → ∞ because µ ∈ L∞. In order to
estimate the second term, we make use of the fact that the sequence
|∂fn| is weakly compact in Ls
loc, see e.g. [12, IV.8.10], and hence |∂fn| is
absolutely equicontinuous in L1
loc, see e.g. [12, IV.8.11]. Thus, for every
ε > 0 there is δ > 0 such that
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 311
∫
E
|∂fn(z)| dx dy < ε, n = 1, 2, . . . ,
whenever E is measurable set in B with |E| < δ. On the other hand, by
the Egoroff theorem, see e.g. [12, III.6.12], µn(z) → µ(z) uniformly on
some set S ⊂ B such that |E| < δ where E = B\S. Now |µn(z)−µ(z)| <
ε on S for large n and consequently
I ′′2 (n) ≤
∫
S
|µ(z) − µn(z)| · |∂fn(z)| dx dy
+
∫
E
|µ(z) − µn(z)| · |∂fn(z)| dx dy
≤ ε
∫
B
|∂fn(z)| dx dy + 2
∫
E
|∂fn(z)| dx dy
≤ ε
(
‖Q‖1/2 · |f(λB)|1/2 + 2
)
for large enough n, i.e. I ′′2 (n) → 0 as n → ∞ because ε > 0 is arbi-
trary. The fact that I3(n) → 0 as n → ∞ is handled similarly. Thus,
∫
B ζ(z) dx dy = 0 for all disks B with B ⊂ D. By the Lebesgue theo-
rem on differentiability of the indefinite integral, see e.g. [34, IV(6.3)],
ζ(z) = 0 a.e. in D.
Remark 2.1. Lemma 2.1 and Lemma 2.2 extend the well known con-
vergence theorem where Q(z) ∈ L∞, see Lemma 4.2 in [6], and [4].
Recall that a doubly-connected domain in the complex plane is called
a ring domain and the modulus of a ring domain E is the number mod E
such that E is conformally equivalent to the annulus {1 < |z| < e mod E}.
We write A = A(r, R; z0), 0 < r < R < ∞, for the annulus r < |z− z0| <
R.
Let Γ be a family of Jordan arcs or curves in the plane. A nonnegative
and Borel measurable function ρ defined in C is called admissible for the
family Γ if the relation
∫
γ
ρ ds ≥ 1 (2.2)
holds for every locally rectifiable γ ∈ Γ. The quantity
M(Γ) = inf
ρ
∫
C
ρ2 dx, (2.3)
312 General Beltrami equations...
where the infimum is taken over all ρ admissible for the family Γ is called
the modulus of the family Γ, see, e.g., [1, p. 16], [24]. It is well known
that this quantity is a conformal invariant. Moreover, in these terms
the conformal modulus of a ring domain E has the representation, see,
e.g., [24],
mod E =
2π
M(Γ)
(2.4)
where Γ is the family of curves joining the boundary components of E
in E. Note also, that this modulus M(Γ) coincides with the conformal
capacity of E. Recall that the reciprocal to M(Γ) is usually called the
extremal length of Γ, however, in what follows, we will don’t make use of
this concept.
The next lemma deals with modulus estimates for quasiconformal
mappings in the plane, cf. [17, 29].
Lemma 2.3. Let f : A → C be a quasiconformal mapping. Then for each
nonnegative measurable functions ρ(t), t ∈ (r, R) and p(θ), θ ∈ (0, 2π),
such that
R
∫
r
ρ(t) dt = 1,
1
2π
2π
∫
0
p(θ) dθ = 1,
the following inequalities hold
2π
[
1
2π
∫∫
A
p2(θ)D−µ,z0
(z)
dx dy
|z − z0|2
]−1
≤ M(f(Γ))
≤
∫∫
A
ρ2(|z − z0|)Dµ,z0
(z) dx dy, (2.5)
where Γ stands for the family of curves joining the boundary components
of A(r, R; z0) in A(r, R; z0),
Dµ,z0
(z) =
∣
∣1 − µ(z)e−2iθ
∣
∣
2
1 − |µ(z)|2 (2.6)
and θ = arg(z − z0).
Remark 2.2. By (2.4), the inequalities (2.5) can be written in the fol-
lowing equivalent form
[
1
2π
∫∫
A
ρ2(|z − z0|)Dµ,z0
(z) dx dy
]−1
≤ mod f(A)
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 313
≤ 1
2π
∫∫
A
p2(θ)D−µ,z0
(z)
dx dy
|z − z0|2
. (2.7)
Proof. Let Γ be a family of curves joining the boundary components of
A = A(r, R; z0) in A, and let ρ satisfies the assumption of the lemma.
Denote by Γ∗ the family of all rectifiable paths γ∗ ∈ f(Γ) for which f−1
is absolutely continuous on every closed subpath of γ∗. Then M(f(Γ)) =
M(Γ∗) by the Fuglede theorem, see e.g. [24, pp. 135 and 170].
Fix γ∗ ∈ Γ∗ and let γ = f−1 ◦ γ∗. Denote by s and s∗ the natural
(length) parameters of γ and γ∗, respectively. Note that the correspon-
dence s∗(s) between the natural parameters is strictly monotone function
and we may assume that s∗(s) is increasing. For γ∗ ∈ Γ∗, the inverse
function s(s∗) has the (N) - property and s∗(s) is differentiable a.e. as a
monotone function. Thus, ds∗/ds 6= 0 a.e. on γ by [28]. Let s be such
that z = γ(s) is a regular point for f and suppose that γ(s) is differ-
entiable at s with ds∗/ds 6= 0. Set r = |z − z0| and let ω be the unit
tangential vector to the curve γ at the point z = γ(s). Then
∣
∣
∣
∣
dr
ds∗
∣
∣
∣
∣
=
dr
ds
/ ds∗
ds
=
|〈ω, ω0〉|
|∂ωf(z)|
where ω0 = (z−z0)/|z−z0|. Let now ρ satisfies the assumption of Lemma
2.3. Without loss of generality, by Lusin theorem, we can assume that ρ
is a Borel function. We set
ρ∗(w) =
{
ρ(|z − z0|)
(
Dµ,z0
(z)
Jf (z)
)1/2}
◦ f−1(w) (2.8)
if f is differentiable and Dµ,z0
(z)/Jf (z) 6= 0 at the point z = f−1(w)
and ρ∗(w) = ∞ otherwise at w ∈ f(A), and ρ∗(w) = 0 outside f(A).
Then ρ∗ is also a Borel function and we will show that ρ∗ is admissible
for the family Γ∗ = f(Γ). Indeed, the function z = γ(s(s∗)) is absolutely
continuous and hence so is r = |z− z0| as a function of the parameter s∗.
Then
∫
γ∗
ρ∗ ds∗ =
∫
γ∗
{
ρ(|z − z0|)
(
Dµ,z0
(z)
Jf (z)
)1/2}
◦ f−1(w) ds∗
≥
∫
γ∗
ρ(r)
∣
∣
∣
∣
dr
ds∗
∣
∣
∣
∣
ds∗ ≥
r2
∫
r1
ρ(r) dr = 1,
because of the inequality
(
Dµ,z0
(z)
Jf (z)
)1/2
◦ f−1(γ∗(s∗)) ≥
∣
∣
∣
dr
ds∗
∣
∣
∣
. (2.9)
314 General Beltrami equations...
Let us verify the later inequality. Since |dr/ds∗| = |〈ω, ω0〉|/|∂ωf(z)| and
∂ωf(z) = fz(z)(1 + µ(z)ω̄2), we see that
min
|ω|=1
|∂ωf(z)|
|〈ω, ω0〉|
= 2|fz| · min
|w|=1
∣
∣
∣
∣
w + a
w + 1
∣
∣
∣
∣
,
where ω0 = (z − z0)/|z − z0|, a = µω̄2
0, w = ω2ω̄2
0 and ω is an arbitrary
unit vector. The Möbius mapping ϕ(w) = w+a
w+1 transforms the unit circle
into a straight line with the unit normal vector ~n = a−1
|a−1| = (ϕ(0) −
ϕ(1))/|ϕ(0) − ϕ(1))|. Then the required distance from the straight line
to the origin is calculated as |〈ϕ(1), ~n〉|. Hence,
min
|ω|=1
|∂ωf(z)|
|〈ω, ω0〉|
= 2|fz|
〈
1 + a
2
,
1 − a
|1 − a|
〉
= |fz|Re
{
(1 + a)(1 − ā)
|1 − a|
}
= |fz|
1 − |µ(z)|2
|1 − µ(z) z̄−z̄0
z−z0
| =
(
Jf (z)
Dµ,z0
(z)
)1/2
and we arrive at the inequality (2.9).
Since f and f−1 are locally absolutely continuous in their domains,
we can perform the following change of variables
∫
f(A)
ρ∗2(w) du dv =
∫
A
ρ2(|z − z0|)Dµ,z0
(z) dx dy.
Thus, we arrive at the inequality
M(f(Γ)) ≤
∫
A
ρ2(|z − z0|)Dµ,z0
(z) dx dy,
completing the first part of the proof.
In order to get the left inequality in (2.5), we take into account that
p(θ) satisfies the assumption of Lemma 2.3 and show that the function
ρ∗(w) =
1
2π
{
p(θ)
|z − z0|
(
D−µ,z0
(z)
Jf (z)
)1/2}
◦ f−1(w) (2.10)
is admissible for the family G∗ = f(G), where G is the family of curve
that separates the boundary components of A in A. Omitting the regu-
larity arguments similar to those of given in the first part of the proof,
we see that
∫
γ∗
ρ∗ ds∗ ≥ 1
2π
∫
γ∗
p(θ)
∣
∣
∣
dθ
ds∗
∣
∣
∣
ds∗ ≥ 1
2π
2π
∫
0
p(θ) dθ = 1
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 315
because of the inequality
1
r
·
(
D−µ,z0
(z)
Jf (z)
)1/2
◦ f−1(γ∗(s∗)) ≥
∣
∣
∣
dθ
ds∗
∣
∣
∣
, r = |z − z0|. (2.11)
Indeed, since
∣
∣
∣
dθ
ds∗
∣
∣
∣
=
∣
∣
∣
dθ
ds
∣
∣
∣
/ ds∗
ds
=
|
√
1 − 〈ω, ω0〉2|/r
|∂ωf(z)| ,
and ∂ωf(z) = fz(z)(1 + µ(z)ω̄2), we see that
min
|ω|=1
|∂ωf(z)|
|
√
1 − 〈ω, ω0〉2|
= 2|fz| · min
|w|=1
∣
∣
∣
w + a
w − 1
∣
∣
∣
,
where ω0 = (z − z0)/|z − z0|, a = µω̄2
0, w = ω2ω̄2
0 and ω is an arbitrary
unit vector. The Möbius conformal mapping ϕ(w) = w+a
w−1 transforms the
unit circle into a straight line with the unit normal vector ~n = 1+a
|1+a| =
(ϕ(−1) − ϕ(0))/|ϕ(−1) − ϕ(0))|. Then the required distance from the
straight line to the origin is calculated as |〈ϕ(−1), ~n〉|. Hence,
min
|ω|=1
|∂ωf(z)|
|
√
1 − 〈ω, ω0〉2|
= 2|fz|
〈
1 − a
2
,
1 + a
|1 + a|
〉
= |fz|Re
{
(1 + a)(1 − ā)
|1 + a|
}
= |fz|
1 − |µ(z)|2
|1 + µ(z) z̄−z̄0
z−z0
| =
(
Jf (z)
D−µ,z0
(z)
)1/2
and we get the inequality (2.11). Performing the change of variable, we
have that
Mf((G)) ≤
∫
f(A)
ρ∗2(w) du dv =
1
4π2
∫
A
p2(θ)D−µ,z0
(z)
dx dy
|z − z0|
.
Noting that Mf((G)) = 1/Mf((Γ)), we arrive at the required left in-
equality (2.5) and thus complete the proof.
Let us consider an application of Lemma 2.3 to the case when the
angular dilatation coefficient Dµ,z0
(z) is dominated by a BMO function.
To this end, we need the following auxiliary result, see [32, Lemma 2.21].
316 General Beltrami equations...
Lemma 2.4. Let Q be a non-negative BMO function in the disk B =
{z : |z| < 1}, and for 0 < t < e−2, let A(t) = {z : t < |z| < e−1}. Then
η(t) :=
∫∫
A(t)
Q(z) dx dy
|z|2(log |z|)2 ≤ c log log 1/t (2.12)
where c is a constant which depends only on the average Q1 of Q over
|z| < e−1 and on the BMO norm ‖Q‖∗ of Q in B.
For the sake of completeness, we give a short proof.
Proof of Lemma 2.4. Fix t ∈ (0, e−2). For n = 1, 2, . . . , let tn = e−n,
An = {z : tn+1 < |z| < tn}, Bn = {z : |z| < tn} and Qn the mean value
of Q(z) in Bn. Now choose an integer N, such that tN+1 ≤ t < tN . Then
A(t) ⊂ A(tN+1) = ∪N+1
n=1 An, and
η(t) ≤
∫∫
A(tN+1)
Q(z)
|z|2(log |z|)2 dx dy = S1 + S2 (2.13)
where
S1 =
N
∑
n=1
∫∫
An
Q(z) − Qn
|z|2(log |z|)2 dx dy (2.14)
and
S2 =
N
∑
n=1
Qn
∫∫
An
dx dy
|z|2(log |z|)2 (2.15)
Since An ⊂ Bn, and for z ∈ An, |z|−2 ≤ πe2/|Bn| and log 1/|z| >
n, it follows that
|S1| ≤
N
∑
n=1
∫∫
An
|Q(z) − Qn|
|z|2(log |z|)2 dx dy
≤ π
N
∑
n=1
e2
n2
(
1
|Bn|
∫∫
Bn
|Q(z) − Qn| dx dy
)
.
Hence,
|S1| ≤ 2πe2‖Q‖∗. (2.16)
Now, note that
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 317
|Qk − Qk−1| =
1
|Bk|
∣
∣
∣
∣
∣
∫∫
Bk
(Q(z) − Qk−1) dx dy
∣
∣
∣
∣
∣
≤ 1
|Bk|
∫∫
Bk
|Q(z) − Qk−1| dx dy
=
e2
|Bk−1|
∫∫
Bk
|Q(z) − Qk−1| dx dy
≤ e2
|Bk−1|
∫∫
Bk−1
|Q(z) − Qk−1| dx dy ≤ e2‖Q‖∗.
Thus, by the triangle inequality,
Qn ≤ Q1 +
n
∑
k=2
|Qk − Qk−1| ≤ Q1 + ne2‖Q‖∗, (2.17)
and, since
∫∫
An
dx dy
|z|2(log |z|)2 ≤ 1
n2
∫∫
An
dx dy
|z|2 =
2π
n2
,
it follows by (2.15), that
S2 ≤ 2π
N
∑
n=1
Qn
n2
≤ 2πQ1
N
∑
n=1
1
n2
+ 2πe2‖Q‖∗
N
∑
1
1
n
. (2.18)
Finally,
∑N
n=1 1/n2 is bounded, and
∑N
n=1 1/n < 1 + log N < 1 +
log log 1/t, and, thus, (2.12) follows from (2.13), (2.16) and (2.18).
Lemma 2.5. Let f : D → C be a quasiconformal mapping with complex
dilatation µ(z) = fz̄(z)/fz(z), such that Kf (z) = Kµ,0(z) ≤ Q(z) ∈
BMO a.e. in D. Then for every annulus A(r, Re−1; z0), r < Re−2,
contained in D,
M(f(Γ)) ≤ c
log log(R/r)
(2.19)
where Γ stands for the family of curves joining the boundary components
of A(r, Re−1; z0) in A(r, Re−1; z0) and c is the constant in Lemma 2.4
associated with the function Q(Rz + z0).
Proof. Since
∫
γ
ds
|z − z0| log(R/|z − z0|)
≥
R/e
∫
r
dt
t log R /t
= log log R /r = a
318 General Beltrami equations...
we see that the function ρ(|z − z0|) = 1/a(|z − z0| log R/|z − z0|) is
admissible for the family Γ. By Lemma 2.3, and the inequality
Dµ,z0
(z) ≤ Kµ,0(z) ≤ Q(z),
we get that
M(f(Γ)) ≤ 1
a2
∫∫
r<|z−z0|<Re−1
Q(z) dx dy
|z − z0|2 log2(R/|z − z0|)
.
Performing the change of variable z 7→ Rz + z0, and making use of
Lemma 2.4, we have
M(f(Γ)) ≤
∫∫
r/R<|z|<e−1
Q(Rz + z0) dx dy
|z|2 log2(1/|z|)
≤ c
log log(R/r)
.
Remark 2.3. Note that it is not possible, in general, to replace the
BMO bound in the previous results by a simpler requirement that the
maximal dilatation itself belongs to BMO. For example, consider the
functions Q(x, y) = 1+| log |y||, (x, y) ∈ R
2 and u(x, y) = Q(x, y) if y > 0
and u(x, y) = 1 if y ≤ 0. Then u ≤ Q and Q ∈ BMO but u does not
belong to BMO.
Lemma 2.6. There exists a universal constant C0 > 0 with the property
that for a ring domain B in C with mod B > C0 which separates a
point z0 from ∞ we can choose an annulus A in B of the form A =
A(r1, r2; z0), r1 < r2, so that mod A ≥ mod B − C0.
For the proof of the above statement, see [18], where the authors
assert that one can take C0 = π−1 log 2(1 +
√
2) = 0.50118 . . . . In fact,
it essentially follows from the famous Teichmüller lemma on his extremal
ring domain.
3. Proof of main theorem
In view of Lemma 2.1 and Lemma 2.2, the problem of the existence
of an W 1,1
loc homeomorphic solution to the Beltrami equation (1.1) can be
reduced, by a suitable approximation procedure, to the problem of nor-
mality of certain families of quasiconformal mappings. By the well-known
Arzela–Ascoli theorem, the latter is related to appropriate oscillation es-
timates.
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 319
Proof of Theorem 1.2. We split the proof of Theorem 1.2 into three
parts. Given µ, ν, we first generate a sequence of quasiconformal map-
pings, corresponding to a suitable truncation of the above Beltrami co-
efficients, and show, making use of Lemma 2.5, that the chosen sequence
is normal with respect to the locally uniform convergence. Then we
prove that the limit mappings are univalent, belong to the Sobolev space
W 1,s
loc (D), s ∈ [1, 2), and satisfy the differential equation (1.1) a.e. in D.
Finally we deduce the regularity properties of the required solution to
the equation (1.1).
n01. Let µ, ν, be Beltrami coefficients defined in D with |µ|+ |ν| < 1
a.e. in D. For n = 1, 2, . . . , we set in Dn = D
⋂
B(n)
µn(z) = µ(z), if |µ(z)| ≤ 1 − 1/n, (3.1)
νn(z) = ν(z), if |ν(z)| ≤ 1 − 1/n, (3.2)
and µn(z) = νn(z) = 0 otherwise, including the points z ∈ B(n) \ Dn.
Here B(n) stands for the disk |z| < n. The coefficients µn, νn now
are defined in the disk B(n) and satisfy the strong ellipticity condition
|µn(z)| + |νn(z)| ≤ qn < 1. Therefore, by Theorem 1.1, there exists a
quasiconformal mapping fn(z) = ωn(z/n)/|ωn(1/n)| of B(n) onto B(Rn)
for some Rn = 1/|ωn(1/n)| > 1 satisfying a.e in B(n) the equation
fnz̄ − µn(z)fnz − νn(z)fnz = 0 (3.3)
and normalized by fn(0) = 0, |fn(1)| = 1. We extend fn over ∂B(n) to
the complex plane C by the symmetry principle. It implies, in particular,
that fn(∞) = ∞. We will call such fn the canonical approximating se-
quence. It follows from (3.3) and the symmetry principle that fn satisfies
a.e. in C the Beltrami equation
fnz̄ = µ∗
n(z)fnz
where
320 General Beltrami equations...
µ∗
n(z) =
{
µ̃n(z), if z ∈ B(n),
µ̃n(n2/z̄)z2/z̄2, if z ∈ C \ B(n),
and
µ̃n(z) = µn(z) + νn(z) · fnz
fnz
.
Note that Kµ,ν(z) ≤ Q(z) a.e. in B(n).
Our immediate task now is to show that the canonical approximating
sequence of quasiconformal mappings fn : C → C forms a normal family
of mappings with respect to the locally uniform convergence in C. To this
end, we first prove that the family is equicontinuous locally uniformly in
C. More precisely, we show that for every compact set E ⊂ C
|fn(z′) − fn(z′′)| ≤ C
(
log
1
|z′ − z′′|
)−α
, (3.4)
for every n ≥ N and z′, z′′ ∈ E such that |z′ − z′′| is small enough. Here
C is an absolute positive constant and α > 0 depends only on E and Q.
Indeed, let E be a compact set of C and z′, z′′ ∈ E be a pair of points
satisfying |z′ − z′′| < e−4. If we choose N such that dist(E, ∂B(N)) > 1,
then we see that the annulus
A = {z ∈ C : |z′ − z′′| < |z − z′| < |z′ − z′′|1/2 · e−1}
is contained in B(N). Moreover, at least one of the points 0 or 1 lies
outside of the annulus A and belongs to the unbounded component of
its complement.
Let Γ be the family of curves joining the circles |z−z′| = |z′−z′′| = r
and |z − z′| = |z′ − z′′|1/2e−1 = Re−1 in A. The complement of the ring
domain fn(A) to the complex plane has the bounded and unbounded
components ∆n and Ωn, respectively. Then, by the well-known Gehring’s
lemma, see [14],
M(fn(Γ)) ≥ 2π
log(λ/δnδ∗n)
where δn and δ∗n stand for the spherical diameters of ∆n and Ωn. Since
for small enough |z′ − z′′| δ∗n ≥ 1/
√
2, we get that
δn ≤
√
2λe−2π/M(fn(Γ))
where λ is an absolute constant. On the other hand, by Lemma 2.5, we
have
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 321
M(fn(Γ)) ≤ c
log log(1/|z′ − z′′|1/2)
(3.5)
where the positive constant c depends only on E and Q. If |z′ − z′′| is
small enough, then 2δn ≥ |fn(z′) − fn(z′′)| and hence
|fn(z′) − fn(z′′)| ≤ 2
√
2λe−2π/M(fn(Γ)). (3.6)
Combining the estimate (3.6) with the inequality (3.5), we arrive at (3.4).
The required normality of the family {fn} with respect to the spher-
ical metric in C now follows by the Ascoli–Arzela theorem, see e.g. [37,
20.4]. Thus, we complete the first part of the proof.
n02. Now we show that the limit mapping f is injective. To this
end, without loss of generality, we may assume that the sequence fn
converges locally uniformly in C to a limit mapping f which is not a
constant because of the chosen normalization. Since the mapping degree
is preserved under uniform convergence, f has degree 1, see e.g., [15].
We now consider the open set V = {z ∈ C : f is locally constant at z}.
First we show that if z0 ∈ C \ V, then f(z) 6= f(z0) for z ∈ C \ {z0}.
Picking a point z∗ 6= z0, we choose a small positive number R so that
|z∗ − z0| > R/e. Then, by Lemma 2.5
mod fn(A(r, R/e; z0) =
2π
M(fn(Γ))
≥ 2π
c
log log(R/r) > C0
for sufficiently small 0 < r < R/e2, where C0 is the constant in Lem-
ma 2.6. By virtue of Lemma 2.6, we can find an annulus An = {w : rn <
|w−fn(z0)| < r′n} in the ring domain fn(A(r, R/e; z0)) for n large enough.
Since f is not locally constant at z0, there exists a point z′ in the disk
|z − z0| < r with f(z0) 6= f(z′). The annulus An separates fn(z0), fn(z′)
from fn(z∗), so we obtain |fn(z′)−fn(z0)| ≤ rn and r′n ≤ |fn(z∗)−fn(z0)|.
In particular, |fn(z′) − fn(z0)| ≤ |fn(z∗) − fn(z0)| for n large enough.
Letting n → ∞, we obtain 0 < |f(z′) − f(z0)| ≤ |f(z∗) − f(z0)|, and
hence f(z∗) 6= f(z0).
We next show that the set V is empty. Indeed, suppose that V has
a non-empty component V0. Then f takes a constant value, say b, in V0.
If z∗ ∈ ∂V0, then by continuity, we have f(z∗) = b. On the other hand,
z∗ /∈ V and therefore f(z) 6= f(z∗) = b for any point z other than z0,
which contradicts the fact that f = b in V0. We conclude that V is empty,
namely, f is not locally constant at any point and hence f(z) 6= f(ζ) if
z 6= ζ. Thus, the injectivity of f follows.
322 General Beltrami equations...
Now we have that the sequence fn of quasiconformal mappings con-
verges locally uniformly in C to a limit function f . On the other hand,
Kµn,νn(z) ≤ Q(z) ∈ Lp(B(n)) for every p > 1 and n = 1, 2, . . . , and
µn(z) → µ(z) and νn(z) → ν(z) a.e. in D and to 0 in C \ D as n → ∞.
Then, by Lemma 2.2, we arrive at the conclusion that the limit mapping
f is homeomorphic solution for the equation ∂f = µ(z)∂f + ν∂f in D of
the class W 1,s
loc (D), s = 2p/(1 + p), and moreover this solution f admits
a conformal extension to C \ D. Furthermore, the infinity is the remov-
able singularity for the limit mapping by Theorem 6.3 in [32]. Thus, the
mapping f admits extension to a self homeomorphism of C, f(∞) = ∞,
which is conformal in C \ D.
n03. The mappings fn, n = 1, 2, . . . , are homeomorphic and therefore
gn := f−1
n → g := f−1 as n → ∞ locally uniformly in C, see [11, p. 268].
By the change of variables, that is correct because fn and gn ∈ W 1,2
loc , we
obtain under large n
∫
D′
∗
|∂gn|2 du dv =
∫
gn(D′
∗
)
dx dy
1 − |µn(z)|2 ≤
∫
D∗
Q(z) dx dy < ∞
for bounded domains D∗ ⊂ C and relatively compact sets D′
∗ ⊂ C with
g(D′
∗) ⊂ D∗. The latter estimate means that the sequence gn is bounded
in W 1,2(D′
∗) for large n and hence g ∈ W 1,2
loc (C). Moreover, ∂gn → ∂g
and ∂gn → ∂g weakly in L2
loc, see e.g. [31, III.3.5]. The homeomorphism
g has (N)−property because g ∈ W 1,2
loc , see e.g. [24, Theorem 6.1 of
Chapter III], and hence Jf (z) 6= 0 a.e., see [28].
Finally, the right inequality in (1.6) follows from (3.4). In order to
get the left inequality we make use of the length-area argument, see,
e.g. [35], p. 75. Let E be a compact set in C and E′ = f(E). Next, let
w′, w′′ be a pair of points in E′ with |w′ − w′′| < 1. Consider the family
of circles {S(w′, r)} centered at w′ of radius r,
r1 = |w′ − w′′| < r < r2 = |w′ − w′′|1/2.
Since g = f−1(w) ∈ W 1,2
loc (C), we can apply the standard oscillation
estimate
r2
∫
r1
osc2(g, S(w′, r)) · dr
r
≤ c
∫∫
|w−w′|<r2
|∇g|2 · dx dy
where S(w′, r) stands for the circle |w − w′| = r. It yields the estimate
B. V. Bojarski, V. V. Gutlyanskĭı and V. I. Ryazanov 323
inf
r∈(r1,r2)
osc(g, S(w′, r)) ≤ c1 log−1/2 1
|w′ − w′′| .
The mapping g is a homeomorphism, so osc(g, B(w′, r))≤osc(gn, S(w′, r))
for every r ∈ (r1, r2) where B(w′, r) = {w : |w − w′| < r}. Thus, we get
the inequality
|g(w′) − g(w′′)| < C1 log−1/2 1
|w′ − w′′| . (3.7)
Setting w′ = f(z′) and w′′ = f(z′′), we arrive at the required estimate
|f(z′) − f(z′′)| > Ce−a/|z′−z′′|2 . (3.8)
The last result can be deduced from Gehring’s oscillation inequality, see,
e.g., [14].
Remark 3.1. The first two parts of the proof for Theorem 1.2 are based
on Lemma 2.1, Lemma 2.2, the modulus estimate
mod fn(A(r, R/e; z0)) ≥ C log log(R/r), (3.9)
as well as on the fact that the right hand side of (3.9) approaches ∞ as
r → 0. Recall that the proof of inequality (3.9) is based on Lemma 2.3
and the estimate (1.5). More refined results, based on Lemma 2.3, can
be obtained for the degenerate Beltrami equation (1.1) if we replace the
basic assumption (1.5) by another one, say, by the inequality
(∣
∣
∣
1 − µ(z) z̄−z̄0
z−z0
∣
∣
∣
+ |ν(z)|
)2
1 − (|µ(z)| + |ν(z)|)2 ≤ Qz0
(z) (3.10)
where Qz0
(z) ∈ BMO for every z0 ∈ D. We also can replace (1.5) by the
inequality
∫∫
D
eH(Kµ,ν(z)) dx dy
(1 + |z|2)2 < M (3.11)
where H stands for a dominating factor of divergence type, see for details
[16]. Notice, that typical choices for H(x) are ηx and ηx/(1+ log+ x) for
a positive constant η. However, we will not pursue these directions here
and have an intention to publish the corresponding results elsewhere.
Acknowledgments. The research of the third author was partially
supported by the Ukrainian State Foundation of Fundamental Investiga-
tions (FFI), Grant number F25.1/055.
324 General Beltrami equations...
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Contact information
Bogdan V. Bojarski Institute of Mathematics
Polish Academy of Sciences,
ul. Sniadeckich 8, P.O. Box 21,
00-956 Warszawa,
Poland
E-Mail: B.Bojarski@impan.gov.pl
326 General Beltrami equations...
Vladimir Ya.
Gutlyanskĭı,
Vladimir I.
Ryazanov
Institute of Applied Mathematics
and Mechanics, NAS of Ukraine,
ul. Roze Luxemburg 74,
83114, Donetsk,
Ukraine
E-Mail: gut@iamm.ac.donetsk.ua,
vlryazanov1@rambler.ru
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