The Beltrami equations and prime ends
We first study the boundary behavior of ring Q-homeomorphisms in terms of Carath´eodory’s prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation ∂f = μ∂f in arbitrary bounded finitely connected domains D of the complex plane C:
Saved in:
| Published in: | Український математичний вісник |
|---|---|
| Date: | 2015 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2015
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/124487 |
| 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: | The Beltrami equations and prime ends / V.Y. Gutlyanskii, V.I. Ryazanov, E. Yakubov // Український математичний вісник. — 2015. — Т. 12, № 1. — С. 27-66. — Бібліогр.: 53 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-124487 |
|---|---|
| record_format |
dspace |
| spelling |
Gutlyanskii, V.Y. Ryazanov, V.I. Yakubov, E. 2017-09-27T06:00:39Z 2017-09-27T06:00:39Z 2015 The Beltrami equations and prime ends / V.Y. Gutlyanskii, V.I. Ryazanov, E. Yakubov // Український математичний вісник. — 2015. — Т. 12, № 1. — С. 27-66. — Бібліогр.: 53 назв. — англ. 1810-3200 2010 MSC. 30C62, 30D40, 37E30, 35A16, 35A23, 35J46, 35J67, 35J70, 35J75, 35Q35. https://nasplib.isofts.kiev.ua/handle/123456789/124487 We first study the boundary behavior of ring Q-homeomorphisms in terms of Carath´eodory’s prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation ∂f = μ∂f in arbitrary bounded finitely connected domains D of the complex plane C: en Інститут прикладної математики і механіки НАН України Український математичний вісник The Beltrami equations and prime ends Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The Beltrami equations and prime ends |
| spellingShingle |
The Beltrami equations and prime ends Gutlyanskii, V.Y. Ryazanov, V.I. Yakubov, E. |
| title_short |
The Beltrami equations and prime ends |
| title_full |
The Beltrami equations and prime ends |
| title_fullStr |
The Beltrami equations and prime ends |
| title_full_unstemmed |
The Beltrami equations and prime ends |
| title_sort |
beltrami equations and prime ends |
| author |
Gutlyanskii, V.Y. Ryazanov, V.I. Yakubov, E. |
| author_facet |
Gutlyanskii, V.Y. Ryazanov, V.I. Yakubov, E. |
| publishDate |
2015 |
| language |
English |
| container_title |
Український математичний вісник |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
We first study the boundary behavior of ring Q-homeomorphisms in terms of Carath´eodory’s prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation ∂f = μ∂f in arbitrary bounded finitely connected domains D of the complex plane C:
|
| issn |
1810-3200 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/124487 |
| citation_txt |
The Beltrami equations and prime ends / V.Y. Gutlyanskii, V.I. Ryazanov, E. Yakubov // Український математичний вісник. — 2015. — Т. 12, № 1. — С. 27-66. — Бібліогр.: 53 назв. — англ. |
| work_keys_str_mv |
AT gutlyanskiivy thebeltramiequationsandprimeends AT ryazanovvi thebeltramiequationsandprimeends AT yakubove thebeltramiequationsandprimeends AT gutlyanskiivy beltramiequationsandprimeends AT ryazanovvi beltramiequationsandprimeends AT yakubove beltramiequationsandprimeends |
| first_indexed |
2025-11-25T15:49:22Z |
| last_indexed |
2025-11-25T15:49:22Z |
| _version_ |
1850517167423881216 |
| fulltext |
Український математичний вiсник
Том 12 (2015), № 1, 27 – 66
The Beltrami equations and prime ends
Vladimir Gutlyanskii, Vladimir Ryazanov,
and Eduard Yakubov
Abstract. We first study the boundary behavior of ring Q-homeo-
morphisms in terms of Carathéodory’s prime ends and then give criteria
to the solvability of the Dirichlet problem for the degenerate Beltrami
equation ∂f = µ∂f in arbitrary bounded finitely connected domains D
of the complex plane C.
2010 MSC. 30C62, 30D40, 37E30, 35A16, 35A23, 35J46, 35J67, 35J70,
35J75, 35Q35.
Key words and phrases. Dirichlet problem, degenerate Beltrami
equations, regular solutions, simply connected domains, pseudoregular
and multi-valued solutions, finitely connected domains, tangent dilata-
tions.
1. Introduction
Let D be a domain in the complex plane C and let µ : D → C be a
measurable function with |µ(z)| < 1 almost everywhere in D. We study
the Beltrami equation
fz̄ = µ(z) fz (1.1)
where fz̄ = ∂f = (fx + ify)/2, fz = ∂f = (fx − ify)/2, z = x + iy, and
fx and fy are partial derivatives of f in x and y, correspondingly.
The classic Dirichlet problem in a Jordan domain D for the uniformly
elliptic Beltrami equation, i.e., when |µ(z)| ≤ k < 1 a.e., is the problem
on the existence of a continuous function f : D → C such thatfz = µ(z) · fz, for a.e. z ∈ D,
lim
z→ζ
Re f(z) = φ(ζ), ∀ ζ ∈ ∂D,
(1.2)
for a prescribed continuous boundary function φ, has been studied long
ago, see, e.g., [2, 52].
Received 1.03.2015
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
28 The Beltrami equations and prime ends
The main goal of this work is to study the Dirichlet problem in an ar-
bitrary bounded finitely connected domain D ⊂ C when the condition of
uniform ellipticity of the Beltrami equations is replaced by the condition
that |µ(z)| < 1 a.e.
The degeneracy of the ellipticity of the Beltrami equations will be
controlled by the dilatation coefficient
Kµ(z) =
1 + |µ(z)|
1− |µ(z)|
(1.3)
as well as by the more refined quantity
KT
µ (z, z0) =
∣∣∣1− z−z0
z−z0
µ(z)
∣∣∣2
1− |µ(z)|2
(1.4)
that is called the tangent dilatation quotient of the Beltrami equation
(1.1) with respect to z0, see, e.g., [1,10,23,34,37,41]. This quantity takes
into account not only the modulus of the complex coefficient µ but also
its argument. Note that
K−1
µ (z) 6 KT
µ (z, z0) 6 Kµ(z) ∀ z ∈ D ∀ z0 ∈ C. (1.5)
Our research is based on the existence theorems of homeomorphic
W 1,1
loc solutions for the degenerate Beltrami equations and on the theory
of prime ends by Carathéodory for such solutions.
New criteria of the existence of homeomorphic W 1,1
loc solutions for the
degenerate Beltrami equations can be found, for instance, in [11,26], see
also the references therein. The boundary behavior of W 1,1
loc homeomor-
phic solutions and the Dirichlet problem for degenerate Beltrami equa-
tions in Jordan domains have been studied, e.g., in [17–19,37]. Concern-
ing the Carathéodory’s theory of prime ends for the conformal mappings,
we refer the reader to [4] and [5, Ch. 9].
Let ED denote the space of prime ends of the domain D and let DP =
D∪ED stand for the completion of the domain D by its prime ends with
the topology described in [5, Section 9.5]. From now on, the continuity
of mappings f : DP → D′
P and the boundary functions φ : ED → R as
functions of the prime end P should be understood with respect to the
given topology. Now the boundary condition is written as
lim
n→∞
Re f(zn) = φ(P ), (1.6)
where the limit is taken over all sequences of points zn ∈ D converging
to the prime end P of the domain D. Note that (1.6) is equivalent to the
condition that
lim
z→P
Re f(z) = φ(P ) (1.7)
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 29
along any ways in D going to the prime end P of the domain D.
Remark 1.1. The Carathéodory topology can be described in terms of
metrics. Really, as is known, every bounded finitely connected domain D
in C can be mapped by a conformal mapping g0 onto a circular domain
D0 whose boundary consists of a finite collection of mutually disjoint
circles and isolated points, see, e.g., Theorem V.6.2 in [9]. Due to the
well-knownWeierstrass theorem, isolated singular points of bounded con-
formal mappings are removable, see Theorem 1.2 in [5]. Hence, isolated
points of ∂D correspond to isolated points of ∂D0 and vice versa.
The mapping theorem allows us to reduce the case to the Carathéodo-
ry theorem, see, e.g., Theorem 9.4 in [5] for simply connected domains.
Thus, we have a natural one-to-one correspondence between points of
∂D0 and prime ends of the domain D, see also Theorem 4.1 in [31]. De-
termine, in DP , the metric ρ0(p1, p2) = |g̃0(p1)− g̃0(p2)| , where g̃0 is the
extension of g0 to DP just mentioned. If g∗ is another conformal map-
ping of the domain D on a circular domain D∗, then the corresponding
metric ρ∗(p1, p2) = |g̃∗(p1)− g̃∗(p2)| generates the same convergence in
DP as the metric ρ0 because g0 ◦ g−1
∗ is a conformal mapping between
D∗ and D0. The latter mapping can be extended to a homeomorphism
between D∗ and D0, see, e.g., Theorem V.6.1′ in [9]. Consequently, the
given metrics induce the same topology in the space DP .
This topology coincides with the topology of prime ends described in
inner terms of the domain D in Section 9.5 of [5]. Later on, we prefer to
apply the description of the topology of prime ends in terms of the given
metrics. We will explore the metrizability ofDP . Note also that the space
DP for every bounded finitely connected domain D in C with the given
topology is compact because the closure of the circular domain D0 is a
compact space and, by construction, g̃0 : DP → D0 is a homeomorphism.
Applying the description of the topology of prime ends given in Sec-
tion 9.5 of [5], we reduce the case of bounded finitely connected domains
to Theorem 9.3 in [5] for simple connected domains and obtain the fol-
lowing statement.
Lemma 1.1. Each prime end P of a bounded finitely connected domain
D in C contains a chain of cross-cuts σm lying on circles S(z0, rm) with
z0 ∈ ∂D and rm → 0 as m→ ∞.
Throughout this work, B(z0, r) = {z ∈ C : |z− z0| < r}, D = B(0, 1),
S(z0, r) = {z ∈ C : |z − z0| = r}, R(z0, r1, r2) = {z ∈ C : r1 < |z − z0| <
r2}.
30 The Beltrami equations and prime ends
2. Regular domains
Recall the following topological notion. A domain D ⊂ C is said to
be locally connected at a point z0 ∈ ∂D if, for every neighborhood U of
the point z0, there is a neighborhood V ⊆ U of z0 such that V ∩ D is
connected. If this condition holds for all z0 ∈ ∂D, then D is said to be
locally connected on ∂D. For a domain that is locally connected on ∂D,
there is a natural one-to-one correspondence between prime ends of D
and points of ∂D, moreover, the topology of prime ends coincides with
the Euclidean topology. Note that every Jordan domain D in C is locally
connected on ∂D, see, e.g., [53, p. 66].
The (conformal) modulus of a family Γ of paths γ in C is the quantity
M(Γ) = inf
ϱ∈admΓ
∫
C
ϱ2(z) dm(z), (2.1)
where a Borel function ϱ : C → [0,∞] is admissible for Γ, write ϱ ∈
admΓ, if ∫
γ
ϱ ds > 1 ∀ γ ∈ Γ. (2.2)
Here s is a natural parameter of the arc length on γ.
Later on, given sets A, B and C in C, ∆(A,B;C) denotes a family
of all paths γ : [a, b] → C joining A and B in C, i.e. γ(a) ∈ A, γ(b) ∈ B
and γ(t) ∈ C for all t ∈ (a, b).
We say that ∂D is weakly flat at a point z0 ∈ ∂D if, for every
neighborhood U of the point z0 and every number P > 0, there exists a
neighborhood V ⊂ U of z0 such that
M(∆(E,F ;D)) > P (2.3)
for all continua E and F in D intersecting ∂U and ∂V . We say that ∂D
is weakly flat if it is weakly flat at each point z0 ∈ ∂D.
We also say that a point z0 ∈ ∂D is strongly accessible if, for every
neighborhood U of the point z0, there exist a compactum E in D, a
neighborhood V ⊂ U of z0 and a number δ > 0 such that
M(∆(E,F ;D)) > δ (2.4)
for all continua F in D intersecting ∂U and ∂V . We say that ∂D is
strongly accessible if each point z0 ∈ ∂D is strongly accessible.
It is easy to see that if a domain D in C is weakly flat at a point
z0 ∈ ∂D, then the point z0 is strongly accessible from D. The following
statement is fundamental, see, e.g., Lemma 5.1 in [21] or Lemma 3.15
in [26].
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 31
Lemma 2.1. If a domain D in C is weakly flat at a point z0 ∈ ∂D, then
D is locally connected at z0.
The notions of strong accessibility and weak flatness at boundary
points of a domain in C defined in [20], see also [21,36], are localizations
and generalizations of the corresponding notions introduced in [24, 25],
cf. with the properties P1 and P2 by Väisälä in [51] and also with the
quasiconformal accessibility and the quasiconformal flatness by Näkki
in [29].
A domain D ⊂ C is called a quasiextremal distance domain, abbr.
QED-domain, see [8], if
M(∆(E,F ;C) 6 K ·M(∆(E,F ;D)) (2.5)
for some K > 1 and all pairs of nonintersecting continua E and F in D.
It is well known, see, e.g., Theorem 10.12 in [51], that
M(∆(E,F ;C)) > 2
π
log
R
r
(2.6)
for any sets E and F in C intersecting all the circles S(z0, ρ), ρ ∈ (r,R).
Hence, a QED-domain has a weakly flat boundary. An example in [26],
Section 3.8, shows that the inverse conclusion is not true even in the case
of simply connected domains in C.
A domain D ⊂ C is called a uniform domain if each pair of points z1
and z2 ∈ D can be joined with a rectifiable curve γ in D such that
s(γ) 6 a · |z1 − z2| (2.7)
and
min
i=1,2
s(γ(zi, z)) 6 b · dist(z, ∂D) (2.8)
for all z ∈ γ, where γ(zi, z) is the portion of γ bounded by zi and z,
see [28]. It is known that every uniform domain is a QED-domain but
there exist QED-domains that are not uniform, see [8]. Bounded convex
domains and bounded domains with smooth boundaries are simple ex-
amples of uniform domains and, consequently, QED-domains, as well as
domains with weakly flat boundaries.
Recall that φ : U → C is said to be a Lipschitz map provided |φ(z1)−
φ(z2)| 6 M · |z1 − z2| for some M < ∞ and for all z1 and z2 ∈ U , and
a bi-Lipschitz map if, in addition, M∗|z1 − z2| 6 |φ(z1) − φ(z2)| for
some M∗ > 0 and for all z1 and z2 ∈ U . One says that D in C is a
Lipschitz domain if every point z0 ∈ ∂D has a neighborhood U that can
be mapped by a bi-Lipschitz homeomorphism φ onto the unit disk D in C
32 The Beltrami equations and prime ends
such that φ(∂D∩U) is the intersection of D with the real axis. Note that
a bi-Lipschitz homeomorphism is quasiconformal and, consequently, the
modulus is quasiinvariant under such a mapping. Hence, the Lipschitz
domains have weakly flat boundaries.
3. BMO, VMO, and FMO functions
A real-valued function u in a domain D in C is said to be of bounded
mean oscillation in D, abbr. u ∈ BMO(D), if u ∈ L1
loc(D) and
∥u∥∗ := sup
B
1
|B|
∫
B
|u(z)− uB| dm(z) <∞, (3.1)
where the supremum is taken over all disks B in D, dm(z) corresponds
to the Lebesgue measure in C and
uB =
1
|B|
∫
B
u(z) dm(z) .
We write u ∈ BMOloc(D) if u ∈ BMO(U) for every relatively compact
subdomain U of D (we also write BMO or BMOloc if it is clear from the
context what D is).
The class BMO was introduced by John and Nirenberg (1961) in
work [16] and soon became an important concept in harmonic analysis,
partial differential equations, and related areas; see, e.g., [12, 35].
A function φ in BMO is said to have vanishing mean oscillation,
abbr. φ ∈ VMO, if the supremum in (3.1) taken over all balls B in
D with |B| < ε converges to 0 as ε → 0. VMO has been introduced
by Sarason in [49]. There are a number of works devoted to the study
of partial differential equations with coefficients of the class VMO, see,
e.g., [6, 15,27,32,33].
Remark 3.1. Note that W 1,2 (D) ⊂ VMO (D) , see, e.g., [3].
Following [14], we say that a function φ : D → R has finite mean
oscillation at a point z0 ∈ D, abbr. φ ∈ FMO(z0), if
lim
ε→0
∫
−
B(z0,ε)
|φ(z)− φ̃ε(z0)| dm(z) <∞ , (3.2)
where
φ̃ε(z0) =
∫
−
B(z0,ε)
φ(z) dm(z) (3.3)
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 33
is the mean value of the function φ(z) over the disk B(z0, ε). Note that
condition (3.2) includes the assumption that φ is integrable in some neigh-
borhood of the point z0. We say also that a function φ : D → R is of
finite mean oscillation in D, abbr. φ ∈ FMO(D) or simply φ ∈ FMO, if
φ ∈ FMO(z0) for all points z0 ∈ D. We write φ ∈ FMO(D) if φ is given
in a domain G in C such that D ⊂ G and φ ∈ FMO(G).
The following statement is obvious by the triangle inequality.
Proposition 3.1. If, for a collection of numbers φε ∈ R, ε ∈ (0, ε0],
lim
ε→0
∫
−
B(z0,ε)
|φ(z)− φε| dm(z) <∞ , (3.4)
then φ is of finite mean oscillation at z0.
In particular, choosing here φε ≡ 0, ε ∈ (0, ε0], we obtain the follow-
ing.
Corollary 3.1. If, for a point z0 ∈ D,
lim
ε→0
∫
−
B(z0,ε)
|φ(z)| dm(z) <∞ , (3.5)
then φ has finite mean oscillation at z0.
Recall that a point z0 ∈ D is called a Lebesgue point of a function
φ : D → R if φ is integrable in a neighborhood of z0 and
lim
ε→0
∫
−
B(z0,ε)
|φ(z)− φ(z0)| dm(z) = 0 . (3.6)
It is known that almost every point in D is a Lebesgue point for every
function φ ∈ L1(D). Thus, we have the following corollary by Proposi-
tion 3.1.
Corollary 3.2. Every locally integrable function φ : D → R has a finite
mean oscillation at almost every point in D.
Remark 3.2. Note that the function φ(z) = log (1/|z|) belongs to BMO
in the unit disk ∆, see, e.g., [35, p. 5], and hence also to FMO. However,
φ̃ε(0) → ∞ as ε → 0, showing that condition (3.5) is only sufficient
but not necessary for a function φ to be of finite mean oscillation at
z0. Clearly, BMO(D) ⊂ BMOloc(D) ⊂ FMO(D) and, as well-known,
BMOloc ⊂ Lp
loc for all p ∈ [1,∞), see, e.g., [16,35]. However, FMO is not
a subclass of Lp
loc for any p > 1 but only of L1
loc. Thus, the class FMO is
much more wide than BMOloc.
34 The Beltrami equations and prime ends
Versions of the next lemma has been first proved for BMO in [40].
For FMO, see works [14,36,42,43] and books [11,26].
Lemma 3.1. Let D be a domain in C and let φ : D → R be a non-
negative function of the class FMO(z0) for some z0 ∈ D. Then∫
ε<|z−z0|<ε0
φ(z) dm(z)(
|z − z0| log 1
|z−z0|
)2 = O
(
log log
1
ε
)
as ε→ 0 (3.7)
for some ε0 ∈ (0, δ0) where δ0 = min(e−e, d0), d0 = inf
z∈∂D
|z − z0|.
4. Beltrami equations and ring Q-homeomorphisms
The following notion was motivated by the ring definition of Gehring
for quasiconformal mappings, see, e.g., [7], and it is closely relevant with
the Beltrami equations. Given a domain D in C and a Lebesgue measur-
able function Q : C → (0,∞), we say that a homeomorphism f : D → C
is a ring Q-homeomorphism at a point z0 ∈ D if
M (∆ (fC1, fC2; fD)) 6
∫
A∩D
Q(z) · η2(|z − z0|) dm(z) (4.1)
for any ring A = A(z0, r1, r2) and arbitrary continua C1 and C2 in D that
belong to the different components of the complement of the ring A in C
including z0 and ∞, correspondingly, and for any Lebesgue measurable
function η : (r1, r2) → [0,∞] such that
r2∫
r1
η(r) dr > 1. (4.2)
The notion was first introduced at inner points of a domain D in
work [41]. The ring Q-homeomorphisms at boundary points of a domain
D have first been considered in works [44,45].
By Lemma 2.2 in [38] or Lemma 7.4 in [26], we obtain the following
criterion for homeomorphisms in C to be ring Q-homeomorphisms, see
also Theorem A.7 in [26].
Lemma 4.1. Let D and D′ be bounded domains in C and Q : C → (0,∞)
be a measurable function. A homeomorphism f : D → D′ is a ring Q-
homeomorphism at z0 ∈ D if and only if
M (∆ (fS1, fS2; fD)) 6
( r2∫
r1
dr
∥Q∥(z0, r)
)−1
∀ r1 ∈ (0, r2) , r2 ∈ (0, d0)
(4.3)
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 35
where Si = S(z0, ri), i = 1, 2, d0 = supz∈D | z − z0| and ∥Q∥(z0, r) is the
L1-norm of Q over D ∩ S(z0, r).
By Theorem 4.1 in [37], every homeomorphic W 1,1
loc solution of the
Beltrami equation (1.1) in a domain D ⊆ C is the so-called lower Q-
homeomorphism at every point z0 ∈ D with Q(z) = KT
µ (z, z0), z ∈ D,
and Q(z) ≡ ε > 0 in C \D. On the other hand, by Theorem 2 in [18] for
a locally integrable Q, if f : D → D′ is a lower Q-homeomorphism at a
point z0 ∈ D, then f is a ring Q-homeomorphism at the point z0. Thus,
we have the following conclusion.
Theorem 4.1. Let f be a homeomorphic W 1,1
loc solution of the Beltrami
equation (1.1) in a domain D ⊆ C and Kµ ∈ L1(D). Then f is a ring
Q-homeomorphism at every point z0 ∈ D with Q(z) = KT
µ (z, z0), z ∈ D.
In fact, it is sufficient to assume here thatKT
µ (z, z0) is integrable along
the circles |z − z0| = r for a.e. small enough r, instead of the condition
Kµ ∈ L1(D) .
5. The continuous extension of ring Q-homeomorphisms
Lemma 5.1. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a ring Qz0-homeomorphism at every point
z0 ∈ ∂D. Suppose that∫
D(z0,ε,ε0)
Qz0(z)·ψ2
z0,ε,ε0(|z−z0|) dm(z) = o
(
I2z0,ε0(ε)
)
as ε→ 0 ∀ z0∈∂D
(5.1)
where D(z0, ε, ε0) = {z ∈ D : ε < |z − z0| < ε0} for every small enough
0 < ε0 < d(z0) = supz∈D |z − z0| and where ψz0,ε,ε0(t) : (0,∞) → [0,∞],
ε ∈ (0, ε0), is a family of (Lebesgue) measurable functions such that
0 < Iz0,ε0(ε) :=
ε0∫
ε
ψz0,ε,ε0(t) dt < ∞ ∀ ε ∈ (0, ε0) .
Then f can be extended to a continuous mapping of DP onto D′
P .
Proof. By Remark 1.1, with no loss of generality, we may assume that
D′ is a circular domain and, thus, D′
P = D′. Let us first prove that the
cluster set
L = C(P, f) :=
{
ζ ∈ C : ζ = lim
n→∞
f(zn), zn → P, zn∈D, n = 1, 2, . . .
}
36 The Beltrami equations and prime ends
consists of a single point ζ0 ∈ ∂D′ for each prime end P of the domain
D.
Note that L ̸= ∅ by compactness of the set D′, and L is a subset
of ∂D′, see, e.g., Proposition 2.5 in [36] or Proposition 13.5 in [26]. Let
us assume that there exist at least two points ζ0 and ζ∗ ∈ L. Set U =
B(ζ0, ρ0) = {ζ ∈ C : |ζ − ζ0| < ρ0}, where 0 < ρ0 < |ζ∗ − ζ0|.
Let σk, k = 1, 2, . . . , be a chain of cross-cuts of D in the prime end
P lying on circles Sk = S(z0, rk) from Lemma 1.1, where z0 ∈ ∂D. Let
Dk, k = 1, 2, . . . be the domains associated with σk. Then there exist
points ζk and ζ∗k in the domains D′
k = f(Dk) such that |ζ0−ζk| < ρ0 and
|ζ0 − ζ∗k | > ρ0 and, moreover, ζk → ζ0 and ζ∗k → ζ∗ as k → ∞. Let Ck be
continuous curves joining ζk and ζ∗k in D′
k. Note that, by construction,
∂U ∩ Ck ̸= ∅, k = 1, 2, . . ..
By the condition of strong accessibility of the point ζ0, there are a
continuum E ⊂ D′ and a number δ > 0 such that
M(∆(E,Ck;D
′)) > δ (5.2)
for all large enough k. Note that C = f−1(E) is a compact subset of D
and hence d0 = dist(z0, C) > 0. Let ε0 ∈ (0, d0) be small enough from
the hypotheses of the lemma. With no loss of generality, we may assume
that rk < ε0 and (5.2) holds for all k = 1, 2, . . ..
Let Γm be a family of all continuous curves in D \ Dm joining the
circle S0 = S(z0, ε0) and σm, m = 1, 2, . . .. Note that, by construction,
Ck ⊂ D′
k ⊂ D′
m for all k > m and, thus, by the principle of minorization,
M(f(Γm)) > δ for all m = 1, 2, . . ..
On the other hand, every function
η(t) = ηm(t) := ψ∗
z0,rm,ε0(t)/Iz0,ε0(rm), m = 1, 2, . . . ,
satisfies condition (4.2) and hence
M(fΓm) 6
∫
D
Q(z) · η2m(z) dm(z) ,
i.e., M(fΓm) → 0 as m→ ∞ in view of (5.1).
The obtained contradiction disproves the assumption that the cluster
set C(P, f) consists of more than one point.
Thus, we have the extension h of f to DP such that C(ED, f) ⊆ ∂D′.
In fact, C(ED, f) = ∂D′. Indeed, if ζ0 ∈ D′, then there is a sequence ζn
in D′ being convergent to ζ0. We may assume with no loss of generality
that f−1(ζn) → P0 ∈ ED, because DP is compact, see Remark 1.1.
Hence, ζ0 ∈ C(P0, f).
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 37
Finally, let us show that the extended mapping h : DP → D′ is
continuous. Indeed, let Pn → P0 in DP . If P0 ∈ D, then the statement
is obvious. If P0 ∈ ED, then, by the last item, we are able to choose
P ∗
n ∈ D such that ρ(Pn, P
∗
n) < 1/n, where ρ is one of the metrics in
Remark 1.1 and |h(Pn)−h(P ∗
n)| < 1/n. Note that, by the first part of the
proof, h(P ∗
n) → h(P0) because P
∗
n → P0. Consequently, h(Pn) → h(P0),
too.
Theorem 5.1. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a ring Qz0-homeomorphism at every point
z0 ∈ ∂D. If
ε(z0)∫
0
dr
||Qz0 ||(r)
= ∞ ∀ z0 ∈ ∂D, (5.3)
where 0 < ε(z0) < d(z0) := sup
z∈D
|z − z0| and
||Qz0 ||(r) :=
∫
D∩S(z0,r)
Qz0 ds (5.4)
then f can be extended to a continuous mapping of DP onto D′
P .
Proof. Indeed, condition (5.3) implies that
ε0∫
0
dr
||Qz0 ||(r)
= ∞ ∀ z0 ∈ ∂D ∀ ε0 ∈ (0, ε(z0)) (5.5)
because the left-hand side in (4.3) is not equal to zero, see Theorem 5.2
in [30], and hence, by Lemma 4.1,
ε(z0)∫
ε0
dr
||Qz0 ||(r)
<∞.
On the other hand, for the functions
ψz0,ε0(t) :=
{
1/||Qz0 ||(t), t ∈ (0, ε0),
0, t ∈ [ε0,∞),
(5.6)
we have, by the Fubini theorem, that∫
D(z0,ε,ε0)
Qz0(z) · ψ2
z0,ε0(|z − z0|) dm(z) =
ε0∫
ε
dr
||Qz0 ||(r)
, (5.7)
38 The Beltrami equations and prime ends
and, consequently, condition (5.1) holds by (5.5) for all z0 ∈ ∂D and
ε0 ∈ (0, ε(z0)).
Here we have used the standard conventions in the integral theory
that a/∞ = 0 for a ̸= ∞ and 0 · ∞ = 0, see, e.g., Section I.3 in [48].
Thus, Theorem 5.1 follows immediately from Lemma 5.1.
6. The extension of the inverse mappings to the boundary
The proof of the extension of the inverse mappings for a ring Q-ho-
meomorphism by prime ends in the plane is based on the following lemma
on the cluster sets.
Lemma 6.1. Let D and D′ be bounded finitely connected domains in
C, and let P0 and P∗ be prime ends of D, P∗ ̸= P0. Denote, by σm,
m = 1, 2, . . ., a chain of cross-cuts in P0 from Lemma 1.1 lying on circles
S(z0, rm), z0 ∈ ∂D, with associated domains dm. Suppose that Q is
integrable over D∩S(z0, r) for a set E of numbers r ∈ (0, δ) of a positive
linear measure, where δ = rm0 and m0 is such that the domain dm0 does
not contain sequences of points converging to P∗. If f : D → D′ is a ring
Q-homeomorphism at the point z0 and ∂D′ is weakly flat, then
C(P0, f) ∩ C(P∗, f) = ∅. (6.1)
Note that, in view of the metrizability of the completion DP of the
domain D with prime ends, see Remark 1.1, the number m0 in Lemma 6.1
always exists.
Proof. Let us choose ε ∈ (0, δ) such that E0 := {r ∈ E : r ∈ (ε, δ)}
has a positive linear measure. Such a choice is possible in view of the
subadditivity of the linear measure and the exhaustion E = ∪En, where
En = {r ∈ E : r ∈ (1/n, δ)} , n = 1, 2, . . .. Note that, by Lemma 4.1 for
S1 = S(z0, ε) and S2 = S(z0, δ),
M (∆ (fS1, fS2; fD)) < ∞ . (6.2)
Let us assume that C0 ∩ C∗ ̸= ∅, where C0 = C(P0, f) and C∗ =
C(P∗, f). By construction, there is m1 > m0 such that σm1 lies on the
circle S(z0, rm1) with rm1 < ε. Let d0 = dm1 and d∗ ⊆ D \ dm0 be a
domain associated with a chain of cross-cuts in the prime end P∗. Let
ζ0 ∈ C1 ∩ C2. Choose ρ0 > 0 such that S(ζ0, ρ0) ∩ f(d0) ̸= ∅ and
S(ζ0, ρ0) ∩ f(d∗) ̸= ∅.
Set Γ = ∆(d0, d∗;D). Correspondingly to (6.2), by the principle of
minorization,
M(f(Γ)) <∞. (6.3)
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 39
Let M0 > M(f(Γ)) be a finite number. By the condition of the lemma,
∂D′ is weakly flat, and, hence, there is ρ∗ ∈ (0, ρ0) such that
M(∆(E,F ;D′)) > M0
for all continua E and F in D′ intersecting the circles S(ζ0, ρ0) and
S(ζ0, ρ∗). However, these circles can be joined by continuous curves c1
and c2 in the domains f(d0) and f(d∗), correspondingly, and, in partic-
ular, for these curves,
M0 6 M(∆(c1, c2;D
′)) 6 M(f(Γ)) . (6.4)
The obtained contradiction disproves the assumption that C0 ∩ C∗ ̸=
∅.
Theorem 6.1. Let D and D′ be bounded finitely connected domains in
C and f : D → D′ be a Qz0-homeomorphism at every point z0 ∈ ∂D
with Qz0 ∈ L1(D ∩ Uz0) for a neighborhood Uz0 of z0. Then f−1 can be
extended to a continuous mapping of D′
P onto DP .
Proof. By Remark 1.1, we may assume with no loss of generality that
D′ is a circular domain, D′
P = D′; C(ζ0, f
−1) ̸= ∅ for every ζ0 ∈ ∂D′
because DP is metrizable and compact. Moreover, C(ζ0, f
−1) ∩D = ∅,
see, e.g., Proposition 2.5 in [36] or Proposition 13.5 in [26].
Let us assume that there exist at least two different prime ends P1
and P2 in C(ζ0, f
−1). Then ζ0 ∈ C(P1, f) ∩ C(P2, f). Let z1 ∈ ∂D be a
point corresponding to P1 from Lemma 1.1. Note that
E = {r ∈ (0, δ) : Qz1 |D∩S(z1,r) ∈ L1(D ∩ S(z1, r))} (6.5)
has a positive linear measure for every δ > 0 by the Fubini theorem, see,
e.g., [48], because Qz1 ∈ L1(D ∩ Uz1). The obtained contradiction with
Lemma 6.1 shows that C(ζ0, f
−1) contains only one prime end of D.
Thus, we have the extension g of f−1 to D′ such that C(∂D′, f−1) ⊆
DP \D. In fact, C(∂D′, f−1) = DP \D. Indeed, if P0 is a prime end of D,
then there is a sequence zn in D being convergent to P0. We may assume
without loss of generality that zn → z0 ∈ ∂D and f(zn) → ζ0 ∈ ∂D′
because D and D′ are compact. Hence, P0 ∈ C(ζ0, f
−1).
Finally, let us show that the extended mapping g : D′ → DP is
continuous. Indeed, let ζn → ζ0 in D′. If ζ0 ∈ D′, then the statement is
obvious. If ζ0 ∈ ∂D′, then we take ζ∗n ∈ D′ such that |ζn− ζ∗n| < 1/n and
ρ(g(ζn), g(ζ
∗
n)) < 1/n, where ρ is one of the metrics in Remark 1.1. Note
that, by construction, g(ζ∗n) → g(ζ0) because ζ∗n → ζ0. Consequently,
g(ζn) → g(ζ0), too.
40 The Beltrami equations and prime ends
Theorem 6.2. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a Qz0-homeomorphism at every point z0 ∈ ∂D
with the condition
ε(z0)∫
0
dr
||Qz0 ||(r)
= ∞, (6.6)
where 0 < ε(z0) < d(z0) = sup
z∈D
|z − z0| and
||Qz0 ||(r) =
∫
D∩S(z0,r)
Qz0 ds. (6.7)
Then f−1 can be extended to a continuous mapping of D′
P onto DP .
Proof. Indeed, condition (6.6) implies that
δ∫
0
dr
||Qz0 ||(r)
= ∞ ∀ z0 ∈ ∂D ∀ δ ∈ (0, ε(z0)) (6.8)
because the left-hand side in (4.3) is not equal to zero, see Theorem 5.2
in [30], and hence, by Lemma 4.1,
ε(z0)∫
δ
dr
||Qz0 ||(r)
< ∞ .
Thus, the set
E = {r ∈ (0, δ) : Qz0 |D∩S(z0,r) ∈ L1(D ∩ S(z0, r))} (6.9)
has a positive linear measure for all z0 ∈ ∂D and all δ ∈ (0, ε(z0)). The
rest of arguments is perfectly similar to one in the proof of Theorem 6.1.
7. The homeomorphic extension
of ring Q-homeomorphisms
Combining Theorems 5.1 and 6.2, we arrive at Theorem 7.1, that is a
counterpart of the classic Carathéodory theorem on conformal mappings
of simply connected domains.
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 41
Theorem 7.1. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a ring Qz0-homeomorphism at every point
z0 ∈ ∂D. If
ε(z0)∫
0
dr
||Qz0 ||(r)
= ∞ ∀ z0 ∈ ∂D, (7.1)
where 0 < ε(z0) < d(z0) := supz∈D |z − z0| and
||Qz0 ||(r) :=
∫
D∩S(z0,r)
Qz0 ds (7.2)
then f can be extended to a homeomorphism of DP onto D′
P .
Corollary 7.1. In particular, the conclusion of Theorem 7.1 holds if
qz0(r) = O
(
log
1
r
)
∀ z0 ∈ ∂D (7.3)
as r → 0, where qz0(r) is the average of Qz0 over the circle |z − z0| = r.
Using Lemma 2.2 in [38], see also Lemma 7.4 in [26], and Theorem 7.1,
we get the following statement, which is a source of new criteria for the
homeomorphic extendability.
Lemma 7.1. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a ring Qz0-homeomorphism at every point
z0 ∈ ∂D, where Qz0 is integrable in a neighborhood of z0. Suppose that∫
D(z0,ε,ε0)
Qz0(z)·ψ2
z0,ε,ε0(|z−z0|) dm(z) = o
(
I2z0,ε0(ε)
)
as ε→ 0 ∀ z0 ∈ ∂D
(7.4)
where D(z0, ε, ε0) = {z ∈ D : ε < |z − z0| < ε0} for every small enough
0 < ε0 < d(z0) = supz∈D |z − z0| and where ψz0,ε,ε0(t) : (0,∞) → [0,∞],
ε ∈ (0, ε0), is a family of (Lebesgue) measurable functions such that
0 < Iz0,ε0(ε) :=
ε0∫
ε
ψz0,ε,ε0(t) dt < ∞ ∀ ε ∈ (0, ε0) .
Then f can be extended to a homeomorphism of DP onto D′
P .
Remark 7.1. In fact, instead of the integrability of Qz0 in a neigh-
borhood of z0, it is sufficient to request that Qz0 be integrable over
D ∩ S(z0, r) for a.e. r ∈ (0, ε0).
42 The Beltrami equations and prime ends
Note that Theorem 7.1 can be deduced also from Lemma 7.1, as it
follows from the proof of Theorem 5.1. Thus, Theorem 7.1 is equivalent
to Lemma 7.1 under the given conditions.
Finally, note that (7.4) holds, in particular, if∫
D(z0,ε0)
Qz0(z) · ψ2(|z − z0|) dm(z) < ∞ ∀ z0 ∈ ∂D, (7.5)
where D(z0, ε0) = {z ∈ D : |z − z0| < ε0} and where ψ(t) : (0,∞) →
[0,∞] is a locally integrable function such that Iz0,ε0(ε) → ∞ as ε → 0.
In other words, for the extendability of f to a homeomorphism of DP
onto D′
P , it suffices for the integrals in (7.5) to be convergent for some
nonnegative function ψ(t) that is locally integrable on (0,∞) but has a
non-integrable singularity at zero.
Choosing ψ(t) := 1
t log 1/t in Lemma 7.1 and applying Lemma 3.1, we
obtain the following proposition.
Theorem 7.2. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a ring Qz0-homeomorphism at every point
z0 ∈ ∂D, where Qz0 has finite mean oscillation at z0. Then f can be
extended to a homeomorphism of DP onto D′
P .
Corollary 7.2. In particular, the conclusion of Theorem 7.2 holds if
lim
ε→0
∫
−
B(z0,ε)
Qz0(z) dm(z) < ∞ ∀ z0 ∈ ∂D. (7.6)
Corollary 7.3. The conclusion of Theorem 7.2 holds if every point z0 ∈
∂D is a Lebesgue point of the function Qz0.
The next statement also follows from Lemma 7.1 under the choice
ψ(t) = 1/t.
Theorem 7.3. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a ring Qz0-homeomorphism at every point
z0 ∈ ∂D. If, for some ε0 = ε(z0) > 0,∫
ε<|z−z0|<ε0
Qz0(z)
dm(z)
|z − z0|2
= o
([
log
1
ε
]2)
as ε→ 0 ∀ z0 ∈ ∂D,
(7.7)
then f can be extended to a homeomorphism of DP onto D′
P .
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 43
Remark 7.2. Choosing the function ψ(t) = 1/(t log 1/t) instead of ψ(t)
= 1/t in Lemma 7.1, (7.7) can be replaced by the more weak condition
∫
ε<|z−z0|<ε0
Qz0(z) dm(z)(
|z − z0| log 1
|z−z0|
)2 = o
([
log log
1
ε
]2)
(7.8)
and (7.3) by the condition in terms of iterated logarithms
qz0(r) = o
(
log
1
r
log log
1
r
)
. (7.9)
Of course, we could give here the whole scale of the corresponding con-
dition of logarithmic type, using suitable functions ψ(t).
Theorem 7.1 has a lot of other fine consequences, for instance:
Theorem 7.4. Let D and D′ be bounded finitely connected domains in C
and let f : D → D′ be a ring Qz0-homeomorphism at every point z0 ∈ ∂D
and ∫
D∩B(z0,ε0)
Φz0 (Qz0(z)) dm(z) < ∞ ∀ z0 ∈ ∂D (7.10)
for ε0 = ε(z0) > 0 and a nondecreasing convex function Φz0 : [0,∞) →
[0,∞) with
∞∫
δ(z0)
dτ
τΦ−1
z0 (τ)
= ∞ (7.11)
for δ(z0) > Φz0(0). Then f is extended to a homeomorphism of DP onto
D′
P .
Indeed, by Theorem 3.1 and Corollary 3.2 in [46], (7.10) and (7.11) im-
ply (7.1) and, thus, Theorem 7.4 is a direct consequence of Theorem 7.1.
Corollary 7.4. In particular, the conclusion of Theorem 7.4 holds if∫
D∩B(z0,ε0)
eα0Qz0 (z) dm(z) < ∞ ∀ z0 ∈ ∂D (7.12)
for some ε0 = ε(z0) > 0 and α0 = α(z0) > 0.
44 The Beltrami equations and prime ends
Remark 7.3. By Theorem 2.1 in [46], see also Proposition 2.3 in [39],
(7.11) is equivalent to each of the conditions from the following series:
∞∫
δ(z0)
H ′
z0(t)
dt
t
= ∞ , δ(z0) > 0 , (7.13)
∞∫
δ(z0)
dHz0(t)
t
= ∞ , δ(z0) > 0 , (7.14)
∞∫
δ(z0)
Hz0(t)
dt
t2
= ∞ , δ(z0) > 0 , (7.15)
∆(z0)∫
0
Hz0
(
1
t
)
dt = ∞ , ∆(z0) > 0 , (7.16)
∞∫
δ∗(z0)
dη
H−1
z0 (η)
= ∞ , δ∗(z0) > Hz0(0) , (7.17)
where
Hz0(t) = log Φz0(t) . (7.18)
Here the integral in (7.14) is understood as the Lebesgue–Stieltjes
integral and the integrals in (7.13) and (7.15)–(7.17) as the ordinary
Lebesgue integrals.
It is necessary to give one more explanation. From the right-hand
sides in conditions (7.13)–(7.17), we have in mind +∞. If Φz0(t) = 0
for t ∈ [0, t∗(z0)], then Hz0(t) = −∞ for t ∈ [0, t∗(z0)] and we complete
the definition H ′
z0(t) = 0 for t ∈ [0, t∗(z0)]. Note that conditions (7.14)
and (7.15) exclude that t∗(z0) belongs to the interval of integrability
because, in the contrary case, the left-hand sides in (7.14) and (7.15)
are either equal to −∞ or indeterminate. Hence, we may assume in
(7.13)–(7.16) that δ(z0) > t0, correspondingly, ∆(z0) < 1/t(z0), where
t(z0) := supΦz0 (t)=0 t, set t(z0) = 0 if Φz0(0) > 0.
The most interesting is condition (7.15) that can be written in the
form:
∞∫
δ(z0)
log Φz0(t)
dt
t2
= ∞ . (7.19)
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 45
Note also that, under every homeomorphism f between domains D
and D′ in C, there is a natural one-to-one correspondence between com-
ponents of their boundaries ∂D and ∂D′, see, e.g., Lemma 5.3 in [14]
or Lemma 6.5 in [26]. Thus, if a bounded domain D in C is finitely
connected and D′ is bounded, then D′ is finitely connected, too.
Finally, note that if a domain D in C is locally connected on its
boundary, then there is a natural one-to-one correspondence between
prime ends of D and boundary points of D. Thus, if D and D′ are,
in addition, locally connected on their boundaries in the theorems of
Section 7, then f is extended to a homeomorphism of D onto D′. We
obtained earlier similar results, when ∂D′ is weakly flat, which is a more
strong condition than that of local connectivity of D′ on its boundary,
see, e.g., [17, 18].
As known, every Jordan domain D in C is locally connected on its
boundary, see, e.g., [53, p. 66]. It is easy to see that the latter implies
that every bounded finitely connected domain D in C whose boundary
consists of mutually disjoint Jordan curves and isolated points is also
locally connected on its boundary.
Conversely, every bounded finitely connected domain D in C that is
locally connected on its boundary has a boundary consisting of mutually
disjoint Jordan curves and isolated points. Indeed, every such a domain
D can be mapped by a conformal mapping f onto the so-called circular
domain D∗ bounded by a finite collection of mutually disjoint circles and
isolated points, see, e.g., Theorem V.6.2 in [9], that is extendable to a
homeomorphism of D onto D∗, see Remark 1.1.
8. The boundary behavior of homeomorphic solutions
This section is devoted to the study of the boundary behavior of
solutions to the Beltrami equations. Combining Theorem 4.1 with the
corresponding results in Sections 6 and 7, we obtain the statements given
below.
Theorem 8.1. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a homeomorphic W 1,1
loc solution of (1.1) with
KT
µ (·, z0) ∈ L1(D ∩ B(z0, ε0)) for every z0 ∈ ∂D. Then f−1 is extended
to a continuous mapping of D′
P onto DP .
Note that any degree of integrability of Kµ does not guarantee a
continuous extendability of the direct mapping f to the boundary, see,
e.g., an example in the proof of Proposition 6.3 in [26]. Conditions for
the continuous extendability have perfectly another nature.
46 The Beltrami equations and prime ends
Theorem 8.2. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a homeomorphic W 1,1
loc solution of the Beltrami
equation (1.1) with the condition
ε0∫
0
dr
||KT
µ ||(z0, r)
= ∞ ∀ z0 ∈ ∂D, (8.1)
where 0 < ε0 = ε(z0) < d(z0) := supz∈D |z − z0| and
||KT
µ ||(z0, r) =
∫
S(z0,r)
KT
µ (z, z0) ds . (8.2)
Then f can be extended to a homeomorphism of DP onto D′
P .
Here and later on, we set that KT
µ is equal to zero outside of the
domain D.
Corollary 8.1. In particular, the conclusion of Theorem 8.2 holds if
kTz0(r) = O
(
log
1
r
)
∀ z0 ∈ ∂D (8.3)
as r → 0 where kTz0(r) is the average of K
T
µ (z, z0) over the circle |z−z0| =
r.
Lemma 8.1. Let D and D′ be bounded finitely connected domains in C
and let f : D → D′ be a homeomorphic W 1,1
loc solution of the Beltrami
equation (1.1) with Kµ ∈ L1(D) and∫
ε<|z−z0|<ε0
KT
µ (z, z0) · ψ2
z0,ε(|z − z0|) dm(z) = o
(
I2z0(ε)
)
∀ z0 ∈ ∂D
(8.4)
as ε→ 0, where 0 < ε0 < supz∈D |z − z0| and ψz0,ε(t) : (0,∞) → [0,∞],
ε ∈ (0, ε0), is a two-parametric family of measurable functions such that
0 < Iz0(ε) :=
ε0∫
ε
ψz0,ε(t) dt < ∞ ∀ ε ∈ (0, ε0) .
Then f can be extended to a homeomorphism of DP onto D′
P .
Theorem 8.3. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a homeomorphic W 1,1
loc solution of the Beltrami
equation (1.1) with KT
µ (z, z0) of finite mean oscillation at every point
z0 ∈ ∂D. Then f can be extended to a homeomorphism of DP onto D′
P .
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 47
Theorem 8.3 remains valid also if the function KT
µ (z, z0) has a domi-
nant of finite mean oscillation in a neighborhood of every point z0 ∈ ∂D.
Corollary 8.2. In particular, the conclusion of Theorem 8.3 holds if
lim
ε→0
∫
−
B(z0,ε)
KT
µ (z, z0) dm(z) < ∞ ∀ z0 ∈ ∂D . (8.5)
Theorem 8.4. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a homeomorphic W 1,1
loc solution of the Beltrami
equation (1.1) with the condition∫
ε<|z−z0|<ε0
KT
µ (z, z0)
dm(z)
|z − z0|2
= o
([
log
1
ε
]2)
∀ z0 ∈ ∂D . (8.6)
Then f can be extended to a homeomorphism of DP onto D′
P .
Remark 8.1. Condition (8.6) can be replaced by the weaker condition∫
ε<|z−z0|<ε0
KT
µ (z, z0) dm(z)(
|z − z0| log 1
|z−z0|
)2 = o
([
log log
1
ε
]2)
∀ z0 ∈ ∂D .
(8.7)
In general, here we are able to give a number of other conditions of
logarithmic type. In particular, condition (8.3) can be replaced, due to
Theorem 8.2, by the weaker condition
kTz0(r) = O
(
log
1
r
log log
1
r
)
. (8.8)
Finally, we complete the series of criteria with the following integral
condition.
Theorem 8.5. Let D and D′ be bounded finitely connected domains in
C and let f : D → D′ be a homeomorphic W 1,1
loc solution of the Beltrami
equation (1.1) with the condition∫
D∩B(z0,ε0)
Φz0
(
KT
µ (z, z0)
)
dm(z) < ∞ ∀ z0 ∈ ∂D (8.9)
for ε0 = ε(z0) > 0 and a nondecreasing convex function Φz0 : [0,∞) →
[0,∞) with
∞∫
δ0
dτ
τΦ−1
z0 (τ)
= ∞ (8.10)
48 The Beltrami equations and prime ends
for δ0 = δ(z0) > Φz0(0). Then f is extended to a homeomorphism of DP
onto D′
P .
Corollary 8.3. In particular, the conclusion of Theorem 8.5 holds if∫
D∩B(z0,ε0)
eα0KT
µ (z,z0) dm(z) < ∞ ∀ z0 ∈ ∂D (8.11)
for some ε0 = ε(z0) > 0 and α0 = α(z0) > 0.
Remark 8.2. Note that condition (8.10) is not only sufficient but also
necessary for a continuous extension to the boundary of all direct map-
pings f with integral restrictions of type (8.9), see, e.g., Theorem 5.1
and Remark 5.1 in [22]. In other words, given Φz0 which does not satisfy
(8.10), one can find a homeomorphicW 1,1
loc solution of (1.1) with condition
(8.9) that is not extended to a homeomorphism of DP onto D′
P .
Recall also that condition (8.10) is equivalent to each of conditions
(7.13)–(7.17).
9. Regular solutions for the Dirichlet problem
Recall that a mapping f : D → C is called discrete if the pre-image
f−1(y) of every point y ∈ C consists of isolated points and open if the
image of every open set U ⊆ D is open in C.
Given φ(P ) ̸≡ const, P ∈ ED, we will say that f is a regular solution
of the Dirichlet problem (1.6) for the Beltrami equation (1.1) if f is a
continuous discrete open mapping f : D → C of the Sobolev class W 1,1
loc
with the Jacobian
Jf (z) = |fz|2 − |fz|2 ̸= 0 a.e., (9.1)
satisfying (1.1) a.e. and the boundary condition (1.6) for all prime ends
of the domain D. For φ(P ) ≡ c ∈ R, P ∈ ED, a regular solution of the
problem is any constant function f(z) = c+ ic′, c′ ∈ R.
Theorem 9.1. Let D be a bounded simply connected domain in C and
let µ : D → D be a measurable function with Kµ ∈ L1
loc and such that
δ(z0)∫
0
dr
||KT
µ ||(z0, r)
= ∞ ∀ z0 ∈ D (9.2)
for some 0 < δ(z0) < d(z0) = sup
z∈D
|z − z0|, where
||KT
µ ||(z0, r) :=
∫
S(z0, r)
KT
µ (z, z0) ds .
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 49
Then the Beltrami equation (1.1) has a regular solution f of the Dirichlet
problem (1.6) for every continuous function φ : ED → R.
Here and later on, we set that KT
µ is equal to zero outside of the
domain D.
Corollary 9.1. Let D be a bounded simply connected domain in C and
let µ : D → D be a measurable function such that
kTz0(ε) = O
(
log
1
ε
)
as ε→ 0 ∀ z0 ∈ D, (9.3)
where kTz0(ε) is the average of the function KT
µ (z, z0) over the circle
S(z0, ε).
Then the Beltrami equation (1.1) has a regular solution f of the
Dirichlet problem (1.6) for every continuous function φ : ED → R.
Remark 9.1. In particular, the conclusion of Theorem 9.1 holds if
KT
µ (z, z0) = O
(
log
1
|z − z0|
)
as z → z0 ∀ z0 ∈ D . (9.4)
Proof of Theorem 9.1. Note that ED cannot consist of a single prime
end. Indeed, all rays going from a point z0 ∈ D to ∞ intersect ∂D
because the domain D is bounded, see, e.g., Proposition 2.3 in [36] or
Proposition 13.3 in [26]. Thus, ∂D contains more than one point and, by
the Riemann theorem, see, e.g., II.2.1 in [9], D can be mapped onto the
unit disk D with a conformal mapping R. However, then there is one-to-
one correspondence between elements of ED and points of the unit circle
∂D by the Carathéodory theorem, see, e.g., Theorem 9.6 in [5].
Let F be a regular homeomorphic solution of Eq. (1.1) in the class
W 1,1
loc which exists in view of condition (9.2), see, e.g., Theorem 5.4 in
work [41] or Theorem 11.10 in book [26].
Note that the domain D∗ = F (D) is simply connected in C, see, e.g.,
Lemma 5.3 in [14] or Lemma 6.5 in [26]. Let us assume that ∂D∗ in C
consists of the single point ∞. Then C \ D∗ also consists of the single
point ∞, i.e., D∗ = C. Indeed, if there is a point ζ0 ∈ C in C \D∗, then,
joining it and any point ζ∗ ∈ D∗ with a segment of a straight line, we
find one more point of ∂D∗ in C, see, e.g., again Proposition 2.3 in [36]
or Proposition 13.3 in [26]. Now, let D∗ denote the exterior of the unit
disk D in C and let κ(ζ) = 1/ζ, κ(0) = ∞, κ(∞) = 0. Consider the
mapping F∗ = κ ◦ F : D̃ → D0, where D̃ = F−1(D∗) and D0 = D \ {0}
is the punctured unit disk. It is clear that F∗ is also a regular homeo-
morphic solution of the Beltrami equation (1.1) in the class W 1,1
loc in the
50 The Beltrami equations and prime ends
bounded two-connected domain D̃ because the mapping κ is conformal.
By Theorem 8.2, there is a one-to-one correspondence between elements
of ED and 0. However, it was shown above that ED cannot consists of
a single prime end. This contradiction disproves the above assumption
that ∂D∗ consists of a single point in C.
Thus, by the Riemann theorem, D∗ can be mapped onto the unit
disk D with a conformal mapping R∗. Note that the function g := R∗ ◦F
is again a regular homeomorphic solution in the Sobolev class W 1,1
loc of
the Beltrami equation (1.1) which maps D onto D. By Theorem 8.2, the
mapping g admits an extension to a homeomorphism g∗ : DP → D.
We find a regular solution of the initial Dirichlet problem (1.6) in the
form f = h◦g, where h is a holomorphic function in D with the boundary
condition
lim
z→ζ
Reh(z) = φ(g−1
∗ (ζ)) ∀ ζ ∈ ∂D .
Note that we have from the right-hand side a continuous function of the
variable ζ.
As known, the analytic function h can be reconstructed in D through
its real part on the boundary up to a pure imaginary additive constant
with the Schwartz formula, see, e.g., § 8, Chapter III, Part 3 in [13],
h(z) =
1
2πi
∫
|ζ|=1
φ ◦ g−1
∗ (ζ) · ζ + z
ζ − z
· dζ
ζ
.
It is easy to see that the function f = h ◦ g is a desired regular solution
of the Dirichlet problem (1.6) for the Beltrami equation (1.1).
Applying Lemma 2.2 in [38], see also Lemma 7.4 in [26], we obtain
the following general lemma immediately from Theorem 9.1.
Lemma 9.1. Let D be a bounded simply connected domain in C and let
µ : D → D be a measurable function with Kµ ∈ L1(D). Suppose that, for
every z0 ∈ D and every small enough ε0 < d(z0) := supz∈D |z−z0|, there
is a family of measurable functions ψz0, ε,ε0 : (0,∞) → [0,∞], ε ∈ (0, ε0)
such that
0 < Iz0,ε0(ε) :=
ε0∫
ε
ψz0, ε,ε0(t) dt < ∞ ∀ ε ∈ (0, ε0) (9.5)
and∫
D(z0, ε, ε0)
KT
µ (z, z0) · ψ2
z0, ε,ε0 (|z − z0|) dm(z) = o(I2z0,ε0(ε)) as ε→ 0,
(9.6)
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 51
where D(z0, ε, ε0) = {z ∈ D : ε < |z − z0| < ε0}. Then the Beltrami
equation (1.1) has a regular solution f of the Dirichlet problem (1.6) for
every continuous function φ : ED → R.
Remark 9.2. In fact, it is sufficient here to request instead of the con-
dition Kµ ∈ L1(D) only a local integrability of Kµ in the domain D and
the condition ||Kµ||(z0, r) ̸= ∞ for a.e. r ∈ (0, ε0) at all z0 ∈ ∂D.
By Lemma 9.1 with the choice ψz0, ε(t) ≡ 1/
(
t log 1
t
)
, we obtain the
following result, see also Lemma 3.1.
Theorem 9.2. Let D be a bounded simply connected domain in C and
let µ : D → D be a measurable function with KT
µ ∈ L1
loc and
KT
µ (z, z0) 6 Qz0(z) ∈ FMO(z0) ∀ z0 ∈ D . (9.7)
Then the Beltrami equation (1.1) has a regular solution f of the Dirichlet
problem (1.6) for every continuous function φ : ED → R.
Remark 9.3. In particular, the hypotheses and the conclusion of The-
orem 9.2 hold if either Qz0 ∈ BMOloc or Qz0 ∈ W1,2
loc because W 1,2
loc ⊂
VMOloc, see, e.g., [3].
By Corollary 3.1, we obtain the following statement from Theorem 9.2.
Corollary 9.2. Let D be a bounded simply connected domain in C and
let µ : D → D be a measurable function with Kµ ∈ L1
loc such that
lim sup
ε→0
∫
−
B(z0, ε)
KT
µ (z, z0) dm(z) < ∞ ∀ z0 ∈ D . (9.8)
Then the Beltrami equation (1.1) has a regular solution f of the Dirichlet
problem (1.6) for every continuous function φ : ED → R.
Remark 9.4. In particular, by (1.5), the conclusion of Theorem 9.2
holds if
Kµ(z) 6 Q(z) ∈ BMO(D) . (9.9)
The next statement follows from Lemma 9.1 under the choice ψ(t) =
1/t, see also Remark 9.2.
Theorem 9.3. Let D be a bounded simply connected domain in C and
let µ : D → D be a measurable function such that∫
ε<|z−z0|<ε0
KT
µ (z, z0)
dm(z)
|z − z0|2
= o
([
log
1
ε
]2)
∀ z0 ∈ D . (9.10)
Then the Beltrami equation (1.1) has a regular solution f of the Dirichlet
problem (1.6) for every continuous function φ : ED → R.
52 The Beltrami equations and prime ends
Remark 9.5. Similarly, choosing ψ(t) = 1/(t log 1/t) instead of ψ(t) =
1/t in Lemma 9.1, we obtain that condition (9.10) can be replaced by
the condition∫
ε<|z−z0|<ε0
KT
µ (z, z0) dm(z)(
|z − z0| log 1
|z−z0|
)2 = o
([
log log
1
ε
]2)
∀ z0 ∈ D .
(9.11)
Here we are able to give a number of other conditions of logarithmic
type. In particular, condition (9.3), due to Theorem 9.1, can be replaced
by the weaker condition
kTz0(r) = O
(
log
1
r
log log
1
r
)
. (9.12)
Finally, by Theorem 9.1, applying also Theorem 3.1 in [46], we come
to the following result.
Theorem 9.4. Let D be a bounded simply connected domain in C and
let µ : D → D be a measurable function with Kµ ∈ L1
loc and∫
D∩B(z0,ε0)
Φz0(K
T
µ (z, z0)) dm(z) < ∞ ∀ z0 ∈ D (9.13)
for ε0 = ε(z0) > 0 and a nondecreasing convex function Φz0 : [0,∞) →
[0,∞) with
∞∫
δ0
dτ
τΦ−1
z0 (τ)
= ∞ (9.14)
for δ0 = δ(z0) > Φz0(0). Then the Beltrami equation (1.1) has a regular
solution f of the Dirichlet problem (1.6) for every continuous function
φ : ED → R.
Remark 9.6. Recall that condition (9.14) is equivalent to each of condi-
tions (7.13)–(7.17). Moreover, condition (9.14) is not only sufficient but
also necessary to have a regular solution of the Dirichlet problem (1.6)
for every Beltrami equation (1.1) with the integral restriction (9.13) for
every continuous function φ : ED → R. Indeed, by the Stoilow theorem
on representation of discrete open mappings, see, e.g., [50], every regular
solution f of the Dirichlet problem (1.6) for the Beltrami equation (1.1)
with Kµ ∈ L1
loc can be represented in the form of composition f = h ◦F,
where h is a holomorphic function and F is a regular homeomorphic
solution of (1.1) in the class W 1,1
loc . Thus, by Theorem 5.1 in [47] on
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 53
the nonexistence of regular homeomorphic solutions of (1.1) in the class
W 1,1
loc , if (9.14) fails, then there is a measurable function µ : D → D
satisfying integral condition (9.13) for which Beltrami equation (1.1) has
no regular solution of the Dirichlet problem (1.6) for any nonconstant
continuous function φ : ED → R.
Corollary 9.3. Let D be a bounded simply connected domain in C and
let µ : D → D be a measurable function with Kµ ∈ L1
loc and∫
D∩B(z0,ε0)
eα0KT
µ (z,z0) dm(z) < ∞ ∀ z0 ∈ D (9.15)
for some ε0 = ε(z0) > 0 and α0 = α(z0) > 0. Then the Beltrami
equation (1.1) has a regular solution f of the Dirichlet problem (1.6) for
every continuous function φ : ED → R.
10. Pseudoregular solutions in multiply
connected domains
As it was probably first noted by B. Bojarski, see, e.g., §6 of Chap-
ter 4 in [52], the Dirichlet problem for the Beltrami equations, generally
speaking, has no regular solution in the class of functions continuous
(single-valued) in C with generalized derivatives in the case of multiply
connected domains D. Hence, the natural question arises: whether solu-
tions exist in wider classes of functions for this case? It is turned out that
solutions for this problem can be found in the class of functions admitting
a certain number (related to connectedness of D) of poles at prescribed
points. Later on, this number will take into account the multiplicity of
these poles from the Stoilow representation.
A discrete open mapping f : D → C of the Sobolev class W 1,1
loc (out-
side of poles) satisfying (1.1) a.e. and the boundary condition (1.6) is
called the pseudoregular solution of the Dirichlet problem if the Jaco-
bian Jf (z) ̸= 0 a.e.
Arguing similarly to the case of simply connected domains and ap-
plying Theorem V.6.2 in [9] on conformal mappings of finitely connected
domains onto circular domains and also Theorems 4.13 and 4.14 in [52],
we obtain the following result.
Theorem 10.1. Let D be a bounded m−connected domain in C with
nondegenerate boundary components, k > m− 1, and let µ : D → D be a
54 The Beltrami equations and prime ends
measurable function with Kµ ∈ L1
loc and
δ(z0)∫
0
dr
||KT
µ ||(z0, r)
= ∞ ∀ z0 ∈ D (10.1)
for some 0 < δ(z0) < d(z0) = supz∈D |z − z0| and
||KT
µ ||(z0, r) :=
∫
S(z0, r)
KT
µ (z, z0) ds .
Then the Beltrami equation (1.1) has a pseudoregular solution f of the
Dirichlet problem (1.6) with k poles at prescribed points in D for every
continuous function φ : ED → R.
Here, as before, we set KT
µ to be extended by zero outside of the
domain D.
Corollary 10.1. Let D be a bounded m−connected domain in C with
nondegenerate boundary components, k > m− 1, and let µ : D → D be a
measurable function with KT
µ ∈ L1
loc and
kTz0(ε) = O
(
log
1
ε
)
as ε→ 0 ∀ z0 ∈ D, (10.2)
where kTz0(ε) is the average of the function KT
µ (z, z0) over the circle
S(z0, ε).
Then the Beltrami equation (1.1) has a pseudoregular solution f of
the Dirichlet problem (1.6) with k poles at prescribed points in D for
every continuous function φ : ED → R.
Remark 10.1. In particular, the conclusion of Theorem 10.1 holds if
KT
µ (z, z0) = O
(
log
1
|z − z0|
)
as z → z0 ∀ z0 ∈ D . (10.3)
Proof of Theorem 10.1. Let F be a regular solution of Eq. (1.1) in the
class W 1,1
loc that exists by condition (10.1), see, e.g., Theorem 5.4 in work
[41] or Theorem 11.10 in book [26]. Note that the domain D∗ = F (D)
is m-connected in C and there is a natural one-to-one correspondence
between components γj of γ = ∂D and components Γj of Γ = ∂D∗,
Γj = C(γj , F ) and γj = C(Γj , F
−1), j = 1, . . . ,m, see, e.g., Lemma 5.3
in [14] or Lemma 6.5 in [26]. Moreover, by Remark 1.1, every subspace
Ej of ED associated with γj consists of more than one prime end, even
it is homeomorphic to the unit circle.
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 55
Next, no one of Γj , j = 1, . . . ,m, is degenerated to a single point.
Indeed, let us assume that Γj0 = {ζ0} first for some ζ0 ∈ C. Let r0 ∈
(0, d0), where d0 = infζ∈Γ\Γj0
|ζ − ζ0|. Then the punctured disk D0 =
{ζ ∈ C : 0 < |ζ − ζ0| < r0} is in the domain D∗ and its boundary
does not intersect Γ \ Γj0 . Set D̃ = F−1(D0). Then, by construction,
D̃ ⊂ D is a 2-connected domain, D̃ ∩ γ \ γj0 = ∅, C(γj0 , F̃ ) = {ζ0}
and C(ζ0, F̃
−1) = γj0 , where F̃ is a restriction of the mapping F̃ to D̃.
However, this contradicts Theorem 8.2 because, as was noted above, Ej0
contains more than one prime end.
Now, let assume that Γj0 = {∞}. Then the component of C \ D∗
associated with Γj0 , see Lemma 5.1 in [14] or Lemma 6.3 in [26], is also
consists of the single point ∞ because if the interior of this component is
not empty, then, choosing there an arbitrary point ζ0 and joining it with
a point ζ∗ ∈ D∗ by a segment of a straight line, we would find one more
point in Γj0 , see, e.g., Proposition 2.3 in [36] or Proposition 13.3 in [26].
Thus, applying, if necessary, an additional stretching (conformal map-
ping), we may assume with no loss of generality that D∗ contains the ex-
teriority D∗ of the unit disk D in C. Set κ(ζ) = 1/ζ, κ(0) = ∞, κ(∞) = 0.
Consider the mapping F∗ = κ ◦ F : D̃ → D0, where D̃ = F−1(D∗) and
D0 = D \ {0} is the punctured unit disk. It is clear that F∗ is also a
homeomorphic solution of the Beltrami equation (1.1) of the class W 1,1
loc
in a 2-connected domain D̃ because the mapping κ is conformal. Con-
sequently, by Theorem 8.2, elements of Ej0 should be in a one-to-one
correspondence with 0. However, it was already noted, Ej0 cannot con-
sists of a single prime end. The obtained contradiction disproves the
assumption that Γj0 = {∞}.
Thus, by Theorem V.6.2 in [9], see also Remark 1.1 in [19], D∗ can be
mapped with a conformal mapping R∗ onto a bounded circular domain
D∗ whose boundary consists of mutually disjoint circles. Note that the
function g := R∗ ◦ F is again a regular homeomorphic solution in the
Sobolev classW 1,1
loc for the Beltrami equation (1.1) that maps D onto D∗.
By Theorem 8.2, the mapping g admits an extension to a homeomor-
phism g∗ : DP → D∗.
Let us find a solution of the initial Dirichlet problem (1.6) in the form
f = h ◦ g, where h is a meromorphic function in D∗ with the boundary
condition
lim
z→ζ
Reh(z) = φ(g−1
∗ (ζ)) ∀ ζ ∈ ∂D∗ (10.4)
and k > m − 1 poles corresponding under the mapping g to those at
prescribed points in D. Note that the function from the right-hand
side in (10.4) is continuous in the variable ζ. Thus, such a function h
56 The Beltrami equations and prime ends
exists by Theorems 4.13 and 4.14 in [52]. It is clear that the function f
associated with h is, by construction, a desired pseudoregular solution of
the Dirichlet problem (1.6) for the Beltrami equation (1.1).
Applying Lemma 2.2 in [38], see also Lemma 7.4 in [26], we obtain
immediately the next lemma from Theorem 10.1.
Lemma 10.1. Let D be a bounded m−connected domain in C with non-
degenerate boundary components, k > m − 1, and let µ : D → D be a
measurable function with Kµ ∈ L1(D). Suppose that, for every z0 ∈ D
and every small enough 0 < ε0 < d(z0) := supz∈D |z−z0|, there is a fam-
ily of measurable functions ψz0, ε,ε0 : (0,∞) → [0,∞], ε ∈ (0, ε0) such
that
0 < Iz0,ε0(ε) :=
ε0∫
ε
ψz0, ε,ε0(t) dt < ∞ ∀ ε ∈ (0, ε0) (10.5)
and∫
ε<|z−z0|<ε0
KT
µ (z, z0)·ψ2
z0, ε,ε0 (|z − z0|) dm(z) = o(I2z0,ε0(ε)) as ε→ 0.
(10.6)
Then the Beltrami equation (1.1) has a pseudoregular solution f of the
Dirichlet problem (1.6) with k poles at prescribed points in D for every
continuous function φ : ED → R.
Remark 10.2. In fact, it is sufficient to assume the local integrability
of Kµ in the domain D and the condition ||Kµ||(z0, r) ̸= ∞ for a.e.
r ∈ (0, ε0) and all z0 ∈ ∂D instead of the condition Kµ ∈ L1(D).
By Lemma 10.1 with the choice ψz0, ε(t) ≡ 1/t log 1
t , we obtain the
following result, see also Lemma 3.1.
Theorem 10.2. Let D be a bounded m−connected domain in C with
nondegenerate boundary components, k > m− 1, and let µ : D → D be a
measurable function with Kµ ∈ L1(D) such that
KT
µ (z, z0) 6 Qz0(z) ∈ FMO(z0) ∀ z0 ∈ D . (10.7)
Then the Beltrami equation (1.1) has a pseudoregular solution f of the
Dirichlet problem (1.6) with k poles at points in D for every continuous
function φ : ED → R.
Remark 10.3. In particular, the conclusion of Theorem 10.2 holds if
either Qz0 ∈ BMOloc or Qz0 ∈ W1,2
loc because W 1,2
loc ⊂ VMOloc, see, e.g.,
[3].
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 57
By Corollary 3.1, we have the next consequence of Theorem 10.2:
Corollary 10.2. Let D be a bounded m−connected domain in C with
nondegenerate boundary components, k > m− 1, and let µ : D → D be a
measurable function with Kµ ∈ L1(D) such that
lim sup
ε→0
∫
−
B(z0, ε)
KT
µ (z, z0) dm(z) < ∞ ∀ z0 ∈ D . (10.8)
Then the Beltrami equation (1.1) has a pseudoregular solution f of the
Dirichlet problem (1.6) with k poles at prescribed points in D for every
continuous function φ : ED → R.
Remark 10.4. In particular, by (1.5), the conclusion of Theorem 10.2
holds if
Kµ(z) 6 Q(z) ∈ BMO(D). (10.9)
The following statement follows from Lemma 10.1 through the choice
ψ(t) = 1/t, see also Remark 10.2.
Theorem 10.3. Let D be a bounded m-connected domain in C with
nondegenerate boundary components, k > m− 1, and let µ : D → D be a
measurable function such that∫
ε<|z−z0|<ε0
KT
µ (z, z0)
dm(z)
|z − z0|2
= o
([
log
1
ε
]2)
∀ z0 ∈ D . (10.10)
Then the Beltrami equation (1.1) has a pseudoregular solution f of the
Dirichlet problem (1.6) with k poles at prescribed points in D for every
continuous function φ : ED → R.
Remark 10.5. Similarly, choosing ψ(t) = 1/(t log 1/t) instead of ψ(t) =
1/t in Lemma 10.1, we obtain that condition (10.10) can be replaced by
the condition∫
ε<|z−z0|<ε0
KT
µ (z, z0) dm(z)(
|z − z0| log 1
|z−z0|
)2 = o
([
log log
1
ε
]2)
∀ z0 ∈ D .
(10.11)
Here we are able to give a number of other conditions of logarithmic type.
In particular, condition (10.2), due to Theorem 10.1, can be replaced by
the weaker condition
kTz0(r) = O
(
log
1
r
log log
1
r
)
. (10.12)
58 The Beltrami equations and prime ends
Finally, by Theorem 10.1, applying also Theorem 3.1 in work [46], we
come to the following result.
Theorem 10.4. Let D be a bounded m−connected domain in C with
nondegenerate boundary components, k > m− 1, and let µ : D → D be a
measurable function with Kµ ∈ L1
loc such that∫
D∩B(z0,ε0)
Φz0(K
T
µ (z, z0)) dm(z) < ∞ (10.13)
for ε0 = ε(z0) > 0 and a nondecreasing convex function Φz0 : [0,∞) →
[0,∞) with
∞∫
δ0
dτ
τΦ−1
z0 (τ)
= ∞ (10.14)
for δ0 = δ(z0) > Φz0(0). Then the Beltrami equation (1.1) has a pseu-
doregular solution f of the Dirichlet problem (1.6) with k poles at pre-
scribed points in D for every continuous function φ : ED → R.
Recall that condition (10.14) is equivalent to every of conditions
(7.13)–(7.17).
Corollary 10.3. Let D be a bounded m−connected domain in C with
nondegenerate boundary components, k > m− 1, and let µ : D → D be a
measurable function with Kµ ∈ L1
loc such that∫
D∩B(z0,ε0)
eα0KT
µ (z,z0) dm(z) < ∞ ∀ z0 ∈ D (10.15)
for some ε0 = ε(z0) > 0 and α0 = α(z0) > 0.
Then the Beltrami equation (1.1) has a pseudoregular solution f of
the Dirichlet problem (1.6) with k poles at prescribed points in D for
every continuous function φ : ED → R.
11. Multivalent solutions in finitely connected domains
In finitely connected domains D in C, in addition to pseudoregular
solutions, the Dirichlet problem for the Beltrami equation (1.1) admits
multivalent solutions in the spirit of the theory of multivalent analytic
functions.
We say that a discrete open mapping f : B(z0, ε0) → C, where
B(z0, ε0) ⊆ D, is a local regular solution of Eq. (1.1) if f ∈ W 1,1
loc ,
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 59
Jf (z) ̸= 0, and f satisfies (1.1) a.e. in B(z0, ε0). The local regular solu-
tions f : B(z0, ε0) → C and f∗ : B(z∗, ε∗) → C of Eq. (1.1) will be called
extensions of each to other if there is a finite chain of such solutions fi :
B(zi, εi) → C, i = 1, . . . ,m, that f1 = f0, fm = f∗ and fi(z) ≡ fi+1(z)
for z ∈ Ei := B(zi, εi) ∩B(zi+1, εi+1) ̸= ∅, i = 1, . . . ,m− 1. A collection
of local regular solutions fj : B(zj , εj) → C, j ∈ J , will be called a mul-
tivalent solution of Eq. (1.1) in D if the disks B(zj , εj) cover the whole
domain D and fj are extensions of each to other through the collection
and the collection is maximal by inclusion. A multivalent solution of Eq.
(1.1) will be called a multivalent solution of the Dirichlet problem for a
prescribed continuous function φ : ED → R if u(z) = Re f(z) = Re fj(z),
z ∈ B(zj , εj), j ∈ J , is a single-valued function in D satisfying the con-
dition limz→P u(z) = φ(P ) along any ways in D going to P ∈ ED.
As above, we assume later on that KT
µ (·, z0) is extended by zero out-
side of the domain D.
The proof of the existence of multivalent solutions of the Dirichlet
problem (1.6) for the Beltrami equation (1.1) in finitely connected do-
mains is reduced on the basis of Section 8 to the Dirichlet problem for
harmonic functions in circular domains, see, e.g., §3 of Chapter VI in [9].
Theorem 11.1. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function with Kµ ∈ L1
loc and
δ(z0)∫
0
dr
||KT
µ ||(z0, r)
= ∞ ∀ z0 ∈ D (11.1)
for some 0 < δ(z0) < d(z0) = sup
z∈D
|z − z0| and
||KT
µ ||(z0, r) :=
∫
S(z0, r)
KT
µ (z, z0) ds .
Then the Beltrami equation (1.1) has a multivalent solution of the Dirich-
let problem (1.6) for every continuous function φ : ED → R.
Proof of Theorem 11.1. Similarly to the first part of Theorem 10.1, it is
proved that there is a regular homeomorphic solution g of the Beltrami
equation (1.1) mapping the domain D onto a circular domain D∗ whose
boundary consists of mutually disjoint circles. By Theorem 8.2, the
mapping g admits an extension to a homeomorphism g∗ : DP → D∗.
60 The Beltrami equations and prime ends
As known, in the circular domain D∗, there is a solution of the Dirich-
let problem
lim
z→ζ
u(z) = φ(g−1
∗ (ζ)) ∀ ζ ∈ ∂D∗ (11.2)
for harmonic functions u, see, e.g., §3 of Chapter VI in [9]. Let B0 =
B(z0, r0) be a disk in the domain D. Then B0 = g(B0) is a simply con-
nected subdomain of the circular domain D∗, where there is a conjugate
function v determined up to an additive constant such that h = u + iv
is a single-valued analytic function. The function h can be extended to,
generally speaking, a multivalent analytic function H along any path in
D∗ because u is given in the whole domain D∗.
Thus, f = H ◦ g is a desired multivalent solution of the Dirichlet
problem (1.6) for the Beltrami equation (1.1).
The hypotheses of the rest theorems and corollaries below yield the
hypotheses of Theorem 11.1, as was shown in the previous section.
Corollary 11.1. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function with Kµ ∈ L1
loc and
kTz0(ε) = O
(
log
1
ε
)
as ε→ 0 ∀ z0 ∈ D, (11.3)
where kTz0(ε) is the average of the function KT
µ (z, z0) over the circle
S(z0, ε).
Then the Beltrami equation (1.1) has a multivalent solution of the
Dirichlet problem (1.6) for every continuous function φ : ED → R.
Remark 11.1. In particular, the conclusion of Theorem 11.1 holds if
KT
µ (z, z0) = O
(
log
1
|z − z0|
)
as z → z0 ∀ z0 ∈ D . (11.4)
Applying Lemma 2.2 in [38], see also Lemma 7.4 in [26], we obtain
the following result immediately from Theorem 11.1.
Lemma 11.1. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function with Kµ ∈ L1(D). Suppose that, for every z0 ∈ D and every
small enough 0 < ε0 < d(z0) := supz∈D |z − z0|, there is a family of
measurable functions ψz0, ε,ε0 : (0,∞) → [0,∞], ε ∈ (0, ε0) such that
0 < Iz0,ε0(ε) :=
ε0∫
ε
ψz0, ε,ε0(t) dt < ∞ ∀ ε ∈ (0, ε0) (11.5)
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 61
and∫
ε<|z−z0|<ε0
KT
µ (z, z0) ·ψ2
z0, ε,ε0 (|z − z0|) dm(z) = o(I2z0,ε0(ε)) as ε→ 0 .
(11.6)
Then the Beltrami equation (1.1) has a multivalent solution of the Dirich-
let problem (1.6) for every continuous function φ : ED → R.
Remark 11.2. In fact, it is sufficient to assume the local integrability
of KT
µ in the domain D and the condition ||KT
µ ||(z0, r) ̸= ∞ for a.e.
r ∈ (0, ε0) and all z0 ∈ ∂D instead of the condition KT
µ ∈ L1(D).
By Lemma 11.1 with the choice ψz0, ε(t) ≡ 1/t log 1
t , we obtain the
following result, see also Lemma 3.1.
Theorem 11.2. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function with Kµ ∈ L1(D) such that
KT
µ (z, z0) 6 Qz0(z) ∈ FMO(z0) ∀ z0 ∈ D . (11.7)
Then the Beltrami equation (1.1) has a multivalent solution of the Dirich-
let problem (1.6) for every continuous function φ : ED → R.
Remark 11.3. In particular, the conclusion of Theorem 11.2 holds if
either Qz0 ∈ BMOloc or Qz0 ∈ W1,2
loc because W 1,2
loc ⊂ VMOloc, see, e.g.,
[3].
By Corollary 3.1, we have the next consequence of Theorem 11.2:
Corollary 11.2. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function with Kµ ∈ L1(D) such that
lim sup
ε→0
∫
−
B(z0, ε)
KT
µ (z, z0) dm(z) < ∞ ∀ z0 ∈ D . (11.8)
Then the Beltrami equation (1.1) has a multivalent solution of the Dirich-
let problem (1.6) for every continuous function φ : ED → R.
Remark 11.4. In particular, by (1.5), the conclusion of Theorem 11.2
holds if
Kµ(z) 6 Q(z) ∈ BMO(D) (11.9)
.
62 The Beltrami equations and prime ends
The following statement follows from Lemma 11.1 through the choice
ψ(t) = 1/t, see also Remark 11.2.
Theorem 11.3. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function such that∫
ε<|z−z0|<ε0
KT
µ (z, z0)
dm(z)
|z − z0|2
= o
([
log
1
ε
]2)
∀ z0 ∈ D . (11.10)
Then the Beltrami equation (1.1) has a multivalent solution of the Dirich-
let problem (1.6) for every continuous function φ : ED → R.
Remark 11.5. Similarly, ψ(t) = 1/(t log 1/t) instead of ψ(t) = 1/t
choosing in Lemma 11.1, we obtain that condition (11.10) can be replaced
by the condition∫
ε<|z−z0|<ε0
KT
µ (z, z0) dm(z)(
|z − z0| log 1
|z−z0|
)2 = o
([
log log
1
ε
]2)
∀ z0 ∈ D .
(11.11)
Here we are able to give a number of other conditions of logarithmic type.
In particular, condition (11.3), due to Theorem 11.1, can be replaced by
the weaker condition
kTz0(r) = O
(
log
1
r
log log
1
r
)
. (11.12)
Finally, by Theorem 11.1, applying also Theorem 3.1 in work [46], we
come to the following result.
Theorem 11.4. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function with Kµ ∈ L1
loc such that∫
D∩B(z0,ε0)
Φz0(K
T
µ (z, z0)) dm(z) < ∞ (11.13)
for ε0 = ε(z0) > 0 and a nondecreasing convex function Φz0 : [0,∞) →
[0,∞) with
∞∫
δ0
dτ
τΦ−1
z0 (τ)
= ∞ (11.14)
for δ0 = δ(z0) > Φz0(0). Then the Beltrami equation (1.1) has a multiva-
lent solution of the Dirichlet problem (1.6) for every continuous function
φ : ED → R.
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 63
Recall that condition (11.14) is equivalent to every of conditions
(7.13)–(7.17).
Corollary 11.3. Let D be a bounded finitely connected domain in C with
nondegenerate boundary components and let µ : D → D be a measurable
function with Kµ ∈ L1
loc such that∫
D∩B(z0,ε0)
eα0KT
µ (z,z0) dm(z) < ∞ ∀ z0 ∈ D (11.15)
for some ε0 = ε(z0) > 0 and α0 = α(z0) > 0.
Then the Beltrami equation (1.1) has a multivalent solution of the
Dirichlet problem (1.6) for every continuous function φ : ED → R.
References
[1] C. Andreian Cazacu, On the length-area dilatation // Complex Var. Theory Appl.,
50 (2005), No. 7–11, 765–776.
[2] B. Bojarski, Generalized solutions of a system of differential equations of the
first order of the elliptic type with discontinuous coefficients // Mat. Sb., 43(85)
(1957), No. 4, 451–503.
[3] H. Brezis, L. Nirenberg, Degree theory and BMO. I. Compact manifolds without
boundaries // Selecta Math. (N.S.), 1 (1995), No. 2, 197–263.
[4] C. Carathéodory, Über die Begrenzung der einfachzusammenhängender Gebiete //
Math. Ann., 73 (1913), 323–370.
[5] E. F. Collingwood, A. J. Lohwater, The Theory of Cluster Sets, Cambridge Tracts
in Math. and Math. Physics, 56, Cambridge: Cambridge Univ. Press, 1966.
[6] F. Chiarenza, M. Frasca, P. Longo, W 2,p-solvability of the Dirichlet problem for
nondivergence elliptic equations with VMO coefficients // Trans. Amer. Math.
Soc., 336 (1993), No. 2, 841–853.
[7] F. W. Gehring, Rings and quasiconformal mappings in space // Trans. Amer.
Math. Soc., 103 (1962), 353–393.
[8] F. W. Gehring, O. Martio, Quasiextremal distance domains and extension of
quasiconformal mappings // J. Anal. Math. 45 (1985), 181–206.
[9] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Transl. of
Math. Monographs 26, Providence, RI: AMS, 1969.
[10] V. Gutlyanskii, O. Martio, T. Sugawa, M. Vuorinen, On the degenerate Beltrami
equation // Trans. Amer. Math. Soc., 357 (2005), 875–900.
[11] V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equa-
tions: A Geometric Approach, Developments in Math. 26, New York: Springer,
2012.
[12] J. Heinonen, T. Kilpelainen, O. Martio, Nonlinear Potential Theory of Degenerate
Elliptic Equations, Oxford: Clarendon Press, 1993.
[13] A. Hurwitz, R. Courant, The Function theory, Nauka, Moscow, 1968 [in Russian].
64 The Beltrami equations and prime ends
[14] A. A. Ignat’ev, V. I. Ryazanov, Finite mean oscillation in the mapping theory //
Ukrainian Math. Bull., 2 (2005), No. 3, 403–424.
[15] T. Iwaniec, C. Sbordone, Riesz transforms and elliptic PDEs with VMO coeffi-
cients // J. Anal. Math., 74 (1998), 183–212.
[16] F. John, L. Nirenberg, On functions of bounded mean oscillation // Comm. Pure
Appl. Math., 14 (1961), 415–426.
[17] D. Kovtonyuk, I. Petkov, V. Ryazanov, On the boundary behaviour of solutions to
the Beltrami equations // Complex Variables and Elliptic Equations, 58 (2013),
No. 5, 647–663.
[18] D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov, R. R. Salimov, Boundary behavior
and Dirichlet problem for Beltrami equations // Algebra and Analysis, 25 (2013),
No. 4, 101–124 [in Russian]; transl in St. Petersburg Math. J., 25 (2014), 587–603.
[19] D. Kovtonyuk, I. Petkov, V. Ryazanov, R. Salimov, On the Dirichlet problem for
the Beltrami equation // J. Anal. Math., 122 (2014), No. 4, 113–141.
[20] D. Kovtonyuk, V. Ryazanov, To the theory of boundaries of space domains //
Proc. Inst. Appl. Math. & Mech. NAS of Ukraine, 13 (2006), 110–120 [in Russian].
[21] D. Kovtonyuk, V. Ryazanov, On the theory of lower Q-homeomorphisms // Ukr.
Mat. Visn., 5 (2008), No. 2, 159–184; transl. in Ukrain. Math. Bull., 5 (2008),
No. 2, 157–181.
[22] D. Kovtonyuk, V. Ryazanov, On the boundary behavior of generalized quasi-
isometries // J. Anal. Math., 115 (2011), 103–120.
[23] O. Lehto, Homeomorphisms with a prescribed dilatation // Lecture Notes in
Math., 118 (1968), 58–73.
[24] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Q-homeomorphisms // Con-
temporary Math., 364 (2004), 193–203.
[25] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, On Q-homeomorphisms // Ann.
Acad. Sci. Fenn. Ser. A1 Math., 30 (2005), 49–69.
[26] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping
Theory, New York: Springer, 2009.
[27] O. Martio, V. Ryazanov, M. Vuorinen, BMO and Injectivity of Space Quasiregular
Mappings // Math. Nachr., 205 (1999), 149–161.
[28] O. Martio, J. Sarvas, Injectivity theorems in plane and space // Ann. Acad. Sci.
Fenn. Ser. A1 Math., 4 (1978/1979), 384–401.
[29] R. Näkki, Boundary behavior of quasiconformal mappings in n-space // Ann.
Acad. Sci. Fenn. Ser. A1 Math., 484 (1970), 1–50.
[30] R. Näkki, Extension of Loewner’s capacity theorem // Trans. Amer. Math. Soc.,
180, 229–236 (1973).
[31] R. Näkki, Prime ends and quasiconformal mappings // J. Anal. Math., 35 (1979),
13–40.
[32] D. K. Palagachev, Quasilinear elliptic equations with VMO coefficients // Trans.
Amer. Math. Soc., 347 (1995), No. 7, 2481–2493.
[33] M. A. Ragusa, Elliptic boundary value problem in vanishing mean oscillation
hypothesis // Comment. Math. Univ. Carolin., 40 (1999), No. 4, 651–663.
[34] E. Reich, H. Walczak, On the behavior of quasiconformal mappings at a point //
Trans. Amer. Math. Soc., 117 (1965), 338–351.
V. Gutlyanskii, V. Ryazanov, and E. Yakubov 65
[35] H. M. Reimann, T. Rychener, Funktionen Beschränkter Mittlerer Oscillation //
Lecture Notes in Math., 487 (1975).
[36] V. Ryazanov, R. Salimov, Weakly flat spaces and boundaries in the mapping
theory // Ukr. Mat. Visn., 4 (2007), No. 2, 199–234; transl. in Ukrain. Math.
Bull. 4 (2007), No. 2, 199–233.
[37] V. Ryazanov, R. Salimov, U. Srebro, E. Yakubov, On Boundary Value Problems
for the Beltrami Equations // Contemporary Math., 591 (2013), 211–242.
[38] V. Ryazanov, E. Sevost’yanov, Equicontinuons classes of ring Q-
homeomorphisms // Sibirsk. Mat. Zh., 48 (2007), No. 6, 1361–1376 [in
Russian]; transl. in Siberian Math. J., 48 (2007), No. 6, 1093–1105.
[39] V. Ryazanov, E. Sevost’yanov, Equicontinuity of mappings quasiconformal in the
mean // Ann. Acad. Sci. Fenn., 36 (2011), 231–244.
[40] V. Ryazanov, U. Srebro, E. Yakubov, BMO-quasiconformal mappings // J. Anal.
Math., 83 (2001), 1–20.
[41] V. Ryazanov, U. Srebro, E. Yakubov, On ring solutions of Beltrami equation //
J. Anal. Math., 96 (2005), 117–150.
[42] V. Ryazanov, U. Srebro, E. Yakubov, Beltrami equation and FMO functions //
Contemp. Math., 382 (2005), 357–364.
[43] V. Ryazanov, U. Srebro, E. Yakubov, Finite mean oscillation and the Beltrami
equation // Israel J. Math., 153 (2006), 247–266.
[44] V. Ryazanov, U. Srebro, E. Yakubov, To strong ring solutions of the Beltrami
equations // Uzbek. Math. J., (2009), No. 1, 127–137.
[45] V. Ryazanov, U. Srebro, E. Yakubov, On strong solutions of the Beltrami equa-
tions // Complex Var. Elliptic Equ., 55 (2010), No. 1-3, 219–236.
[46] V. Ryazanov, U. Srebro, E. Yakubov, Integral conditions in the mapping theory //
Ukr. Mat. Visn., 7 (2010), No. 1, 73–87; transl. in Math. Sci. J., 173 (2011), No. 4,
397–407.
[47] V. Ryazanov, U. Srebro, E. Yakubov, Integral conditions in the theory of the
Beltrami equations // Complex Var. Elliptic Equ., 57 (2012), No. 12, 1247–1270.
[48] S. Saks, Theory of the Integral, New York: Dover, 1964.
[49] D. Sarason, Functions of vanishing mean oscillation // Trans. Amer. Math. Soc.,
207 (1975), 391–405.
[50] S. Stoilow, Lecons sur les Principes Topologue de le Theorie des Fonctions Ana-
lytique, Paris: Gauthier-Villars, 1938.
[51] J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes
in Math., 229. Berlin: Springer, 1971.
[52] I. N. Vekua, Generalized analytic functions, London: Pergamon Press, 1962.
[53] R. L. Wilder, Topology of Manifolds, New York: AMS, 1949.
66 The Beltrami equations and prime ends
Contact information
Vladimir
Gutlyanskii,
Vladimir Ryazanov
Institute of Applied Mathematics
and Mechanics, NAS of Ukraine,
84100 Slavyansk,
Dondass region,
Ukraine
E-Mail: vladimirgut@mail.ru,
vlryazanov1@rambler.ru,
vl_ryazanov1@mail.ru
Eduard Yakubov Holon Institute of Technology,
Holon, Israel
E-Mail: yakubov@hit.ac.il,
eduardyakubov@gmail.com
|