On the theory of the Beltrami equation
We study ring homeomorphisms and, on this basis, obtain a series of theorems on existence of the so-called ring solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami equations is formulated extending earlier results.
Збережено в:
| Дата: | 2006 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2006
|
| Назва видання: | Український математичний журнал |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/165535 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the theory of the Beltrami equation / V.I. Ryazanov, U. Srebro, E. Yakubov // Український математичний журнал. — 2006. — Т. 58, № 11. — С. 1571–1583. — Бібліогр.: 51 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-165535 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1655352020-02-15T01:27:33Z On the theory of the Beltrami equation Ryazanov, V.I. Srebro, U. Yakubov, E. Статті We study ring homeomorphisms and, on this basis, obtain a series of theorems on existence of the so-called ring solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami equations is formulated extending earlier results. Вивчаються кільцеві гомеоморфізми, i на цій підставі отримано низку теорем про існування так званих кільцевих розв'язків вироджених рівнянь Бельтрамі. Сформульовано загальне твердження про існування розв'язків рівнянь Бельтрамі, що узагальнює більш ранні результати. 2006 Article On the theory of the Beltrami equation / V.I. Ryazanov, U. Srebro, E. Yakubov // Український математичний журнал. — 2006. — Т. 58, № 11. — С. 1571–1583. — Бібліогр.: 51 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165535 517.5 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Ryazanov, V.I. Srebro, U. Yakubov, E. On the theory of the Beltrami equation Український математичний журнал |
| description |
We study ring homeomorphisms and, on this basis, obtain a series of theorems on existence of the so-called ring
solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami
equations is formulated extending earlier results. |
| format |
Article |
| author |
Ryazanov, V.I. Srebro, U. Yakubov, E. |
| author_facet |
Ryazanov, V.I. Srebro, U. Yakubov, E. |
| author_sort |
Ryazanov, V.I. |
| title |
On the theory of the Beltrami equation |
| title_short |
On the theory of the Beltrami equation |
| title_full |
On the theory of the Beltrami equation |
| title_fullStr |
On the theory of the Beltrami equation |
| title_full_unstemmed |
On the theory of the Beltrami equation |
| title_sort |
on the theory of the beltrami equation |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165535 |
| citation_txt |
On the theory of the Beltrami equation / V.I. Ryazanov, U. Srebro, E. Yakubov // Український математичний журнал. — 2006. — Т. 58, № 11. — С. 1571–1583. — Бібліогр.: 51 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT ryazanovvi onthetheoryofthebeltramiequation AT srebrou onthetheoryofthebeltramiequation AT yakubove onthetheoryofthebeltramiequation |
| first_indexed |
2025-07-14T18:52:17Z |
| last_indexed |
2025-07-14T18:52:17Z |
| _version_ |
1837649507617079296 |
| fulltext |
UDC 517.5
V. Ryazanov (Inst. Appl. Math. and Mech. Nat. Acad. Sci. Ukraine, Donetsk),
U. Srebro (Technion — Israel Inst. Technol., Haifa, Israel),
E. Yakubov (Holon Acad. Inst. Technol., Israel)
ON THE THEORY OF THE BELTRAMI EQUATION
DO TEORI} RIVNQNNQ BEL\TRAMI
We study ring homeomorphisms and, on this basis, obtain a series of theorems on existence of the so-called ring
solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami
equations is formulated extending earlier results. In particular, we give new existence criteria of homeomorphic
solutions f of the class W 1,1
loc with f−1 ∈ W 1,2
loc in terms of tangential dilatations and functions of finite mean
oscillation. The ring solutions also satisfy additional capacity inequalities.
Vyvçagt\sq kil\cevi homeomorfizmy, i na cij pidstavi otrymano nyzku teorem pro isnuvannq tak zvanyx
kil\cevyx rozv’qzkiv vyrodΩenyx rivnqn\ Bel\trami. Sformul\ovano zahal\ne tverdΩennq pro isnu-
vannq rozv’qzkiv rivnqn\ Bel\trami, wo uzahal\ng[ bil\ß ranni rezul\taty. Zokrema, navedeno novi
kryteri] isnuvannq homeomorfnyx rozv’qzkivf klasu W 1,1
loc z f−1 ∈ W 1,2
loc u terminax tanhencial\nyx
dylatacij ta funkcij skinçennoho seredn\oho kolyvannq. Kil\cevi rozv’qzky zadovol\nqgt\ takoΩ
dodatkovi [mnisni nerivnosti.
1. Introduction. As known, the Beltrami equation plays an important role in the mapping
theory. The main goal of this paper is to present general principles which allow to obtain
variety of conditions for the existence of homeomorphic ACL solutions in the degenerate
case. Our existence theorems are proved by an approximation method.
Let D be a domain in the complex plane C, i.e., open and connected subset of C, and
let µ : D → C be a measurable function with |µ(z)| < 1 a.e. The Beltrami equation is
the equation of the form
fz = µ(z) · fz (1)
where fz = ∂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 function µ is called the complex
coefficient and
Kµ(z) =
1 + |µ(z)|
1 − |µ(z)| (2)
the maximal dilatation or in short the dilatation of the equation (1). The Beltrami equa-
tion (1) is said to be degenerate if
ess supKµ(z) = ∞. (3)
An ACL homeomorphism f : D → C is called a ring solution of the Beltrami equa-
tion (1) if f satisfies (1) a.e., f−1 ∈ W 1,2
loc and f is a ring Q-homeomorphism at every
point z0 ∈ D with Q(z) = KTµ (z, z0) where the function
KTµ (z, z0) =
∣∣∣∣1 − z − z0
z − z0
µ(z)
∣∣∣∣
2
1 − |µ(z)|2 (4)
is called the tangential dilatation with respect to z0, cf. [1 – 3]. For the definition of ring
Q-homeomorphisms, see Section 2. The terms have been first introduced in [4].
Recall that a mapping f : D → C is absolutely continuous on lines, abbr. f ∈ ACL,
if, for every closed rectangle R in D whose sides are parallel to the coordinate axes, f |R
is absolutely continuous on almost all line segments in R which are parallel to the sides of
c© V. RYAZANOV, U. SREBRO, E. YAKUBOV, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11 1571
1572 V. RYAZANOV, U. SREBRO, E. YAKUBOV
R. In particular, f is ACL if it belongs to the Sobolev class W 1,1
loc , see e.g. [5, p. 8]. Note
that, if f ∈ ACL, then f has partial derivatives fx and fy a.e. and, thus, by the well-known
Gehring – Lehto theorem every ACL homeomorphism f : D → C is totally differentiable
a.e., see [6] or [7, p. 128]. For a sense-preserving ACL homeomorphism f : D → C, the
Jacobian Jf (z) = |fz|2−|fz|2 is nonnegative a.e., see [7, p. 10]. In this case, the complex
dilatation µf of f is the ratio µ(z) = fz/fz, if fz �= 0 and µ(z) = 0 otherwise, and the
dilatation Kf of f is Kµ(z), see (2), and, thus, |µ(z)| ≤ 1 and Kµ(z) ≥ 1 a.e.
Note that every homeomorphic ACL solution f of the Beltrami equation with Kµ ∈
∈ L1
loc belongs to the class W 1,1
loc as in all our theorems. Indeed,
|fz| + |fz| = K1/2
µ (z)J1/2
f (z)
and by Hölder’s inequality, on every compact set C ⊂ D,
‖fz‖1 ≤ ‖fz‖1 ≤ ‖Kµ‖1/2
1 A(f(C))1/2
where A(f(C)) is the area of the set f(C). Hence f ∈ W 1,1
loc , see e.g. [5, p. 8]. Similarly,
it is easy to show that, if Kµ ∈ Lploc, p ∈ [1,∞], then f ∈ W 1,s
loc where s = 2p/(1 + p) ∈
∈ [1, 2].
Note also that the condition f−1 ∈ W 1,2
loc given in the definition of a ring solution
implies that a.e. point z is a regular point for the mapping f, i.e., f is differentiable at z
and Jf (z) �= 0, see Remark 2. Finally, note that the condition Kµ ∈ L1
loc is necessary for
a homeomorphic ACL solution f of (1) to have the property g = f−1 ∈ W 1,2
loc because
this property implies that∫
C
Kµ(z) dxdy ≤ 4
∫
C
dxdy
1 − |µ(z)|2 = 4
∫
f(C)
|gw|2 dudv < ∞
for every compact set C ⊂ D.
Recall that a real valued function ϕ ∈ L1
loc(D) is said to be of bounded mean oscilla-
tion in D, abbr. ϕ ∈ BMO(D) or simply ϕ ∈ BMO, if
‖ϕ‖∗ = sup
B⊂D
−
∫
B
∣∣ϕ(z) − ϕB
∣∣ dxdy < ∞
where the supremum is taken over all disks B in D and
ϕB = −
∫
B
ϕ(z)dxdy =
1
|B|
∫
B
ϕ(z) dxdy
is the mean value of the function ϕ over B. It is well-known that L∞(D) ⊂ BMO(D) ⊂
⊂ Lploc(D) for all 1 ≤ p < ∞, see e.g. [8]. A function ϕ in BMO is said to have vanishing
mean oscillation, abbr. ϕ ∈ VMO, if the above supremum taken over all disks B in D
with |B| < ε converges to 0 as ε → 0.
The BMO space was introduced by John and Nirenberg, see [9], and soon became
one of the main concepts in harmonic analysis, complex analysis and partial differential
equations. BMO functions are related in many ways to quasiconformal and quasiregular
mappings, see e.g. [10 – 15], and to mappings with finite distortion, see e.g. [16 – 19].
VMO has been introduced by Sarason, see [20]. A large number of papers are devoted to
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
ON THE THEORY OF THE BELTRAMI EQUATION 1573
the study of existence, uniqueness and properties of solutions for various kind of differen-
tial equations and, in particular, of elliptic type with VMO coefficients, see e.g. [21 – 24].
In this connection, it should be noted that by a recent result of Brezis and Nirenberg in
[25] the Sobolev class W 1,2
loc is a subclass of VMO, see also [26].
Let D be a domain in the complex plane C. We say that a function ϕ : D → R has
finite mean oscillation at a point z0 ∈ D, abbr. f of FMO at z0, if
dϕ(z0) = lim
ε→0
−
∫
D(z0,ε)
|ϕ(z) − ϕε(z0)|dxdy < ∞ (5)
where
ϕε(z0) = −
∫
D(z0,ε)
ϕ(z)dxdy < ∞ (6)
is the mean value of the function ϕ(z) over the disk D(z0, ε) with small ε > 0. Thus, the
notion includes the assumption that ϕ is integrable in some neighborhood of the point z0.
We call dϕ(z0) the dispersion of the function ϕ at the point z0. We say that a function
ϕ : D → R is of finite mean oscillation in D, abbr. ϕ ∈ FMO(D) or simply ϕ ∈ FMO,
if ϕ has a finite dispersion at every point z ∈ D. We call also the number
dϕ(z0, ε0) = sup
ε∈(0,ε0]
−
∫
D(z0,ε)
∣∣ϕ(z) − ϕε(z0)
∣∣ dxdy < ∞ (7)
the maximal dispersion of the function ϕ in the disk D(z0, ε0).
Below we use the notations D(r) = D(0, r) = {z ∈ C : |z| < r} and
A(ε, ε0) = {z ∈ C : ε < |z| < ε0}. (8)
Lemma 1. Let D ⊂ C be a domain such that D(e−1) ⊂ D, and let ϕ : D → R be
a nonnegative function. If ϕ is integrable in D(e−1) and of FMO at 0, then∫
A(ε,e−1)
ϕ(z) dxdy(
|z| log 1
|z|
)2 ≤ C log log
1
ε
(9)
for ε ∈ (0, e−e) where
C = 2π
[
2ϕ0 + 3e2d0
]
, (10)
ϕ0 is the mean value of ϕ over the disk D(e−1) and d0 is the maximal dispersion of ϕ in
D(e−1).
Versions of this lemma have been first established for BMO functions in [27, 28] and
then for FMO functions in [29] and [30, p. 251] (Lemma 2.15).
Conditions for the existence and uniqueness of ACL homeomorphic solutions for the
Beltrami equation can be given in terms of the maximal dilatation Kµ(z). It was shown
for instance that if Kµ(z) has a BMO or FMO majorant, then the Beltrami equation
(1) has a homeomorphic ACL solution, see e.g. [27, 28, 30, 31], cf. [32, 33]. Various
conditions for the existence of solutions for the Beltrami equation have been formulated
in terms of integral and measure constraints on Kµ, see e.g. [32 – 39]. These conditions
assume either exponential integrability or at least high local integrability of the dilatation
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1574 V. RYAZANOV, U. SREBRO, E. YAKUBOV
Kµ. Here, as in [2 – 4], the existence criteria are expressed in terms of the tangential
dilatations KTµ (z, z0).
In [30], we proved that if Kµ(z) has a majorant Q(z) of finite mean oscillation in D,
then (1) has a homeomorphic ACL solution. Now we prove a stronger result, namely,
the existence of a ring solution of (1) where the assumption on an FMO majorant for Kµ
in D is replaced by the weaker condition that every point z0 ∈ D has a neighborhood
Uz0 and a function Qz0 : Uz0 → [0,∞] which is of finite mean oscillation at z0 such that
KTµ (z, z0) ≤ Qz0(z) for all z ∈ Uz0 , see Theorem 4 in Section 3 below. This as well as
other new existence theorems established here are based on a general existence principle,
Lemma 3. Some of these existence theorems are expressed in terms of the mean and
the logarithmic mean of the tangential dilatation KTµ (z, z0) over infinitesimal disks and
annuli centered at z0, see e.g. Theorems 5 and 6. We also use Lemma 3 in a new proof of
an extension of Lehto’s theorem, Theorem 7, that we have first established in [4]. Finally,
note that a series of interesting distortion theorems have been formulated in various terms
in Section 2, see Remark 3 in the end of the work.
2. Distortion estimates for ring Q-homeomorphisms. Below we use the stan-
dard conventions a/∞ = 0 for a �= ∞ and a/0 = ∞ if a > 0 and 0 · ∞ = 0, see
e.g. [40, p. 6].
Given a measurable function Q : D → [1,∞], we say that a homeomorphism f : D →
→ C is a Q-homeomorphism if
M(fΓ) ≤
∫
D
Q(z) · ρ2(z)dxdy (11)
holds for every path family Γ in D and each ρ ∈ adm Γ. This term was introduced in [41 –
43].
Recall that, given a family of paths Γ in C, a Borel function ρ : C → [0,∞] is called
admissible for Γ, abbr. ρ ∈ adm Γ, if∫
γ
ρ(z) |dz| ≥ 1
for each γ ∈ Γ. The modulus of Γ is defined by
M(Γ) = inf
ρ∈adm Γ
∫
C
ρ2(z)dxdy.
We say that a property P holds for almost every (a.e.) path γ in a family Γ if the subfamily
of all paths in Γ for which P fails has modulus zero. In particular, almost all paths in C
are rectifiable.
Given a domain D and two sets E and F in C, Γ(E,F,D) denotes the family of all
paths γ : [a, b] → C which join E and F in D, i.e., γ(a) ∈ E, γ(b) ∈ F and γ(t) ∈ D
for a < t < b. We set Γ(E,F ) = Γ(E,F,C) if D = C. A ring domain, or shortly a
ring in C is a doubly connected domain R in C. Let R be a ring in C. If C1 and C2 are
the connected components of C \R, we write R = R(C1, C2). The capacity of R can be
defined by
capR(C1, C2) = M(Γ(C1, C2, R)), (12)
see e.g. [44]. Note that
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
ON THE THEORY OF THE BELTRAMI EQUATION 1575
M(Γ(C1, C2, R)) = M(Γ(C1, C2)), (13)
see e.g. Theorem 11.3 in [45].
Motivated by the ring definition of quasiconformality in [46], we introduce the fol-
lowing notion that localizes and extends the notion of a Q-homeomorphism. Let D be
a domain in C, z0 ∈ D, r0 ≤ dist(z0, ∂D) and Q : D(z0, r0) → [0,∞] a measurable
function in the disk
D(z0, r0) = {z ∈ C : |z − z0| < r0}.
Set
A(r1, r2, z0) = {z ∈ C : r1 < |z − z0| < r2},
C(z0, ri) = {z ∈ C : |z − z0| = ri}, i = 1, 2.
We say that a homeomorphism f : D → C is a ring Q-homeomorphism at the point z0 if
capR(fC1, fC2) ≤
∫
A
Q(z) · η2
(
|z − z0|
)
dxdy (14)
for every annulus A = A(r1, r2, z0), 0 < r1 < r2 < r0, and for every measurable
function η : (r1, r2) → [0,∞] such that
r2∫
r1
η(r)dr = 1. (15)
Note that every Q-homeomorphism f : D → C is a ring Q-homeomorphism at each
point z0 ∈ D. The next proposition gives other conditions on f which force it to be a ring
Q-homeomorphism, see Theorem 2.17 in [4].
Proposition 1. Let f : D → C be a sense-preserving homeomorphism of the class
W 1,2
loc such that f−1 ∈ W 1,2
loc . Then at every point z0 ∈ D the mapping f is a ring Q-
homeomorphism with Q(z) = KTµ (z, z0) where µ(z) = µf (z).
If f is a plane W 1,2
loc homeomorphism with a locally integrable Kf (z), then f−1 ∈
∈ W 1,2
loc , see e.g. [47]. Hence we obtain the following consequence of Proposition 1.
Corollary 1. Let f : D → C be a sense-preserving homeomorphism of the class
W 1,2
loc and suppose that Kf (z) is integrable in a disk D(z0, r0) ⊂ D for some z0 ∈ D and
r0 > 0. Then f is a ring Q-homeomorphism at the point z0 ∈ D with Q(z) = KTµ (z, z0)
where µ(z) = µf (z).
Note also the convergence theorem which plays an important role in our scheme for
deriving the existence theorems of the Beltrami equation, see Theorem 2.22 in [4].
Proposition 2. Let fn : D → C, n = 1, 2, . . . , be a sequence of ring Q-homeomor-
phisms at a point z0 ∈ D. If fn converge locally uniformly to a homeomorphism f : D →
→ C, then f is also a ring Q-homeomorphism at the point z0.
For points z, ζ ∈ C, the spherical (chordal) distance s(z, ζ) between z and ζ is
given by
s(z, ζ) =
|z − ζ|
(1 + |z|2) 1
2 (1 + |ζ|2) 1
2
if z �= ∞ �= ζ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1576 V. RYAZANOV, U. SREBRO, E. YAKUBOV
s(z,∞) =
1
(1 + |z|2) 1
2
if z �= ∞.
Given a set E ⊂ C, δ(E) denotes the spherical diameter of E, i.e.,
δ(E) = sup
z1,z2∈E
s(z1, z2).
Given a number ∆ ∈ (0, 1), a domain D ⊂ C, a point z0 ∈ D, a number r0 ≤
≤ dist (z0, ∂D), and a measurable function Q : D(z0, r0) → [0,∞], let R∆
Q denote the
class of all ring Q-homeomorphisms f : D → C at z0 such that
δ
(
C \ f(D)
)
≥ ∆. (16)
Next, we introduce the classes B∆
Q and F∆
Q of certain qc mappings. Let B∆
Q denote the
class of all quasiconformal mappings f : D → C satisfying (16) such that
KTµ (z, z0) =
∣∣∣∣1 − z − z0
z − z0
µ(z)
∣∣∣∣
2
1 − |µ(z)|2 ≤ Q(z) a.e. in D(z0, r0) (17)
where µ = µf . Similarly, let F∆
Q denote the class of all quasiconformal mappings f : D →
→ C satisfying (16) such that
Kµ(z) =
1 + |µ(z)|
1 − |µ(z)| ≤ Q(z) a.e. in D(z0, r0) . (18)
Remark 1. Note that
F∆
Q ⊂ B∆
Q ⊂ R∆
Q. (19)
Lemma 2. Let f ∈ R∆
Q and ψε : (0,∞) → [0,∞], 0 < ε < r0, a one parameter
family of measurable functions such that
0 < I(ε) =
r0∫
ε
ψε(t)dt < ∞, ε ∈ (0, r0). (20)
Then, for all ζ ∈ D(z0, r0),
s
(
f(ζ), f(z0)
)
≤ 32
∆
exp
(
− 2π
ω(|ζ − z0|)
)
(21)
where
ω(ε) =
1
I2(ε)
∫
A(ε)
Q(z)ψ2
ε
(
|z − z0|
)
dxdy (22)
and A(ε) = A(ε, r0, z0) =
{
z ∈ C : ε < |z − z0| < r0
}
.
Proof. Set C = {z ∈ C : |z − z0| = |ζ − z0|} and C0 = {z ∈ C : |z − z0| = r0}.
Then
s
(
f(ζ), f(z0)
)
≤ 32
∆
exp
(
− 2π
capR(fC, fC0)
)
for every f ∈ R∆
Q by Lemma 3.21 in [4]. Thus, choosing η(r) = ψε(r)/I(ε), r ∈ (ε, r0),
in (14), we come to (21).
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
ON THE THEORY OF THE BELTRAMI EQUATION 1577
Theorem 1. Let f ∈ R∆
Q and ψ : [0,∞] → [0,∞] a measurable function such that
0 <
r0∫
ε
ψ(t) dt < ∞, ε ∈ (0, r0). (23)
Suppose that ∫
ε<|z−z0|<r0
Q(z)ψ2(|z − z0|)dxdy ≤ C
r0∫
ε
ψ(t)dt (24)
for all ε ∈ (0, r0). Then
s(f(ζ), f(z0)) ≤
32
∆
exp
−2π
C
r0∫
|ζ−z0|
ψ(t)dt
(25)
whenever ζ ∈ D(z0, r0).
In particular, Theorem 1 which is an immediate consequence of Lemma 2 makes
possible to estimate the distortion in terms of the mean value of Q over balls and its
maximal dispersion, see (7).
Theorem 2. Let f ∈ R∆
Q. Then
s
(
f(ζ), f(z0)
)
≤ 32
∆
(
log
eε0
|ζ − z0|
)− 1
β0
(26)
for every point ζ ∈ D(z0, e
1−eε0) where
β0 = 2q0 + 3e2d0, (27)
q0 is the mean value and d0 the maximal dispersion of Q(z) in D(z0, ε0).
Proof. With no loss of generality we may assume that Q has finite mean oscilla-
tion at z0 and integrable over a disk D(z0, ε0) because otherwise (26) is trivial. The
mean value and the dispersion of a function over disks are invariant under linear trans-
formations w = (z − z0)/eε0. Hence, (26) follows by Lemma 1 and Theorem 1 with
ψ(t) = 1/(t log 1/t).
Choosing in Lemma 2 a special functional parameter ψ(t) we obtain also the follow-
ing distortion theorem for ring Q-homeomorphisms in terms of the mean value of Q over
spheres.
Theorem 3. Let f ∈ R∆
Q. Then
s
(
f(ζ), f(z0)
)
≤ 32
∆
exp
−
r0∫
|ζ−z0|
dr
rq(r)
(28)
for all ζ ∈ D(z0, r0) where q(r) is the mean of Q(z) over the circle |z − z0| = r.
Proof. With no loss of generality we may assume that the integral I �= 0 in (28)
because otherwise (28) is trivial and that I �= ∞ because otherwise we can replace Q(z)
by Q(z) + δ with arbitrarily small δ > 0 and then take the limit as δ → 0 in (28). Note
that R∆
Q ⊂ R∆
Q+δ. The condition I �= ∞ implies, in particular, that q(r) �= 0 a.e. in
(ε, r0), ε = |ζ − z0|. Set
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1578 V. RYAZANOV, U. SREBRO, E. YAKUBOV
ψ(t) =
{
1/
[
tq(t)
]
, t ∈ (ε, r0),
0, otherwise.
Then ∫
A
Q(z)ψ2
(
|z − z0|
)
dxdy = 2πI
where A = A(ε, r0, z0) =
{
z ∈ C : ε < |z − z0| < r0
}
.
Another consequence of Lemma 2 can be formulated in terms of the logarithmic mean
of Q over an annulus A(ε) = A(ε, ε0, z0) =
{
z ∈ C : ε < |z − z0| < ε0
}
which is
defined by
MQ
log(ε) = −
ε0∫
ε
q(t) d log t :=
1
log ε0/ε
ε0∫
ε
q(t)
dt
t
(29)
where q(t) denotes the mean value of Q over the circle |z − z0| = t. Choosing in the
expression (21) ψε(t) = 1/t for 0 < ε < ε0, and setting ε = |ζ − z0| we have the
following statement.
Corollary 2. Let Q : D(z0, r0) → [0,∞], r0 ≤ dist (z0, ∂D), be a measurable
function, ε0 ∈ (0, r0) and ∆ > 0. If f ∈ R∆
Q, then
s
(
f(ζ), f(z0)
)
≤ 32
∆
( |ζ − z0|
ε0
)1/MQ
log(|ζ−z0|)
(30)
for all ζ ∈ D(z0, ε0).
3. A general existence lemma and corollaries.
Lemma 3. Let µ : D → C be a measurable function with
∣∣µ(z)
∣∣ < 1 a.e. and
Kµ ∈ L1
loc. Suppose that for every z0 ∈ D there exist ε0 ≤ dist (z0, ∂D) and a family of
measurable functions ψz0,ε : (0,∞) → (0,∞), ε ∈ (0, ε0), such that
0 < Iz0(ε) :=
ε0∫
ε
ψz0,ε(t)dt < ∞, (31)
and ∫
ε<|z−z0|<ε0
KTµ (z, z0)ψ2
z0,ε
(
|z − z0|
)
dxdy = o
(
I2
z0(ε)
)
(32)
as ε → 0. Then the Beltrami equation (1) has a ring solution fµ.
Proof. Fix z1 and z2 in D. For n ∈ N, define µn : D → C by letting µn(z) =
= µ(z) if
∣∣µ(z)
∣∣ ≤ 1 − 1/n and 0 otherwise. Let fn : D → C be a homeomorphic
ACL solution of (1), with µn instead of µ, which fixes z1 and z2. Such fn exists by
the well-known existence theorem in the nondegenerate case, see e.g. [48, p. 98], cf. [7,
p. 185 and 194]. By Proposition 1 and Lemma 2, in view of (32), the sequence fn is
equicontinuous and hence by the Arzela – Ascoli theorem, see e.g. [49, p. 267], and [50,
p. 382], it has a subsequence, denoted again by fn, which converges locally uniformly
to some nonconstant mapping f in D. Then, by Theorem 3.1 and Corollary 5.12 in [27]
on convergence, f is K(z)-qc with K(z) = Kµ(z) and f satisfies (1) a.e. Thus, f is
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
ON THE THEORY OF THE BELTRAMI EQUATION 1579
a homeomorphic ACL solution of (1). Moreover, by Propositions 1 and 2 f is a ring
Q-homeomorphism, see (14), with Q(z) = KTµ (z, z0) at every point z0 ∈ D.
Since the locally uniform convergence fn → f of the sequence fn is equivalent to
the continuous convergence, i.e., fn(zn) → f(z0) if zn → z0, see [49, p. 268], and since
f is injective, it follows that gn = f−1
n → f−1 = g continuously, and hence locally
uniformly. By a change of variables which is permitted because fn and gn are in W 1,2
loc ,
see e.g. [7, p. 121, 128, 130, and 150], we obtain that for large n∫
B
|∂gn|2 dudv =
∫
gn(B)
dxdy
1 − |µn(z)|2
≤
∫
B∗
Q(z) dxdy < ∞ (33)
where B∗ and B are relatively compact domains in D and in f(D), respectively, such
that g(B̄) ⊂ B∗. The relation (33) implies that the sequence gn is bounded in W1,2(B),
and hence f−1 ∈ W1,2
loc(f(D)), see e.g. [51, p. 319].
Remark 2. If fµ is as in Lemma 3, then f−1
µ is locally absolutely continuous and
preserves nulls sets, and fµ is regular a.e., i.e., differentiable with Jfµ
(z) > 0 a.e. Indeed,
the assertion about f−1
µ follows from the fact that f−1
µ ∈ W1,2
loc, see [7, p. 131 and 150]. As
an ACL mapping fµ has a.e. partial derivatives and hence by [6] it has a total differential
a.e. Let E denote the set of points of D where fµ is differentiable and Jfµ(z) = 0, and
suppose that |E| > 0. Then |fµ(E)| > 0, since E = f−1
µ (fµ(E)) and f−1
µ preserves null
sets. Clearly f−1
µ is not differentiable at any point of fµ(E), contradicting the fact that
f−1
µ is differentiable a.e.
Corollary 3. Let µ : D → C be a measurable function with |µ(z)| < 1 a.e., Kµ ∈
∈ L1
loc, and let ψ : (0,∞) → (0,∞) be a measurable function such that for all 0 < t1 <
< t2 < ∞
0 <
t2∫
t1
ψ(t)dt < ∞,
t2∫
0
ψ(t)dt = ∞. (34)
Suppose that for every z0 ∈ D there is ε0 < dist(z0, ∂D) such that
∫
ε<|z−z0|<ε0
Kµ(z)ψ2
(
|z − z0|
)
dxdy ≤ O
ε0∫
ε
ψ(t)dt
(35)
as ε → 0. Then (1) has a ring solution.
Lemmas 1 and 3 yield the following theorem by choosing
ψz0,ε(t) =
1
t log 1
t
. (36)
Theorem 4. Let µ : D → C be a measurable function with
∣∣µ(z)
∣∣ < 1 a.e. and
Kµ ∈ L1
loc. Suppose that every point z0 ∈ D has a neighborhood Uz0 such that
KTµ (z, z0) ≤ Qz0(z) a.e. (37)
for some function Qz0(z) of finite mean oscillation at the point z0 in the variable z. Then
the Beltrami equation (1) has a ring solution.
The following theorem is an immediate consequence of Theorem 4.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1580 V. RYAZANOV, U. SREBRO, E. YAKUBOV
Theorem 5. Let µ : D → C be a measurable function with
∣∣µ(z)
∣∣ < 1 a.e. and
Kµ ∈ L1
loc. Suppose that at every point z0 ∈ D
lim
ε→0
−
∫
D(z0,ε)
∣∣∣∣1 − z − z0
z − z0
µ(z)
∣∣∣∣
2
1 − |µ(z)|2 dxdy < ∞. (38)
Then the Beltrami equation (1) has a ring solution fµ.
Applying Lemma 3 with ψ(t) = 1/t, we have also the following statement which
is formulated in terms of the logarithmic mean, see (29), of KTµ (z, z0) over the annuli
A(ε) =
{
z ∈ C : ε < |z − z0| < ε0
}
for a fixed ε0 = δ(z0) ≤ dist (z0, ∂D).
Theorem 6. Let µ : D → C be a measurable function with
∣∣µ(z)
∣∣ < 1 a.e. and
Kµ ∈ L1
loc. If at every point z0 ∈ D the logarithmic mean of KTµ over A(ε) does not
converge to ∞ as ε → 0, i.e.,
lim inf
ε→0
M
KT
µ
log (ε) < ∞ , (39)
then the Beltrami equation (1) has a ring solution.
Corollary 4. Let µ : D → C be a measurable function with
∣∣µ(z)
∣∣ < 1 a.e. and
Kµ ∈ L1
loc. Denote by qTz0(t) the mean value of KTµ (z, z0) over the circle C = {z ∈
∈ C : |z − z0| = t}. If
δ(z0)∫
0
qTz0(t)
dt
t
< ∞ (40)
at every point z0 ∈ D for some δ(z0) > 0, then (1) has a ring solution.
Lehto considers in [2] degenerate Beltrami equations in the special case where the
singular set Sµ
Sµ =
{
z ∈ C : lim
ε→0
‖Kµ‖L∞(D(z,ε)) = ∞
}
of the complex coefficient µ in (1) is of measure zero, and shows that, if for every z0 ∈ C
and every r1 and r2 ∈ (0,∞) the integral
r2∫
r1
dr
r(1 + qTz0(r))
, r2 > r1,
is positive and tends to ∞ as either r1 → 0 or r2 → ∞ where
qTz0(r) =
1
2π
2π∫
0
∣∣1 − e−2iϑµ(z0 + reiϑ)
∣∣2
1 −
∣∣µ(z0 + reiϑ)
∣∣2 dϑ,
then there exists a homeomorphism f : C → C which is ACL in C \ Sµ and satisfies (1)
a.e. Note that the integrand here is the tangential dilatation KTµ (z, z0), see (4).
We present now an extension of Lehto’s existence theorem which enables to derive
many other existence theorems as it was shown in [4]. In this extension we prove the
existence of a ring solution in a domain D ⊂ C which by the definition is ACL in D and
not only in D \ Sµ. Note that, in the following theorem, the situation where Sµ = D is
possible. Note also that the condition (41) in the following theorem is weaker than the
condition in Lehto’s existence theorem.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
ON THE THEORY OF THE BELTRAMI EQUATION 1581
Theorem 7. Let D be a domain in C and let µ : D → C be a measurable function
with
∣∣µ(z)
∣∣ < 1 a.e. and Kµ ∈ L1
loc. Suppose that at every point z0 ∈ D
δ(z0)∫
0
dr
rqTz0(r)
= ∞ (41)
where ε0 = δ(z0) < dist (z0, ∂D) and qTz0(r) is the mean of KTµ (z, z0) over |z−z0| = r.
Then the Beltrami equation (1) has a ring solution.
Proof. Theorem 7 follows from Lemma 3 by special choosing the functional param-
eter
ψz0,ε(t) ≡ ψz0(t) : =
{
1/
[
tqTz0(t)
]
, t ∈ (0, ε0),
0, otherwise.
Corollary 5. If Kµ ∈ L1
loc and at every point z0 ∈ D
qTz0(r) = O
(
log
1
r
)
as r → 0, (42)
then (1) has a ring solution.
Since KTµ (z, z0) ≤ Kµ(z) we obtain, in particular, as a consequence of Theorem 7
the following result which is due to Miklyukov and Suvorov [38] for the case Kµ ∈ Lploc,
p > 1.
Corollary 6. If Kµ ∈ Lploc for p ≥ 1 and (41) holds for Kµ(z) instead of KTµ (z, z0)
for every point z0 ∈ D, then (1) has a W 1,s
loc homeomorphic solution with s = 2p/(p+1).
Since the maximal dilatation dominates the tangential dilatation, all the above re-
sults obviously imply similar existence theorems in terms of conditions on the maximal
dilatation established earlier in [30] which are important particular cases, see also the
survey [39].
Remark 3. Note that all the distortion estimates for Q-homeomorphisms given in
Section 2 can be applied to ring solutions from the above existence theorems. Indeed,
in these theorems, for every domain D ⊂ C and every pairs of points z1 and z2 ∈ D
and w1 and w2 ∈ C, there exist ring solutions with the normalization f(z1) = w1 and
f(z2) = w2. The estimates hold for such solutions with Q(z) = KTµ (z, z0) and ∆ =
= min
(
1/(1 + |w1|2
)1/2
, 1/(1 + |w2|2)1/2) and, in particular, if w1 = 0 and w2 = w0,
then ∆ = 1/
(
1 + |w0|2
)1/2
. For ring solutions with
∣∣f(z1)
∣∣ and
∣∣f(z2)
∣∣ ≤ R < ∞,
∆ = 1/(1 + R2)1/2.
1. Reich E., Walczak H. On the behavior of quasiconformal mappings at a point // Trans. Amer. Math. Soc.
– 1965. – 117. – P. 338 – 351.
2. Lehto O. Homeomorphisms with a prescribed dilatation // Lect. Notes Math. – 1968. – 118. – P. 58 – 73.
3. Gutlyanskii V., Martio O., Sugawa T., Vuorinen M. On the degenerate Beltrami equation // Trans. Amer.
Math. Soc. – 2005. – 357. – P. 875 – 900.
4. Ryazanov V., Srebro U., Yakubov E. On ring solutions of Beltrami equation // J. Anal. Math. – 2005. – 96.
– P. 117 – 150.
5. Maz’ya V. Sobolev classes. – Berlin; New York: Springer, 1985.
6. Gehring F. W., Lehto O. On the total differentiability of functions of a comlex variable // Ann. Acad. Sci.
Fenn. Math. AI. – 1959. – 272. – P. 1 – 9.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1582 V. RYAZANOV, U. SREBRO, E. YAKUBOV
7. Lehto O., Virtanen K. Quasiconformal mappings in the plane. – New York etc.: Springer, 1973.
8. Reimann H. M., Rychener T. Funktionen Beschrankter Mittlerer Oscillation. – Berlin etc.: Springer, 1975.
9. John F., Nirenberg L. On functions of bounded mean oscillation // Communs Pure and Appl. Math. –
1961. – 14. – P. 415 – 426.
10. Astala K. A remark on quasiconformal mappings and BMO-functions // Mich. Math. J. – 1983. – 30. –
P. 209 – 212.
11. Astala K., Gehring F. W. Injectivity, the BMO norm and the universal Teichmuller space // J. Anal. Math.
– 1986. – 46. – P. 16 – 57.
12. Gehring F. W. Characteristic Properties of Quasidisk. – Montreal: Presses Univ. de Montreal, 1982.
13. Jones P. M. Extension theorems for BMO // Indiana Univ. Math. J. – 1980. – 29. – P. 41 – 66.
14. Martio O., Ryazanov V., Vuorinen M. BMO and injectivity of space quasiregular mappings // Math. Nachr.
– 1999. – 205. – P. 149 – 161.
15. Reimann H. M. Functions of bounded mean oscillation and quasiconformal mappings // Comment. math.
helv. – 1974. – 49. – P. 260 – 276.
16. Astala K., Iwaniec T., Koskela P., Martin G. Mappings of BMO-bounded distortion // Math. Ann. – 2000.
– 317. – P. 703 – 726.
17. Iwaniec T., Koskela P., Martin G. Mappings of BMO-distortion and Beltrami type // J. Anal. Math. – 2002.
– 88. – P. 337 – 381.
18. Iwaniec T., Martin G. Geometrical function theory and nonlinear analysis. – Oxford: Clarendon Press,
2001.
19. Sastry S. Boundary behavior of BMO-QC automorphisms // Isr. J. Math. – 2002. – 129. – P. 373 – 380.
20. Sarason D. Functions of vanishing mean oscillation // Trans. Amer. Math. Soc. – 1975. – 207. – P. 391 –
405.
21. Chiarenza F., Frasca M., Longo P. W 2,p-solvability of the Dirichlet problem for nondivergence elliptic
equations with VMO coefficients // Ibid. – 1993. – 336, # 2. – P. 841 – 853.
22. Iwaniec T., Sbordone C. Riesz transforms and elliptic PDEs with VMO coefficients // J. Anal. Math. –
1998. – 74. – P. 183 – 212.
23. Palagachev D. K. Quasilinear elliptic equations with VMO coefficients // Trans. Amer. Math. Soc. – 1995.
– 347, # 7. – P. 2481 – 2493.
24. Ragusa M. A. Elliptic boundary value problem in vanishing mean oscillation hypothesis // Comment.
math. Univ. Carol. – 1999. – 40, # 4. – P. 651 – 663.
25. Brezis H., Nirenberg L. Degree theory and BMO. I. Compact manifolds without boundaries // Selecta
Math. (N. S.). – 1995. – 1, # 2. – P. 197 – 263.
26. Cianchi A., Pick L. Sobolev embeddings into BMO, VMO, and L∞ // Ark. mat. – 1998. – 36, # 2. –
P. 317 – 340.
27. Ryazanov V., Srebro U., Yakubov E. BMO-quasiconformal mappings // J. Anal. Math. – 2001. – 83. –
P. 1 – 20.
28. Ryazanov V., Srebro U., Yakubov E. Plane mappings with dilatation dominated by functions of bounded
mean oscillation // Sib. Adv. Math. – 2001. – 11, # 2. – P. 94 – 130.
29. Ignat’ev A., Ryazanov V. Finite mean oscillation in the mapping theory // Ukr. Math. Bull. – 2005. – 2,
# 3. – P. 403 – 424.
30. Ryazanov V., Srebro U., Yakubov E. Finite mean oscillation and the Beltrami equation // Isr. J. Math. –
2006. – 153. – P. 247 – 266.
31. Ryazanov V., Srebro U., Yakubov E. The Beltrami equation and FMO functions // Contemp. Math. – 2005.
– 382. – P. 247 – 266.
32. David G. Solutions de l’equation de Beltrami avec |µ|∞ = 1 // Ann. Acad. Sci. Fenn. Ser. AI. Math. AI.
– 1988. – 13, # 1. – P. 25 – 70.
33. Tukia P. Compactness properties of µ-homeomorphisms // Ibid. – 1991. – 16, # 1. – P. 47 – 69.
34. Brakalova M. A., Jenkins J. A. On solutions of the Beltrami equation // J. Anal. Math. – 1998. – 76. –
P. 67 – 92.
35. Kruglikov V. I. The existence and uniqueness of mappings that are quasiconformal in the mean // Metric
Questions of the Theory of Functions and Mappings. – Kiev: Naukova Dumka, 1973. – P. 123 – 147.
36. Martio O., Miklyukov V. On existence and uniqueness of the degenerate Beltrami equation // Complex
Variables Theory Appl. – 2004. – 49. – P. 647 – 656.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
ON THE THEORY OF THE BELTRAMI EQUATION 1583
37. Miklyukov V. M., Suvorov G. D. On existence and uniqueness of quasiconformal mappings with unbounded
characteristics // Investigations in the Theory of Functions of Complex Variables and its Applications. –
Kiev: Inst. Math. Acad. Sci. USSR, 1972.
38. Pesin I. N. Mappings quasiconformal in the mean // Dokl. Akad. Nauk SSSR. – 1969. – 187, # 4. –
P. 740 – 742.
39. Srebro U., Yakubov E. The Beltrami equation // Complex Analysis, Geometry, Function Theory. – Ams-
terdam: Elsevier, 2005. – Vol. 2. – P. 555 – 597.
40. Saks S. Theory of the integral. – New York: Dover Publ. Inc., 1964.
41. Martio O., Ryazanov V., Srebro U., Yakubov E. Q-homeomorphisms // Contemp. Math. – 2004. – 364. –
P. 193 – 203.
42. Martio O., Ryazanov V., Srebro U., Yakubov E. Mappings with finite length distortion // J. Anal. Math. –
2004. – 93. – P. 215 – 236.
43. Martio O., Ryazanov V., Srebro U., Yakubov E. On Q-homeomorphisms // Ann. Acad. Sci. Fenn. – 2005.
– 30. – P. 49 – 69.
44. Gehring F. W. A remark on the moduli of rings // Comment. math. helv. – 1961. – 36. – P. 42 – 46.
45. Vaisala J. Lectures on n-dimensional quasiconformal mappings // Lect. Notes Math. – Berlin etc.:
Springer, 1971. – 229.
46. Gehring F. W. Rings and quasiconformal mappings in space // Trans. Amer. Math. Soc. – 1962. – 103. –
P. 353 – 393.
47. Heinonen J., Koskela P. Sobolev mappings with integrable dilatations // Arch. Ration. Mech. and Anal. –
1993. – 125. – P. 81 – 97.
48. Ahlfors L. V. Lectures on quasiconformal mappings. – Princeton etc.: D. Van Nostrand Company, Inc.,
1966.
49. Dugundji J. Topology. – Boston: Allyn and Bacon, Inc., 1966.
50. Dunford N., Schwartz J. T. Linear operators. Pt I: General theory. – New York; London: Int. Publ., Inc.,
1957.
51. Reshetnyak Yu. G. Space mappings with bounded distortion // Transl. Math. Monographs, AMS. – 1989.
– 73.
Received 29.06.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
|