The Beltrami equations and lower Q-homeomorphisms
In this article it is shown that each homeomorphic W1,1loc solution to the Beltrami equation ∂f = μ∂f is the so-called lower Q-homeomorphism with Q(z) = Kμ(z) where Kμ(z) is dilatation quotient of this equation. It is developed on this base the theory of the boundary behavior and the removability of...
Збережено в:
| Дата: | 2010 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2010
|
| Назва видання: | Труды Института прикладной математики и механики |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/123958 |
| 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: | The Beltrami equations and lower Q-homeomorphisms / D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2010. — Т. 21. — С. 114-117. — Бібліогр.: 13 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-123958 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1239582017-09-16T03:03:43Z The Beltrami equations and lower Q-homeomorphisms Kovtonyuk, D.A. Petkov, I.V. Ryazanov, V.I. In this article it is shown that each homeomorphic W1,1loc solution to the Beltrami equation ∂f = μ∂f is the so-called lower Q-homeomorphism with Q(z) = Kμ(z) where Kμ(z) is dilatation quotient of this equation. It is developed on this base the theory of the boundary behavior and the removability of singularities of such solutions. В работе показано, что любое гомеоморфное W1,1loc решение уравнения Бельтрами ∂f = μ∂f является так называемым нижним <3-гомеоморфизмом с Q(z) = Kμ(z) где Kμ(z) - коэффициент дилатации этого уравнения. На этой основе развита теория граничного поведения и устранимость сингулярностей таких решений. У роботі показано, що будь-який гомеоморфний W1,1loc розв'язок рівняння Бельтрамі ∂f = μ∂f є так званим нижнім (^-гомеоморфізмом зQ(z) = Kμ(z), де Kμ(z) - коефіцієнт дилатації цього рівняння. На цій основі розвинуто теорію граничної поведінки і усунення сингулярностей таких розв'язків. 2010 Article The Beltrami equations and lower Q-homeomorphisms / D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2010. — Т. 21. — С. 114-117. — Бібліогр.: 13 назв. — англ. 1683-4720 http://dspace.nbuv.gov.ua/handle/123456789/123958 517.5 en Труды Института прикладной математики и механики Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this article it is shown that each homeomorphic W1,1loc solution to the Beltrami equation ∂f = μ∂f is the so-called lower Q-homeomorphism with Q(z) = Kμ(z) where Kμ(z) is dilatation quotient of this equation. It is developed on this base the theory of the boundary behavior and the removability of singularities of such solutions. |
| format |
Article |
| author |
Kovtonyuk, D.A. Petkov, I.V. Ryazanov, V.I. |
| spellingShingle |
Kovtonyuk, D.A. Petkov, I.V. Ryazanov, V.I. The Beltrami equations and lower Q-homeomorphisms Труды Института прикладной математики и механики |
| author_facet |
Kovtonyuk, D.A. Petkov, I.V. Ryazanov, V.I. |
| author_sort |
Kovtonyuk, D.A. |
| title |
The Beltrami equations and lower Q-homeomorphisms |
| title_short |
The Beltrami equations and lower Q-homeomorphisms |
| title_full |
The Beltrami equations and lower Q-homeomorphisms |
| title_fullStr |
The Beltrami equations and lower Q-homeomorphisms |
| title_full_unstemmed |
The Beltrami equations and lower Q-homeomorphisms |
| title_sort |
beltrami equations and lower q-homeomorphisms |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/123958 |
| citation_txt |
The Beltrami equations and lower Q-homeomorphisms / D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2010. — Т. 21. — С. 114-117. — Бібліогр.: 13 назв. — англ. |
| series |
Труды Института прикладной математики и механики |
| work_keys_str_mv |
AT kovtonyukda thebeltramiequationsandlowerqhomeomorphisms AT petkoviv thebeltramiequationsandlowerqhomeomorphisms AT ryazanovvi thebeltramiequationsandlowerqhomeomorphisms AT kovtonyukda beltramiequationsandlowerqhomeomorphisms AT petkoviv beltramiequationsandlowerqhomeomorphisms AT ryazanovvi beltramiequationsandlowerqhomeomorphisms |
| first_indexed |
2025-07-09T00:36:18Z |
| last_indexed |
2025-07-09T00:36:18Z |
| _version_ |
1837127570962776064 |
| fulltext |
ISSN 1683-4720 Труды ИПММ НАН Украины. 2010. Том 21
UDK 517.5
c©2010. D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov
THE BELTRAMI EQUATIONS AND LOWER Q-HOMEOMORPHISMS
In this article it is shown that each homeomorphic W 1,1
loc solution to the Beltrami equation ∂f = µ ∂f
is the so-called lower Q-homeomorphism with Q(z) = Kµ(z) where Kµ(z) is dilatation quotient of this
equation. It is developed on this base the theory of the boundary behavior and the removability of
singularities of such solutions.
Key words:Beltrami equations, lower Q-homeomorphism
1. Introduction. In this paper we present applications of our results on the so-called
lower Q-homeomorphisms in the monograph [9] to the study of the boundary behavior
of solutions for the Beltrami equations with degeneration.
Let D be a domain in the complex plane C, i.e., a connected and open subset of C,
and let µ : D → C be a measurable function with |µ(z)| < 1 a.e. (almost everywhere) in
D. 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 dilatation quotient for the equation (1). The Beltrami equation (1) is said to be
degenerate if ess sup Kµ(z) = ∞.
The existence theorem for homeomorphic W 1,1
loc solutions was established to many
degenerate Beltrami equations, see, e.g., the recent monographs [1] and [9] and the surveys
[6] and [13].
A continuous mapping γ of an open subset ∆ of the real axis R or a circle into D
is called a dashed line, see, e.g., Section 6.3 in [9]. Recall that every open set ∆ in R
consists of a countable collection of mutually disjoint intervals. This is the motivation
for the term.
Given a family Γ of dashed lines γ in complex plane C, a Borel function % : C→ [0,∞]
is called admissible for Γ, write % ∈ admΓ, if
∫
γ
% ds > 1 (3)
for every γ ∈ Γ. The (conformal) modulus of Γ is the quantity
M(Γ) = inf
%∈admΓ
∫
C
%2(z) dm(z) (4)
114
The Beltrami equations and lower Q-homeomorphisms
where dm(z) corresponds to the Lebesgue measure in C. We say that a property P
holds for a.e. (almost every) γ ∈ Γ if a subfamily of all lines in Γ for which P fails has
the modulus zero, cf. [3]. Later on, we also say that a Lebesgue measurable function
% : C → [0,∞] is extensively admissible for Γ, write % ∈ ext admΓ, if (3) holds for
a.e. γ ∈ Γ, see, e.g., Section 9.2 in [9].
The following concept was motivated by Gehring’s ring definition of quasiconforma-
lity in [4]. Given domains D and D′ in C = C ∪ {∞}, z0 ∈ D \ {∞}, and a measurable
function Q : D → (0,∞), we say that a homeomorphism f : D → D′ is a lower
Q-homeomorphism at the point z0 if
M(fΣε) > inf
%∈ext adm Σε
∫
D∩Rε
%2(x)
Q(x)
dm(x) (5)
for every ring
Rε = {z ∈ C : ε < |z − z0| < ε0}, ε ∈ (0, ε0), ε0 ∈ (0, d0),
where
d0 = sup
z∈D
|z − z0|,
and Σε denotes the family of all intersections of the circles
S(r) = S(z0, r) = {z ∈ C : |z − z0| = r}, r ∈ (ε, ε0),
with the domain D.
The notion can be extended to the case z0 = ∞ ∈ D in the standard way by applying
the inversion T with respect to the unit circle in C, T (x) = z/|z|2, T (∞) = 0, T (0) = ∞.
Namely, a homeomorphism f : D → D′ is a lower Q-homeomorphism at ∞ ∈ D
if F = f ◦ T is a lower Q∗-homeomorphism with Q∗ = Q ◦ T at 0. We also say that
a homeomorphism f : D → C is a lower Q-homeomorphism in ∂D if f is a lower
Q-homeomorphism at every point z0 ∈ ∂D.
Further we show that each homeomorphic W 1,1
loc solution of the Beltrami equation
(1) is a lower Q-homeomorphism with Q(z) = Kµ(z) and, thus, the whole theory of the
boundary behavior in [7], see also Chapter 9 in [9], can be applied to such solutions.
In other words, in the plane this holds for homeomorphisms with finite distortion by
Iwaniec, see, e.g., related references in the monographs [1] and [9].
2. The main result.
Theorem. Let f be a homeomorphic W 1,1
loc solution of the Beltrami equation (1).
Then f is a lower Q-homeomorphism at each point z0 ∈ D with Q(z) = Kµ(z).
Proof. Let B be a (Borel) set of all points z in D where f has a total differential with
Jf (z) 6= 0 a.e. It is known that B is the union of a countable collection of Borel sets Bl,
l = 1, 2, . . . , such that fl = f |Bl
is a bi-Lipschitz homeomorphism, see e.g. Lemma 3.2.2 in
[2]. With no loss of generality, we may assume that the Bl are mutually disjoint. Denote
also by B∗ the set of all points z ∈ D where f has a total differential with f ′(z) = 0.
115
D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov
Note that the set B0 = D \ (B∪B∗) has the Lebesgue measure zero in C by Gehring-
Lehto-Menchoff theorem, see [5] and [11]. Hence by Theorem 2.11 in [8], see also Lemma
9.1 in [9], length(γ∩B0) = 0 for a.e. paths γ in D. Let us show that length(f(γ)∩f(B0)) =
0 for a.e. circle γ centered at z0.
The latter follows from absolute continuity of f on closed subarcs of γ ∩ D for a.e.
such circle γ. Indeed, the class W 1,1
loc is invariant with respect to local quasi-isometries,
see e.g. Theorem 1.1.7 in [10], and the functions in W 1,1
loc is absolutely continuous on lines,
see e.g. Theorem 1.1.3 in [10]. Applying say the transformation of coordinates log(z−z0),
we come to the absolute continuity on a.e. such circle γ.
Thus, length(γ∗ ∩ f(B0)) = 0 where γ∗ = f(γ) for a.e. circle γ centered at z0. Now,
let %∗ ∈ adm f(Γ) where Γ is the collection of all dashed lines γ ∩ D for such circles γ
and %∗ ≡ 0 outside f(D). Set % ≡ 0 outside D and
%(z) : = %∗(f(z)) (|fz|+ |fz̄|) for a.e. z ∈ D
Arguing piecewise on Bl, we have by Theorem 3.2.5 under m = 1 in [2] that
∫
γ
% ds >
∫
γ∗
%∗ ds∗ > 1 for a.e. γ ∈ Γ
because length(f(γ) ∩ f(B0)) = 0 and length(f(γ) ∩ f(B∗)) = 0 for a.e. γ ∈ Γ,
consequently, % ∈ ext adm Γ.
On the other hand, again arguing piecewise on Bl, we have the inequality
∫
D
%2(x)
Kµ(z)
dm(z) 6
∫
f(D)
%2
∗(w) dm(w)
because %(z) = 0 on B∗. Consequently, we obtain that
M(fΓ) > inf
%∈ext admΓ
∫
D
%2(z)
Kµ(z)
dm(z) ,
i.e., f is really a lower Q-homeomorphism with Q(z) = Kµ(z).
1. Astala K., Iwaniec T. and Martin G.J. Elliptic differential equations and quasiconformal mappings
in the plane. – Princeton Math. Ser., v. 48, Princeton Univ. Press, Princeton, 2009.
2. Federer H. Geometric Measure Theory. – Springer-Verlag, Berlin 1969.
3. Fuglede B. Extremal length and functional completion // Acta Math. – 1957. – V. 98. – P. 171-219.
4. Gehring F.W. Rings and quasiconformal mappings in space // Trans. Amer. Math. Soc. – 1962. –
V. 103. – P. 353-393.
5. Gehring F.W., Lehto O. On the total differentiability of functions of a complex variable // Ann.
Acad. Sci. Fenn. A1. Math. – 1959. – V. 272. – P. 1-9.
6. Gutlyanskii V., Ryazanov V., Srebro U. and Yakubov E. On recent advances in the degenerate
Beltrami equations // Ukraininan Math. Bull. – 2010. – V. 7, no. 4. – P. 467-515.
7. Kovtonyuk D., Ryazanov V. On the theory of lower Q-homeomorphisms // Ukrainian Math. Bull.
– 2008. – V. 5, no. 2. – P. 157-181.
116
The Beltrami equations and lower Q-homeomorphisms
8. Kovtonyuk D., Ryazanov V. On the theory of mappings with finite area distortion // J. Anal. Math.
– 2008. – V. 104. – P. 291-306.
9. Martio O., Ryazanov V., Srebro U., Yakubov E. Moduli in Modern Mapping Theory. – Springer,
New York, 2009.
10. Maz’ya V. Sobolev Classes. – Springer-Verlag, Berlin, 1985.
11. Menchoff D. Sur les differentielles totales des fonctions univalentes // Math. Ann. – 1931. – V. 105.
– P. 75-85.
12. Ryazanov V., Srebro U. and Yakubov E. BMO-quasiconformal mappings // J. d’Anal. Math. – 2001.
– V. 83. – P. 1-20.
13. Srebro U. and Yakubov E. The Beltrami equation // Handbook in Complex Analysis. Geometric
function theory. – 2005. – V. 2. – P. 555-597.
Д.А. Ковтонюк, И.В. Петков, В.И. Рязанов
Уравнения Бельтрами и нижние Q-гомеоморфизмы.
В работе показано, что любое гомеоморфное W 1,1
loc решение уравнения Бельтрами ∂f = µ ∂f яв-
ляется так называемым нижним Q-гомеоморфизмом с Q(z) = Kµ(z), где Kµ(z) – коэффициент
дилатации этого уравнения. На этой основе развита теория граничного поведения и устранимость
сингулярностей таких решений.
Ключевые слова: Уравнения Бельтрами, нижние Q-гомеоморфизмы
Д.О. Ковтонюк, I.В. Пєтков, В.I. Рязанов
Рiвняння Бельтрамi та нижнi Q-гомеоморфiзми.
У роботi показано, що будь-який гомеоморфний W 1,1
loc розв’язок рiвняння Бельтрамi ∂f = µ ∂f
є так званим нижнiм Q-гомеоморфiзмом з Q(z) = Kµ(z), де Kµ(z) – коефiцiєнт дилатацiї цього
рiвняння. На цiй основi розвинуто теорiю граничної поведiнки i усунення сингулярностей таких
розвязкiв.
Ключовi слова: Рiвняння Бельтрамi, нижнi Q-гомеоморфiзми
Ин-т прикл. математики и механики НАН Украины, Донецк
denis kovtonyuk@bk.ru, igorpetkov@list.ru,
vlryazanov1@rambler.ru
Received 10.12.10
117
|