The Dirichlet problem for a Beltrami equation of the second type
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irk-123456789-857652015-08-20T03:01:35Z The Dirichlet problem for a Beltrami equation of the second type Bojarski, B.V. Gutlyanskiĭ, V.Ya. Ryazanov, V.I. Математика 2013 Article The Dirichlet problem for a Beltrami equation of the second type / B.V. Bojarski, V.Ya. Gutlyanskiĭ, V.I. Ryazanov // Доповiдi Нацiональної академiї наук України. — 2013. — № 6. — С. 29–34. — Бібліогр.: 15 назв. — англ. 1025-6415 http://dspace.nbuv.gov.ua/handle/123456789/85765 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Математика Математика Bojarski, B.V. Gutlyanskiĭ, V.Ya. Ryazanov, V.I. The Dirichlet problem for a Beltrami equation of the second type Доповіді НАН України |
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Article |
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Bojarski, B.V. Gutlyanskiĭ, V.Ya. Ryazanov, V.I. |
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Bojarski, B.V. Gutlyanskiĭ, V.Ya. Ryazanov, V.I. |
| author_sort |
Bojarski, B.V. |
| title |
The Dirichlet problem for a Beltrami equation of the second type |
| title_short |
The Dirichlet problem for a Beltrami equation of the second type |
| title_full |
The Dirichlet problem for a Beltrami equation of the second type |
| title_fullStr |
The Dirichlet problem for a Beltrami equation of the second type |
| title_full_unstemmed |
The Dirichlet problem for a Beltrami equation of the second type |
| title_sort |
dirichlet problem for a beltrami equation of the second type |
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Видавничий дім "Академперіодика" НАН України |
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2013 |
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Математика |
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http://dspace.nbuv.gov.ua/handle/123456789/85765 |
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The Dirichlet problem for a Beltrami equation of the second type / B.V. Bojarski, V.Ya. Gutlyanskiĭ, V.I. Ryazanov // Доповiдi Нацiональної академiї наук України. — 2013. — № 6. — С. 29–34. — Бібліогр.: 15 назв. — англ. |
| series |
Доповіді НАН України |
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2025-07-06T13:07:10Z |
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2025-07-06T13:07:10Z |
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| fulltext |
UDC 517.5
B.V. Bojarski,
Corresponding Member of the NAS of Ukraine V. Ya. Gutlyanskĭı,
V. I. Ryazanov
The Dirichlet problem for a Beltrami equation of the
second type
Сriteria on the existence of regular solutions of the Dirichlet problem for the degenerate Beltrami
equation ∂f = ν∂f in a Jordan domain of the complex plane C are given.
Let D be a domain in the complex plane C. Throughout this paper, we use the notations z =
= x+ iy, B(z0, r) : = {z ∈ C : |z − z0| < r} for z0 ∈ C and r > 0, B(r) : = B(0, r), B : = B(1),
and C : = C
⋃
∞.
The purpose of this paper is to study the Dirichlet problem
fz = ν(z) · fz, z ∈ D,
lim
z→ζ
Ref(z) = ϕ(ζ), ∀ ζ ∈ ∂D,
(1)
in a Jordan domain D of the complex plane C with continuous boundary data ϕ(ζ). Here, ν(z)
stands for a measurable coefficient satisfying the inequality |ν(z)| < 1 a. e. in D. The degeneracy
of the ellipticity for Beltrami equations of the second type fz = ν(z) · fz is controlled by the
dilatation coefficient
Kν(z) :=
1 + |ν(z)|
1− |ν(z)|
∈ L1
loc. (2)
Note that the Beltrami equations of the second type take a key part in many problems of
mathematical physics, see, e. g., [11].
We will look for a solution as a continuous, discrete, and open mapping f : D → C of the
Sobolev class W 1,1
loc and such that the Jacobian Jf (z) 6= 0 a. e. in D. Such a solution will be called
a regular solution of the Dirichlet problem (1) in a domain D. Recall that a mapping f : D → C
is called discrete if the preimage f−1(y) consists of isolated points for every y ∈ C and open if f
maps every open set U ⊆ D onto an open set in C.
For the uniformly elliptic case, i. e. when Kν(z) 6 K < ∞ a. e. in D, the Dirichlet problem
was studied in [1]. The solvability of the Dirichlet problem for the degenerate Beltrami equations
of the first type
fz = µ(z) · fz (3)
was studied in [5, 8], and [12]. Recall that the problem of the existence of homeomorphic solutions
for equation (3) was solved in the uniformly elliptic case where ‖µ‖∞ < 1 long ago, see, e. g., [1].
The existence problem for the degenerate Beltrami equations (3), when Kµ /∈ L∞, is currently
an active area of research, see, e. g., survey [7] and monograph [8] and references therein. A series
of criteria on the existence of regular solutions for a Beltrami equation of the second type were
© B. V. Bojarski, V.Ya. Gutlyanskĭı, V. I. Ryazanov, 2013
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №6 29
given in our recent papers [3, 4]. There, a homeomorphism f ∈W 1,1
loc (D) is called a regular solution
of the equation if f satisfies the equation a. e. in D, and Jf (z) = |fz|
2 − |fz|
2 6= 0 a. e. in D.
To derive criteria of the existence of regular solutions for the Dirichlet problem (1) in a Jordan
domain D ⊂ C, we make use of the approximate procedure based on the existence theorems in
the case Kν ∈ L∞ given in [1] and convergence theorems for the Beltrami equations of the second
type when Kν ∈ L1
loc established in [3]. The Schwarz formula
f(z) = i Im f(0) +
1
2πi
∫
|ζ|=1
Re f(ζ) ·
ζ + z
ζ − z
dζ
ζ
, (4)
that allows one to recover an analytic function f in the unit disk B by its real part ϕ(ζ) = Ref(ζ)
on the boundary of B up to a purely imaginary additive constant c = iImf(0), see, e. g., Section 8,
Chapter III, Part 3 in [6], and the Arzela–Ascoli theorem combined with moduli techniques are
also used.
2. On BMO, VMO, and FMO functions. Recall that a real-valued function u in a
domain D in C is said to be a bounded mean oscillation in D, abbr. u ∈ BMO(D), if u ∈ L1
loc(D)
and
‖u‖∗ := sup
B
1
|B|
∫
B
|u(z)− uB | dxdy <∞, (5)
where the supremum is taken over all disks B in D and
uB =
1
|B|
∫
B
u(z) dxdy.
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 [10] and soon became an
important concept in harmonic analysis, the theory of partial differential equations, and related
areas.
A function u in BMO is said to have a vanishing mean oscillation, abbr. u ∈ VMO, if the
supremum in (5) taken over all balls B in D with |B| < ε converges to 0 as ε→ 0. VMO has been
introduced by Sarason in [15]. There exist a number of papers devoted to the study of partial
differential equations with coefficients of the class VMO.
Following [9], we say that a function u : D → R has a finite mean oscillation at a point
z0 ∈ D if
lim
ε→0
−
∫
B(z0,ε)
|u(z)− ũε(z0)| dxdy <∞, (6)
where ũε(z0) ũε(z0) = −
∫
B(z0,ε)
u(z) dxdy is the mean value of the function u(z) over the disk
B(z0, ε) with small ε > 0. We also say that a function u : D → R has a finite mean oscillation
in D, abbr. u ∈ FMO(D) or simply u ∈ FMO, if relation (6) holds at every point z0 ∈ D.
Remark 1. Clearly, BMO ⊂ FMO. There exist the examples showing that FMO is not BMOloc,
see, e. g., [8]. By definition, FMO ⊂ L1
loc, but FMO is not a subset of Lp
loc for any p > 1 in
comparison with BMOloc ⊂ Lp
loc for all p ∈ [1,∞).
30 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №6
Proposition 1. If, for some collection of numbers uε ∈ R, ε ∈ (0, ε0],
lim
ε→0
−
∫
B(z0,ε)
|u(z)− uε| dxdy <∞, (7)
then u has a finite mean oscillation at z0.
Corollary 1. If, for a point z0 ∈ D,
lim
ε→0
−
∫
B(z0,ε)
|u(z)| dxdy <∞ (8)
then u has a finite mean oscillation at z0.
Remark 2. Note that the function u(z) = log(1/|z|) belongs to BMO in the unit disk B and
hence also to FMO. However, ũε(0) → ∞ as ε→ 0, showing that condition (8) is only sufficient,
but not necessary for a function u to have a finite mean oscillation at z0.
Lemma 1. Let u : D → R be a nonnegative function with finite mean oscillation at 0 ∈ D,
and let u be integrable in B(0, e−1) ⊂ D. Then
∫
A(ε,e−1)
u(z) dxdy
(
|z| log 1
|z|
)2 6 C · log log
1
ε
, ∀ ε ∈ (0, e−e). (9)
Here, we use the notation A(ε, ε0) = {z ∈ C : ε < |z| < ε0}.
3. The main lemma. The following lemma is the main tool for deriving criteria on the
existence of regular solutions for the Dirichlt problem with degenerate Beltrami equations of the
second type in a Jordan domain D ⊂ C.
Lemma 2. Let D be a Jordan domain in C with 0 ∈ D and let ν : D → C be a measurable
function with Kν ∈ L1(D). Suppose that, for every z0 ∈ D, there exist ε0 = ε(z0) > 0 and a
family of measurable functions ψz0,ε : (0,∞) → (0,∞), ε ∈ (0, ε0), such that
0 < Iz0(ε) :=
ε0∫
ε
ψz0,ε(t) dt <∞, (10)
and such that
∫
ε<|z−z0|<ε0
Kν(z) · ψ
2
z0,ε
(|z − z0|) dxdy = o(I2z0(ε)) (11)
as ε → 0. Then the Dirichlet problem (1) has a regular solution f with Im f(0) = 0 for each
nonconstant continuous function ϕ : ∂D → R.
Here, we assume that ν is extended by zero outside the domain D.
Corollary 2. Let D be a Jordan domain in C with 0 ∈ D and let ν : B → C be a measurable
function with Kν ∈ L1(B). Suppose that, for every z0 ∈ B and some ε0 > 0,
∫
ε<|z−z0|<ε0
Kν(z) · ψ
2(|z − z0|) dxdy 6 O
( ε0∫
ε
ψ(t) dt
)
(12)
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №6 31
as ε → 0, where ψ : (0,∞) → (0,∞) is a measurable function such that
ε0∫
0
ψ(t) dt = ∞, 0 <
ε0∫
ε
ψ(t) dt <∞, ∀ ε ∈ (0, ε0). (13)
Then the Dirichlet problem (1) has a regular solution f with Im f(0) = 0 for each nonconstant
continuous function ϕ : ∂D → R.
4. Existence theorems. Everywhere further, we assume that the function ν : D → C is
extended by zero outside the domain D. In particular, by Lemmas 1 and 2 with ψz0,ε(t) ≡
≡ 1/(t log 1/t), we have the following result.
Theorem 1. Let D be a Jordan domain in C with 0 ∈ D and let ν : D → B be a measurable
function such that Kν(z) 6 Q(z) ∈ FMO. Then the Dirichlet problem (1) has a regular solution f
with Im f(0) = 0 for each nonconstant continuous function ϕ : ∂D → R.
Now, Theorem 1 and Corollary 1 yield the following conclusion.
Corollary 3. In particular, if
lim
ε→0
−
∫
B(z0,ε)
1 + |ν(z)|
1− |ν(z)|
dxdy <∞, ∀ z0 ∈ D, (14)
then the Dirichlet problem (1) in a Jordan domain D, 0 ∈ D, has a regular solution f with
Im f(0) = 0 for each nonconstant continuous function ϕ : ∂D → R.
Similarly, choosing the function ψz0,ε(t) ≡ 1/t, in Lemma 2, we come to the following
statement.
Theorem 2. Let D be a Jordan domain in C with 0 ∈ D and let ν : D → B be a measurable
function such that Kν ∈ L1
loc(D). Suppose that
∫
ε<|z−z0|<ε0
Kν(z)
dxdy
|z − z0|2
= o
([
log
1
ε
]2)
, ∀ z0 ∈ D, (15)
as ε → 0 for some ε0 = δ(z0). Then the Dirichlet problem (1) has a regular solution f with
Im f(0) = 0 for each nonconstant continuous function ϕ : ∂D → R.
Remark 3. Choosing the function ψ(t) = 1/(t log 1/t) instead of ψ(t) = 1/t in Lemma 2, we
are able to replace (15) by
∫
ε<|z−z0|<ε0
Kν(z) dxdy(
|z − z0| log
1
|z − z0|
)2 = o
([
log log
1
ε
]2)
. (16)
In general, we are able to give here the whole scale of the corresponding conditions in log, using
functions ψ(t) of the form 1/(t log 1/t · log log 1/t · · · · · log . . . log 1/t).
Theorem 3. Let D be a Jordan domain in C with 0 ∈ D, let ν : D → B be a measurable
function, Kν ∈ L1(D), and let kz0(r) be the mean value of Kν(z) over the circle |z − z0| = r.
Suppose that
δ(z0)∫
0
dr
rkz0(r)
= ∞, ∀ z0 ∈ D. (17)
32 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №6
Then the Dirichlet problem (1) has a regular solution f with Imf(0) = 0 for each nonconstant
continuous function ϕ : ∂D → R.
Theorem 3 also follows from Lemma 2 by a special choice of the functional parameter
ψz0,ε(t) ≡ 1/[tkz0(t)].
Corollary 4. In particular, the conclusion of Theorem 3 holds if
kz0(r) = O
(
log
1
r
)
as r → 0, ∀ z0 ∈ D. (18)
In fact, it is clear that condition (17) yields the whole scale of conditions in terms of log with
the use of functions of the form log 1/r ·log log 1/r · · · log · · · log 1/r on the right-hand side in (18).
By Theorem 3.1 in [13], see also Theorem 3.17 in [14], it follows that conditions (19) and (20)
below yield condition (17). Thus, by Theorem 3, we obtain the following significant result.
Theorem 4. Let D be a Jordan domain in C with 0 ∈ D, and let µ and ν : D → B be
measurable functions such that
∫
D
Φ(Kν(z)) dxdy <∞, (19)
where Φ: [0,∞] → [0,∞] is a non-decreasing convex function satisfying the condition
∞∫
δ
dτ
τΦ−1(τ)
= ∞ (20)
for some δ > Φ(0). Then the Dirichlet problem (1) has a regular solution f with Im f(0) = 0 for
each nonconstant continuous function ϕ : ∂D → R.
Remark 4. Note also that we may assume in Theorem 4 that the function Φ(t) is not convex
and non-decreasing on the whole segment [0,∞], but only on a segment [T,∞] for some T ∈
∈ (1,∞).
Corollary 5. In particular, the conclusion of Theorem 4 holds if, for some α > 0,
∫
D
eαKν(z)dxdy <∞. (21)
Finally we would like to note that our approach makes it possible to study the Dirichlet
problem for the degenerate Beltrami equations also in finitely connected domains of the complex
plane.
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Received 25.12.2012Institute of Mathematics of Polish Academy
of Sciences, Warsaw, Poland
Institute of Applied Mathematics and Mechanics,
NAS of Sciences of Ukraine, Donetsk, Ukraine
Б.В. Боярський, член-кореспондент НАН України В.Я. Гутлянський,
В.И. Рязанов
Задача Дiрiхле для рiвняння Бельтрамi другого роду
Для виродженого рiвняння Бельтрамi ∂f = ν∂f доведено критерiї iснування регулярного
розв’язку задачи Дiрiхле у довiльнiй жордановiй областi комплексної площини C.
Б.В. Боярский, член-корреспондент НАН Украины В.Я. Гутлянский,
В.И. Рязанов
Задача Дирихле для уравнения Бельтрами второго рода
Для вырожденного уравнения Бельтрами ∂f = ν∂f доказаны критерии существования ре-
гулярного решения задачи Дирихле в произвольной жордановой области комплексной плос-
кости C.
34 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №6
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