Toward the theory of the Dirichlet problem for the Beltrami equations
The Dirichlet problem for the degenerate Beltrami equations in arbitrary finitely connected domains is studied. In terms of the tangent dilatations, a series of criteria for the existence of regular solutions in arbitrary simply connected domains, as well as pseudoregular and multivalent solution...
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irk-123456789-979432016-04-06T03:02:41Z Toward the theory of the Dirichlet problem for the Beltrami equations Gutlyanskii, V.Ya. Ryazanov, V.I. Yakubov, E. Математика The Dirichlet problem for the degenerate Beltrami equations in arbitrary finitely connected domains is studied. In terms of the tangent dilatations, a series of criteria for the existence of regular solutions in arbitrary simply connected domains, as well as pseudoregular and multivalent solutions in arbitrary finitely connected domains without degenerate boundary components, are formulated. Вивчається задача Дiрiхле для вироджених рiвнянь Бельтрамi в довiльних скiнченнозв’язних областях. У термiнах дотичних дилатацiй сформульовано цiлий ряд критерiїв iснування регулярних розв’язкiв цiєї проблеми в довiльних обмежених однозв’язних областях, а також псевдорегулярних i багатозначних розв’язкiв в довiльних обмежених скiнченнозв’язних областях без вироджених граничних компонентiв. Изучается задача Дирихле для вырожденных уравнений Бельтрами в произвольных конечносвязных областях. В терминах касательных дилатаций сформулирован целый ряд критериев существования регулярных решений этой проблемы в произвольных ограниченных односвязных областях, а также псевдорегулярных и многозначных решений в произвольных ограниченных конечносвязных областях без вырожденных граничных компонент. 2015 Article Toward the theory of the Dirichlet problem for the Beltrami equations / V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov // Доповiдi Нацiональної академiї наук України. — 2015. — № 11. — С. 23-29. — Бібліогр.: 10 назв. — англ. 1025-6415 http://dspace.nbuv.gov.ua/handle/123456789/97943 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Математика Математика Gutlyanskii, V.Ya. Ryazanov, V.I. Yakubov, E. Toward the theory of the Dirichlet problem for the Beltrami equations Доповіді НАН України |
| description |
The Dirichlet problem for the degenerate Beltrami equations in arbitrary finitely connected
domains is studied. In terms of the tangent dilatations, a series of criteria for
the existence of regular solutions in arbitrary simply connected domains, as well as
pseudoregular and multivalent solutions in arbitrary finitely connected domains without degenerate boundary components, are formulated. |
| format |
Article |
| author |
Gutlyanskii, V.Ya. Ryazanov, V.I. Yakubov, E. |
| author_facet |
Gutlyanskii, V.Ya. Ryazanov, V.I. Yakubov, E. |
| author_sort |
Gutlyanskii, V.Ya. |
| title |
Toward the theory of the Dirichlet problem for the Beltrami equations |
| title_short |
Toward the theory of the Dirichlet problem for the Beltrami equations |
| title_full |
Toward the theory of the Dirichlet problem for the Beltrami equations |
| title_fullStr |
Toward the theory of the Dirichlet problem for the Beltrami equations |
| title_full_unstemmed |
Toward the theory of the Dirichlet problem for the Beltrami equations |
| title_sort |
toward the theory of the dirichlet problem for the beltrami equations |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2015 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/97943 |
| citation_txt |
Toward the theory of the Dirichlet problem for the Beltrami equations / V.Ya. Gutlyanskii, V.I. Ryazanov, E. Yakubov // Доповiдi Нацiональної академiї наук України. — 2015. — № 11. — С. 23-29. — Бібліогр.: 10 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT gutlyanskiivya towardthetheoryofthedirichletproblemforthebeltramiequations AT ryazanovvi towardthetheoryofthedirichletproblemforthebeltramiequations AT yakubove towardthetheoryofthedirichletproblemforthebeltramiequations |
| first_indexed |
2025-07-07T05:49:07Z |
| last_indexed |
2025-07-07T05:49:07Z |
| _version_ |
1836966058104193024 |
| fulltext |
UDC 517.5
Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskii,
V. I. Ryazanov, E. Yakubov
Toward the theory of the Dirichlet problem for the
Beltrami equations
The Dirichlet problem for the degenerate Beltrami equations in arbitrary finitely con-
nected domains is studied. In terms of the tangent dilatations, a series of criteria for
the existence of regular solutions in arbitrary simply connected domains, as well as
pseudoregular and multivalent solutions in arbitrary finitely connected domains without
degenerate boundary components, are formulated.
Keywords: Beltrami equations, Dirichlet problem, prime ends, regular solutions, simply con-
nected domains, finitely connected domains, pseudoregular and multivalent solutions.
The purpose of this paper is to give a brief presentation of results in our paper [1] on the Dirichlet
problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains.
Let D be a domain in the complex plane C, and let µ : D → C be a measurable function with
|µ(z)| < 1 a. e. (almost everywhere) in D. We study the Beltrami equation
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 with respect to x and y, correspondingly.
The classical Dirichlet problem in a Jordan domain D for the uniformly elliptic Beltrami
equation, i. e., when |µ(z)| 6 k < 1 a. e., is the problem of the existence of a continuous function
f : D → C such that
fz = µ(z)fz for a.e. z ∈ D,
lim
z→ζ
Re f(z) = φ(ζ) ∀ζ ∈ ∂D,
(2)
for a continuous function φ : ∂D → R. It was studied long ago, see, e. g., [2, 3].
The degeneracy of the ellipticity of the Beltrami equation will be controlled by the dilatation
coefficient
Kµ(z) =
1 + |µ(z)|
1− |µ(z)|
, (3)
as well as by the more refined quantity, see, e. g., [4–6],
KT
µ (z, z0) =
∣∣∣∣1− z − z0
z − z0
µ(z)
∣∣∣∣2
1− |µ(z)|2
, (4)
© V.Ya. Gutlyanskii, V. I. Ryazanov, E. Yakubov, 2015
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №11 23
taking not only the modulus of the complex coefficient µ but also its argument into account.
Note that
K−1
µ (z) 6 KT
µ (z, z0) 6 Kµ(z) ∀z ∈ D ∀z0 ∈ C. (5)
Our research is based on new existence theorems of homeomorphic W 1,1
loc solutions for the
degenerate Beltrami equations in [7] and on the theory of prime ends by Carathéodory, see,
e. g., [8]. The boundary behavior ofW 1,1
loc homeomorphic solutions, as well as the Dirichlet problem
for the degenerate Beltrami equations in Jordan domains, has been already studied, see, e. g., [5]
and references therein.
Let ED denote the space of prime ends of a domain D in C, and let DP = D
∪
ED stand for
the completion of the domain D by its prime ends with the topology described in [8], 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 for the Dirichlet problem is written as
lim
n→∞
Re f(zn) = φ(P ), (6)
where the limit is taken over all sequences of points zn ∈ D converging to the prime end P .
It was established in [5] that every homeomorphic W 1,1
loc solution of the Beltrami equation (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. We established in [1] that it is also the so-called ringQ-homeomorphism
at every point z0 ∈ D with Q(z) = KT
µ (z, z0), z ∈ D, if Kµ ∈ L1(D) or KT
µ (z, z0) is integrable
along the circles |z − z0| = r for a. e. small enough r at every z0 ∈ D. In other words, the
latter means that the homeomorphic W 1,1
loc solutions of the Beltrami equation (1) satisfy certain
inequalities in terms of a conformal modulus for families of curves that is the main geometric
tool in the mapping theory.
Then in [1], we developed the theory of the boundary behavior with respect to prime ends for
ring Q-homeomorphisms that form a wider class than lower Q-homeomorphisms and, in parti-
cular, established far-reaching generalizations of the Carathéodory theorem on a homeomorphic
extension of conformal mappings to the boundary in prime ends. This is a basis to develop, in [1],
the theory of the boundary behavior with respect to prime ends for generalized homeomorphic
solutions to the degenerate Beltrami equation (1). Finally, the latter makes possible to reduce the
Dirichlet problem for the degenerate Beltrami equations (1) to the case of analytic and harmonic
functions in circular domains.
In what follows, we use the notations B(z0, r) := {z ∈ C : |z− z0| < r} for z0 ∈ C and r > 0,
S(z0, r) := {z ∈ C : |z − z0| = r}, D := B(0, 1), and C := C
∪
{∞}.
1. On BMO and FMO functions. Recall that 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| dxdy <∞, (7)
where the supremum is taken over all disks B in D, and
uB =
1
|B|
∫
B
u(z) dxdy.
24 ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №11
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 paper [9] and soon became
an important concept in harmonic analysis, partial differential equations, and related areas.
Following [10], we say that a function u : D → R has finite mean oscillation at a point z0 ∈ D if
lim
ε→0
−
∫
B(z0,ε)
|u(z)− ũε(z0)| dxdy <∞, (8)
where
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 is of finite mean oscillation in D, abbr. u ∈ FMO(D) or simply u ∈ FMO,
if (8) holds at every point z0 ∈ D.
Remark 1. Clearly, BMO ⊂ FMO. There exist examples showing that FMO is not BMO loc,
see, e. g., [7]. By definition, FMO ⊂ L1
loc, but FMO is not a subset of Lp
loc for any p > 1 in
comparison with BMO loc ⊂ Lp
loc for all p ∈ [1,∞).
Proposition 1. If, for some collection of numbers uε ∈ R, ε ∈ (0, ε0],
lim
ε→0
−
∫
B(z0,ε)
|u(z)− uε| dxdy <∞, (9)
then u is of finite mean oscillation at z0.
Сorollary 1. If, for a point z0 ∈ D,
lim
ε→0
−
∫
B(z0,ε)
|u(z)| dxdy <∞, (10)
then u has 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 (10) is only sufficient
but not necessary for a function u to be of finite mean oscillation at z0.
2. The Dirichlet problem in simply connected domains. Given a continuous function
φ(P ) ̸≡ const, P ∈ ED, we will say that f is a regular solution of the Dirichlet problem (6) for
the Beltrami equation (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., (11)
satisfying (1) a. e. and the boundary condition (6) for all prime ends of the domain D.
Recall that a mapping f : D → C is called discrete if f−1(y) for every point y ∈ C consists
of isolated points and open if the image of every open set U ⊆ D is open in C.
For φ(P ) ≡ c ∈ R, P ∈ ED, a regular solution of the problem is any constant function
f(z) = c + ic′, c′ ∈ R.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №11 25
Theorem 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 (12)
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.
Then the Beltrami equation (1) has a regular solution f of the Dirichlet problem (6) for every
continuous function φ : ED → R.
Here and below, we set that KT
µ is equal to zero outside of the domain D.
Corollary 2. 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, (13)
where kTz0(ε) is the average of the function KT
µ (z, z0) over the circle S(z0, ε).
Then the Beltrami equation (1) has a regular solution f of the Dirichlet problem (6) for every
continuous function φ : ED → R.
Remark 3. In particular, the conclusion of Theorem 1 holds if
KT
µ (z, z0) = O
(
log
1
|z − z0|
)
as z → z0 ∀z0 ∈ D. (14)
Theorem 2. Let D be a bounded simply connected domain in C, let µ : D → D be a measurable
function with Kµ ∈ L1
loc, and let
KT
µ (z, z0) 6 Qz0(z) ∈ FMO(z0) ∀z0 ∈ D. (15)
Then the Beltrami equation (1) has a regular solution f of the Dirichlet problem (6) for every
continuous function φ : ED → R.
Corollary 3. 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. (16)
Then the Beltrami equation (1) has a regular solution f of the Dirichlet problem (6) for every
continuous function φ : ED → R.
Remark 4. In particular, by (5), the conclusion of Theorem 2 holds if
Kµ(z) 6 Q(z) ∈ BMO(D). (17)
26 ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №11
Theorem 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. (18)
Then the Beltrami equation (1) has a regular solution f of the Dirichlet problem (6) for every
continuous function φ : ED → R.
Remark 5. Here, we are able to give a number of other conditions of logarithmic type. In
particular, condition (18) 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, (19)
and condition (13) can be replaced by the weaker condition
kTz0(r) = O
(
log
1
r
log log
1
r
)
. (20)
Theorem 4. Let D be a bounded simply connected domain in C, let µ : D → D be a measurable
function with Kµ ∈ L1
loc, and let∫
D∩B(z0,ε0)
Φz0(K
T
µ (z, z0)) dm(z) <∞ ∀z0 ∈ D (21)
for ε0 = ε(z0) > 0 and a nondecreasing convex function Φz0 : [0,∞) → [0,∞) with
∞∫
δ0
dτ
τΦ−1
z0 (τ)
= ∞ (22)
for δ0 = δ(z0) > Φz0(0). Then the Beltrami equation (1) has a regular solution f of the Dirichlet
problem (6) for every continuous function φ : ED → R.
Remark 6. Moreover, it was shown by us that condition (22) is not only sufficient but also
necessary to have a regular solution of the Dirichlet problem (6) for every Beltrami equation (1)
with the integral restrictions (21) and every continuous function φ : ED → R.
Corollary 4. Let D be a bounded simply connected domain in C, let µ : D → D be a
measurable function with Kµ ∈ L1
loc, and let∫
D∩B(z0,ε0)
eα0KT
µ (z,z0)dm(z) <∞ ∀z0 ∈ D (23)
for some ε0 = ε(z0) > 0 and α0 = α(z0) > 0. Then the Beltrami equation (1) has a regular
solution f of the Dirichlet problem (6) for every continuous function φ : ED → R.
3. The Dirichlet problem in multiply connected domains. As was probably first
noted by B. Bojarski, see, e. g., Sect. 6 of Chapter 4 in [3], the Dirichlet problem for the
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №11 27
Beltrami equations, generally speaking, has no regular solution in the class of functions conti-
nuous (single-valued) in C with generalized derivatives in the case of multiply connected domai-
ns D. Hence, the natural question arose: Do solutions exist in wider classes of functions in
this case? It turned out that the solutions of this problem can be found in the class of functi-
ons admitting a certain number (related to the connectedness of D) of poles at prescribed
points. This number should involve the multiplicity of these poles from the Stoilow repre-
sentation.
A discrete open mapping f : D → C of the Sobolev class W 1,1
loc (outside of poles) satisfying (1)
a. e. and the boundary condition (6) are called a pseudoregular solution of the Dirichlet problem
if the Jacobian Jf (z) ̸= 0 a. e.
It was demonstrated in [1] that, under the same conditions on the complex coefficient µ as in
Section 2, in bounded m — connected domains with nondegenerate boundary components, for
every prescribed integer k > m− 1, the Beltrami equation (1) has a pseudoregular solution f of
the Dirichlet problem (6) with k poles at prescribed points in D for every continuous function
φ : ED → R.
It was also shown in [1] that, under the same conditions in finitely connected doma-
ins, the Dirichlet problem (6) for the Beltrami equations (1) admits multivalent solutions in
the spirit of the theory of multivalent analytic functions in addition to pseudoregular soluti-
ons.
We say that the discrete open mapping f : B(z0, ε0) → C, where B(z0, ε0) ⊆ D, is a local
regular solution of Eq. (1) if f ∈ W 1,1
loc , Jf (z) ̸= 0, and f satisfies (1) a. e. in B(z0, ε0). The
local regular solutions f : B(z0, ε0) → C and f∗ : B(z∗, ε∗) → C of Eq. (1) will be called the
extensions of each to other if there is a finite chain of solutions fi : B(zi, εi) → C, i = 1, . . . ,m,
such 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
multivalent solution of Eq. (1) in D if the disks B(zj , εj) cover the whole domain D, and if fj are
extensions of one to another through the collection, and the collection is maximal by inclusion.
A multivalent solution of Eq. (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 condition lim
z→P
u(z) = φ(P ) along any ways
in D going to P ∈ ED.
References
1. Gutlyanskii V., Ryazanov V., Yakubov E. Ukr. Mat. Visn., 2015, 12, No 1: 27–66 (in Russian).
2. Bojarski B. Mat. Sbornik, 1957, 43(85), No 4: 451–503 (in Russian); English transl. in Rep. Univ.
Jyväskylä, Dept. Math. Stat., 2009, 118: 1–64.
3. Vekua I. N. Generalized Analytic Functions, London: Pergamon Press, 1962.
4. Gutlyanskii V., Martio O., Sugawa T., Vuorinen M. Trans. Amer. Math. Soc., 2005, 357: 875–900.
5. Ryazanov V., Salimov R., Srebro U., Yakubov E. Contemp. Math., 2013, 591: 211–242.
6. Ryazanov V., Srebro U., Yakubov E. J. Anal. Math., 2005, 96: 117–150.
7. Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami Equation: A Geometric Approach,
Developments in Mathematics, Vol. 26, New York: Springer, 2012.
8. Collingwood E. F., Lohwater A. J. The Theory of Cluster Sets, Cambridge Tracts in Math. and Math.
Physics, Vol. 56, Cambridge: Cambridge Univ. Press, 1966.
9. John F., Nirenberg L. Commun. Pure Appl. Math., 1961, 14: 415–426.
28 ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №11
10. Ignat’ev A., Ryazanov V. Ukr. Mat. Visn., 2005, 2, No 3: 395–417 (in Russian); transl. in Ukrainian Math.
Bull., 2005, 2, No 3: 403–424.
Received 17.06.2015Institute of Applied Mathematics and Mechanics
of the NAS of Ukraine, Sloviansk
Holon Institute of Technology, Israel
Член-кореспондент НАН України В.Я. Гутлянський, В. I. Рязанов, Е. Якубов
До теорiї задачi Дiрiхле для рiвнянь Бельтрамi
Iнститут прикладної математики i механiки НАН України, Слов’янськ
Холонський технологiчний iнститут, Iзраїль
Вивчається задача Дiрiхле для вироджених рiвнянь Бельтрамi в довiльних скiнченнозв’я-
зних областях. У термiнах дотичних дилатацiй сформульовано цiлий ряд критерiїв iсну-
вання регулярних розв’язкiв цiєї проблеми в довiльних обмежених однозв’язних областях,
а також псевдорегулярних i багатозначних розв’язкiв в довiльних обмежених скiнченно-
зв’язних областях без вироджених граничних компонентiв.
Ключовi слова: рiвняння Бельтрамi, задача Дiрiхле, простi кiнцi, регулярнi розв’язки, одно-
зв’язнi областi, скiнченнозв’язнi областi, псевдорегулярнi та багатозначнi розв’язки.
Член-корреспондент НАН Украины В.Я. Гутлянский, В. И. Рязанов,
Э. Якубов
К теории задачи Дирихле для уравнений Бельтрами
Институт прикладной математики и механики НАН Украины, Славянск
Холонский технологический институт, Израиль
Изучается задача Дирихле для вырожденных уравнений Бельтрами в произвольных коне-
чносвязных областях. В терминах касательных дилатаций сформулирован целый ряд кри-
териев существования регулярных решений этой проблемы в произвольных ограниченных
односвязных областях, а также псевдорегулярных и многозначных решений в произвольных
ограниченных конечносвязных областях без вырожденных граничных компонент.
Ключевые слова: уравнения Бельтрами, задача Дирихле, простые концы, регулярные ре-
шения, односвязные области, конечносвязные области, псевдорегулярные и многозначные
решения.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №11 29
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