On the Dirichlet problem for Beltrami equations with sources in simply connected domains
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3
ОПОВІДІ
НАЦІОНАЛЬНОЇ
АКАДЕМІЇ НАУК
УКРАЇНИ
МАТЕМАТИКА
MATHEMATICS
ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2024. № 1: 3—12
C i t a t i o n: Gutlyanskiĭ V.Ya., Nesmelova O.V., Ryazanov V.I., Yakubov E. On the Dirichlet problem for Beltrami equations
with sources in simply connected domains. Dopov. nac. akad. nauk Ukr. 2024. No 1. P. 3—12. https://doi.org/10.15407/
dopovidi2024.01.003
© Publisher PH «Akademperiodyka» of the NAS of Ukraine, 2024. Th is is an open access article under the CC BY-NC-
ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
https://doi.org/10.15407/dopovidi2024.01.003
UDC 517.5
V.Ya. Gutlyanskiĭ1, 2, https://orcid.org/0000-0002-8691-4617
O.V. Nesmelova 1, 2, 4, https://orcid.org/0000-0003-2542-5980
V.I. Ryazanov 1, 2, https://orcid.org/0000-0002-4503-4939
E. Yakubov 3, https://orcid.org/0000-0002-2744-1338
1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk, Ukraine
2 Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine
3 Holon Institute of Technology, Holon, Israel
4 Donbas State Pedagogical University, Slovyansk, Ukraine
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com, yakubov@hit.ac.il
On the Dirichlet problem for beltrami equations
with sources in simply connected domains
In this paper, we present our recent results on the solvability of the Dirichlet problem Re ( ) ( )z as z
z D D with continuous boundary data D R for degenerate Beltrami equations ( ) ( )z zz z ,
( ) 1z a.e., with sources D C that belong to the class ( )pL D , 2p , and have compact supports in D . In
the case of locally uniform ellipticity of the equations, we formulate, in arbitrary simply connected domains D of the
complex plane C a series of eff ective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on
singularities of the equations at the boundary for existence of locally Hölder continuous solutions in the class 1 2
loc ( )W D of
the Dirichlet problem with their representation through the so-called generalized analytic functions with sources.
Keywords: Dirichlet problem, nonhomogeneous degenerate Beltrami equations, generalized analytic functions with
sources, BMO (bounded mean oscillation), FMO (fi nite mean oscillation), singularities at the boundary.
1. Introduction. Let D be a domain in the complex plane C We investigate the Dirichlet problem
lim Re ( ) ( )
z
z D
(1)
see Chapter 4 in [1], with continuous boundary data D R in arbitrary bounded simply
connected domains D for the Beltrami equation
( ) ( )z zz z z D (2)
4 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2024. No. 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, E. Yakubov
with a source D C in pL , 2p , where D C is a measurable function with ( ) 1z
a.e., ( ) 2z x yi , ( ) 2z x yi , z x iy , x and y are partial derivatives of the
function in x and y , respectively.
For the case || || 1 , (2) was initially introduced by L. Ahlfors and L. Bers in the paper [2].
Here, we examine the case of locally uniform ellipticity of the equation (2) when its dilatation
quotient K is bounded only locally in D ,
1 ( )
( )
1 ( )
z
K z
z
(3)
i.e., if K L on each compact set in D but admits singularities at the boundary. A point D
is called a singular point of the equation (2) if K L on each neighborhood of the point.
Here we present the corresponding results in the subject, as proven in our recent paper [3].
Specifi cally, we demonstrate that if D is an arbitrary bounded simply connected domain in C ,
then the Dirichlet problem (1) for the equation (2) has a locally Hölder continuous solution in
class 1 2
loc ( )W D for a broad range of singularities of (2) at the boundary. Futhermore, this solution
is unique up to an additive pure imaginary constant, and it can be expressed through suitable gen-
eralized analytic functions with sources.
In this connection, recall that the Vekua monograph [1] was devoted to generalized analytic
functions, i.e., continuous complex valued functions ( )H z of one complex variable z x iy of
class 1 1
locW in a domain D satisfying the equations
( ) 2z z x yH aH bH S i (4)
with complex valued coeffi cients ( )pa b S L D , 2p .
Th e paper [4] was devoted to boundary value problems with measurable data for the spacial
case of generalized analytic functions H with sources S D C , when 0a b ,
( ) ( )z H z S z z D (5)
in regular enough domains D .
2. Th e Main Existence Lemma. It is well known that the homogeneous Beltrami equation
( )z zf z f (6)
is the basic equation in analytic theory of quasiconformal and quasiregular mappings in the plane
with numerous applications in nonlinear elasticity, gas fl ow, hydrodynamics and other sections of
natural sciences.
Th e equation (6) is termed degenerate if esssup ( )K z . It is known that if K is bounded,
then the equation has homeomorphic solutions in 1 2
locW called quasiconformal mappings (see his-
toric comments in [5]). Recently, criteria for existence of homeomorphic solutions in 1 1
locW were also
established for degenerate Beltrami equations; refer to papers [6]—[9] and monographs [10, 11].
Th ese criteria were formulated both in terms of K and the quantity
0
0
2
0 2
1 ( )
( )
1 ( )
z z
z zT
z
K z z
z
(7)
5ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2024. № 1
On the Dirichlet problem for Beltrami equations with sources in simply connected domains
called tangent dilatation quotient of the Beltrami equations with respect to a point 0z C . Note that
–1
0 0( ) ( ) ( )TK z K z z K z z D z C (8)
Let D be a domain in the complex plane C . A function f D C in the Sobolev class 1 1
locW
is called a regular solution of the Beltrami equation (6) if f satisfi es (6) a.e., and its Jacobian
2 2( ) 0f z zJ z f f a.e. By Lemma 3 and Remark 2 in [8], we have the following statement on
the existence of regular homeomorphic solutions f in C for the Beltrami equation (6).
Proposition 1. Let C C be a measurable function with ( ) 1z a.e. and 1 loc ( )K L C
Suppose that, for each 0z C with some 0 0( ) 0z
0 0
0 0
2 2
0 0( ) ( ) ( ) ( ( )) as 0T
z z
z z
K z z z z dm z o I
(9)
for a family of measurable functions
0 0(0 ) (0 )z 0(0 ) with
0
0 0 0( ) ( ) (0 )z zI t dt
(10)
Th en the Beltrami equation (6) has a regular homeomorphic solution f .
Here ( )dm z corresponds to the Lebesgue measure in C and by (8) TK can be replaced by
K . We call such solutions f of (6) conformal mappings. It is assumed here and further that
the dilatation quotients 0( )TK z z and ( )K z are extended by 1 outside of the domain D .
Lemma 1. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support in D , D C be a measurable function with ( ) 1z a.e., K be locally bounded
in D , 1 ( )K L D and conditions (9) and (10) hold for all 0z D .
Th en the Beltrami equation (2) with the source has a locally Hölder continuous solution
in the class 1 2
locW of the Dirichlet problem (1) in D for each continuous function D R that is
unique up to an additive pure imaginary constant.
Moreover, h f , where f C C is a conformal mapping with extended by zero
outside D and h D C is a generalized analytic function in ( )D f D with the source S of the
class ( )pL D
for some (2 )p p ,
–1zfS f
J
(11)
where 2 2
z zJ f f is the Jacobian of f , that satisfi es the Dirichlet condition
–1lim Re ( ) ( ) with Dw
h w D f
(12)
Remark 1. In turn, the generalized analytic function h with the source S by Th eorem 1.16 in
[1] has the representation h A H , where
1 ( )( ) – ( )
D
SH w dm w C
w
(13)
6 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2024. No. 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, E. Yakubov
with wH S , and A is a holomorphic function in D with the Dirichlet condition
lim Re ( ) ( ) with Re Dw
A w D H
(14)
Note that H is Hölder continuous in D with 1 2 p by Th eorem 1.19 and
1 ( )p
DH W D
by Th eorems 1.36 and 1.37 in [1]. Also note that f and –1f are locally quasi-
conformal mappings in D and D , respectively.
Th e proof of Lemma 1 is based on known results about the existence of a harmonic function
u D R , satisfying the Dirichlet condition
lim ( ) ( )
w
u w D
(15)
see Corollary 4.1.8 and Th eorem 4.2.1 in [12], the existence of a holomorphic function
A u iv D C (unique up to an additive pure imaginary constant) in an arbitrary bounded
simple connected domain D and the factorization h f of solutions of (2) in terms of suitable
generalized analytic functions with sources, see Lemma 1 and Remark 2 in [13] for the uniformly
elliptic case. Th e existence of the given conformal mapping f follows from Proposition 1; see
further details of the proof in [3].
Remark 2. Note that if the family of the functions
0 0
( ) ( )z zt t , 0z D , in Lemma 1
is independent on the parameter , then the condition (9) implies that
0
( )zI as 0 .
Th is follows immediately from arguments by contradiction, apply for it (8) and the condition
1 ( )K L D . Note also that (9) holds, in particular, if, for some 0 0( )z ,
0
0 0
2
0 0 0( ) ( ) ( )T
z
z z
K z z z z dm z z D
(16)
and
0
( )zI as 0 . In other words, for the solvability of the Dirichlet problem (1) in D for
the Beltrami equations with sources (2) for all continuous boundary functions , it is suffi cient that
the integral in (16) converges for some nonnegative function
0
( )z t that is locally integrable over
0(0 ] but has a nonintegrable singularity at 0 . Th e functions 0log ( )e z z , (0 1) , z D ,
0z D , and ( ) 1 ( log( ))t t e t , (0 1)t , show that the condition (16) is compatible with the
condition
0
( )zI as 0 . Furthermore, the condition (9) shows that it is suffi cient for the
solvability of the Dirichlet problem even that the integral in (16) is divergent but in a controlled way.
3. Th e main existence integral criteria. Lemma 1 enables us to derive several eff ective inte-
gral criteria for the solvability of the Dirichlet problem for Beltrami equations with sources.
Firstly, recall that a real-valued function u in a domain D in C is said to be of bounded mean
oscillation in D , abbr. BMO( )u D , if 1
loc ( )u L D and
1|| || sup ( ) ( )B
B B
u u z u dm z
B
(17)
where the supremum is taken over all discs B in D and
1 ( ) ( )B
B
u u z dm z
B
7ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2024. № 1
On the Dirichlet problem for Beltrami equations with sources in simply connected domains
We write loc( )BMOu D if BMO( )u U for each relatively compact subdomain U of D .
We also write sometimes for short BMO and locBMO , respectively.
Th e class BMO was introduced by John and Nirenberg (1961) in the paper [14] and soon be-
came an important concept in harmonic analysis, partial diff erential equations, and related areas.
Following [15], we say that a function u D R has fi nite mean oscillation at the point
0z D , abbr. 0FMO( )u z , if
0
lim
0( ) 0( ) ( ) ( )B z u z z dm zu (18)
where
0( )zu
0( ) ( ) ( )B z u z dm z (19)
is the mean value of the function ( )u z over disk 0 0( ) { }B z z C z z . Note that the
condition (18) includes the assumption that u is integrable in some neighborhood of the point
0z . We say also that a function u D R is of fi nite mean oscillation in D , abbr. FMO( )u D or
simply FMOu , if 0FMO( )u z for all points 0z D .
Th e following statement is obvious by the triangle inequality.
Proposition 2. If, for a collection of numbers u R , 0(0 ] ,
0
lim
0( ) ( ) ( )B z u z u dm z (20)
then u is of fi nite mean oscillation at 0z .
Recall that a point 0z D is called a Lebesgue point of a function u D R if u is integrable
in a neighborhood of 0z and
0
lim
0( ) 0( ) ( ) ( ) 0B z u z u z dm z (21)
Th us, we have by Proposition 2 the next corollary.
Corollary 1. Every locally integrable function u D R has a fi nite mean oscillation at almost
every point in D .
Remark 4. Th e latter shows that the FMO condition is very weak. Clearly,
locBMO( ) ( ) FMO( )BMOD D D and as well-known loc locBMO pL for all [1 )p , see, e.g.,
[14]. However, FMO is not a subclass of loc
pL for any 1p but only of 1
locL , see Examples 2.3.1 in
[10]. Th us, the class FMO is much more wider than locBMO .
Versions of the next lemma have been fi rst proved for the class BMO in [7]. For the FMO case,
see the paper [15] and the monographs [10] and [11].
Lemma 2. Let D be a domain in C and let u D R be a non-negative function of the class
0FMO( )z for some 0z D . Th en
0 0 0
21
0
( ) ( ) 1loglog as 0
( log )z z z z
u z dm z
O
z z
(22)
for some 0 0(0 ) where –
0 0min( )ee d , 0 0sup
z D
d z z
.
8 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2024. No. 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, E. Yakubov
We assume further that the dilatation quotients 0( )TK z z and ( )K z are extended by 1
outside the domain D .
Choosing ( ) 1 ( log(1 ))t t t in Lemma 1, see also Remark 1, we obtain by Lemma 2 the
following result with the FMO type criterion.
Th eorem 1. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D ,
1 ( )K L D ,
00( ) ( )T
zK z z Q z a.e. in
0zU for each point 0z D , a neighborhood
0zU of 0z , a
function
0 0
[0 ]z zQ U in the class 0FMO( )z .
Th en the Beltrami equation (2) with the source has a locally Hölder continuous solution
in the class 1 2
locW of the Dirichlet problem (1) in D for each continuous function D R that is
unique up to an additive pure imaginary constant.
Furthermore, h f , h A H , where f C C is a conformal mapping with ex-
tended by zero outside of D , H D C is a generalized analytic function in ( )D f D with the
source S calculated in (11) and A is a holomorphic function in D with the Dirichlet condition (14).
In particular, choosing 0 , 0(0 ] in Proposition 2, we obtain:
Corollary 2. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D ,
1 ( )K L D and, for each point 0z D ,
0
lim
0( ) 0( ) ( )T
B z K z z dm z (23)
Th en all the conclusions of Th eorem 1 on solutions for the Dirichlet problem (1) with arbitrary
continuous boundary data D R to the Beltrami equation (2) with the source hold.
Since 0( ) ( )TK z z K z for all z and 0z C , we also obtain the following consequences of
Th eorem 1 with the BMO-type criterion.
Corollary 3. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with
compact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded
in D and K have a dominant Q BMO loc in a neighborhood of D . Th en the conclusions of
Th eorem 1 hold.
Corollary 4. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D
and K have a dominant Q FMO in a neighborhood of D . Th en the conclusions of Th eorem 1 hold.
Similarly, choosing in Lemma 1 the function ( ) 1t t , see also Remark 1, we come to the
next statement with the Calderon-Zygmund type criterion.
Th eorem 2. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D ,
1( )K L D and, for each point 0z D and 0 0( ) 0z ,
0 0
2
0 2
0
( ) 1( ) log as 0T
z z
dm z
K z z o
z z
(24)
Th en the Beltrami equation (2) with the source has a locally Hölder continuous solution
in the class 1 2
locW of the Dirichlet problem (1) in D for each continuous function D R that is
unique up to an additive pure imaginary constant.
9ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2024. № 1
On the Dirichlet problem for Beltrami equations with sources in simply connected domains
Furthermore, h f , h A H , where f C C is a conformal mapping with
extended by zero outside D , H D C is a generalized analytic function in ( )D f D with
the source S calculated in (11) and A is a holomorphic function in D with the Dirichlet condi-
tion (14).
Remark 5. Choosing in Lemma 1 the function ( ) 1 ( log1 )t t t instead of ( ) 1t t , we
are able to replace (24) by the conditions
0 0 0
2
0
021
0
( ) ( ) 1loglog
( log )
T
z z z z
K z z dm z
o z D
z z
(25)
as 0 for some 0 0( ) 0z . More generally, we would be able to give here the whole scale
of the corresponding conditions in log using functions ( )t of the form 1 ( log1 loglog1t t
log log1 )t … … t .
Choosing in Lemma 1 the functional parameter
0 0( ) 1 [ ( )]T
z t tk z t , where 0( )Tk z r is
the integral mean of 0( )TK z z over the circle 0 0( ) { }S z r z C z z r , we obtain the Lehto
type criterion.
Th eorem 3. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D ,
1 ( )K L D and, for each point 0z D and 0 0( ) 0z ,
0
00 ( )T
dr
rk z r
(26)
Th en the Beltrami equation (2) with the source has a locally Hölder continuous solution
in the class 1 2
locW of the Dirichlet problem (1) in D for each continuous function D R that is
unique up to an additive pure imaginary constant.
Moreover, h f , h A H , where f C C is a conformal mapping with ex-
tended by zero outside D , H D C is a generalized analytic function in ( )D f D with the
source S calculated in (11) and A is a holomorphic function in D with the Dirichlet condition (14).
Corollary 5. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D ,
1 ( )K L D and, for each point 0z D ,
0
1( ) log as 0Tk z O
(27)
Th en all conclusions of Th eorem 3 on solutions for the Dirichlet problem (1) with arbitrary con-
tinuous boundary data D R to the Beltrami equation (2) with the source hold.
Remark 6. In particular, the conclusions of Th eorem 3 hold if
0 0 0
0
1( ) log asTK z z O z z z D
z z
(28)
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2024. No. 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, E. Yakubov
Moreover, the condition (27) can be replaced by the series of weaker conditions
0 0
1 1 1( ) log loglog log logTk z O … … z D
(29)
Combining Th eorems 2.5 and 3.2 in [9] with Th eorems 3 we obtain the following signifi cant
result with the Orlicz type criterion.
Th eorem 4. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D ,
1 ( )K L D and, for each point 0z D and a neighborhood
0zU of 0z ,
0
0
0( ( )) ( )
z
T
z
U
K z z dm z (30)
where
0
(0 ] (0 ]z is a convex non-decreasing function such that
0
0
02
( )
log ( ) for some ( ) 0z
z
dtt z
t
(31)
Th en the Beltrami equation (2) with the source has a locally Hölder continuous solution
in the class 1 2
locW of the Dirichlet problem (1) in D for each continuous function D R that is
unique up to an additive pure imaginary constant.
Moreover, h f , h A H , where f C C is a conformal mapping with ex-
tended by zero outside D , H D C is a generalized analytic function in ( )D f D with the
source S calculated in (11) and A is a holomorphic function in D with the Dirichlet condition (14).
Corollary 6. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D ,
1 ( )K L D and, for each point 0z D , a neighborhood
0zU of 0z and 0( ) 0z ,
0 0
0
( ) ( ) ( )
T
z
z K z z
U
e dm z (32)
Th en all conclusions of Th eorem 4 on solutions for the Dirichlet problem (1) with continuous
data D R to the Beltrami equation (2) with the source hold.
Corollary 7. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D
and, for a neighborhood U of D ,
( ( )) ( )
U
K z dm z (33)
where (0 ] (0 ] is a convex non-decreasing function with, for 0 ,
2log ( ) dtt
t
(34)
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2024. № 1
On the Dirichlet problem for Beltrami equations with sources in simply connected domains
Th en all conclusions of Th eorem 4 on solutions for the Dirichlet problem (1) with continuous
data D R to the Beltrami equation (2) with the source hold.
Remark 7. By Th eorems 2.5 and 5.1 in [9], condition (34) is not only suffi cient but also nec-
essary to have the regular solutions of the Dirichlet problem (1) in D for arbitrary Beltrami
equations with sources (2), satisfying the integral constraints (33), for all continuous functions
D R because such solutions have the representation through regular homeomorphic solu-
tions f f of the homogeneous Beltrami equation (6) from Proposition 1.
Corollary 8. Let D be a bounded simply connected domain in C , ( )pL D , 2p , with com-
pact support, D C be a measurable function with ( ) 1z a.e., K be locally bounded in D
and, for a neighborhood U of D and 0 ,
( ) ( )K z
U
e dm z (35)
Th en all conclusions of Th eorem 4 on solutions for the Dirichlet problem (1) with continuous data
D R to the Beltrami equation (2) with the source hold.
Acknowledgements. Th e fi rst 3 authors are partially supported by the Grant EFDS-FL2-08 of
the found of the European Federation of Academies of Sciences and Humanities (ALLEA).
REFERENCES
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Publishing Co., Inc., Reading, Mass.
2. Ahlfors, L. V. & Bers, L. (1960). Riemann’s mapping theorem for variable metrics. Ann. Math., 2, No. 72,
pp. 385-404. https://doi.org/10.2307/1970141
3. Gutlyanskii, V., Nesmelova, O., Ryazanov, V. & Yakubov, E. (2023). The Dirichle problem for the Beltrami
equations with sources. Ukr. Mat. Visn., 20, No. 1, pp. 24-59; translated in (2023). J. Math. Sci. (N.Y.), 273,
No. 3, pp. 351—376; see also arXiv:2305.16331v2 [math.CV]. https://doi.org/10.1007/s10958-023-06503-0
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Society (EMS). https://doi.org/10.4171/122
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12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2024. No. 1
V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, E. Yakubov
13. Gutlyanskii, V., Nesmelova, O., Ryazanov, V. & Yakubov, E. (2022). On the Hilbert problem for semi-linear
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Received 13.07.2023
В.Я. Гутлянський 1, 2, https://orcid.org/0000-0002-8691-4617
О.В. Нєсмєлова 1, 2, 4, https://orcid.org/0000-0003-2542-5980
В.І. Рязанов 1, 2, https://orcid.org/0000-0002-4503-4939
Е. Якубов 3, https://orcid.org/0000-0002-2744-1338
1 Інститут прикладної математики і механіки НАН України, Слов’янськ, Україна
2 Інститут математики НАН України, Київ, Україна
3 Інститут технологій Холона, Холон, Ізраїль
4 Донбаський державний педагогічний університет, Слов’янськ, Україна
E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com, yakubov@hit.ac.il
ПРО ЗАДАЧУ ДІРІХЛЕ ДЛЯ РІВНЯНЬ БЕЛЬТРАМІ
З ДЖЕРЕЛАМИ В ОДНОЗВ’ЯЗАНИХ ОБЛАСТЯХ
У цій статті ми представляємо наші нещодавно отримані результати про розв’язність задачі Діріхле
Re ( ) ( )z для z z D D з неперервними граничними даними D R для виродже-
них рівнянь Бельтрамі ( ) ( )z zz z , ( ) 1z м.в., з джерелами D C , що належать до класу
( )pL D , 2p , з компактними носіями в D. У випадку локально рівномірної еліптичності рівнянь сформу-
льовано у довільних однозв’язаних областях D комплексної площини C низку ефективних інтегральних
критеріїв, типу BMO, FMO, Кальдерона-Зигмунда, Лехто та Орлича, сингулярності рівнянь на границі для
існування локально неперервних за Гельдером розв’язків у класі 1 2
loc ( )W D задачі Діріхле з представленням
їх через так звані узагальнені аналітичні функції з джерелами.
Ключові слова: задача Діріхле, неоднорідні вироджені рівняння Бельтрамі, узагальнені аналітичні функції з
джерелами, BMO, обмежені середні коливання, FMO, скінченні середні коливання, сингулярності на границі.
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| id | nasplib_isofts_kiev_ua-123456789-202282 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-12-07T16:44:36Z |
| publishDate | 2024 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. Yakubov, E. 2025-03-10T15:24:37Z 2024 On the Dirichlet problem for Beltrami equations with sources in simply connected domains / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov, E. Yakubov // Доповіді Національної академії наук України. — 2024. — № 1. — С. 3-12. — Бібліогр.: 15 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/202282 517.5 DOI: doi.org/10.15407/dopovidi2024.01.003 The first 3 authors are partially supported by the Grant EFDS-FL2-08 of the found of the European Federation of Academies of Sciences and Humanities (ALLEA). en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика On the Dirichlet problem for Beltrami equations with sources in simply connected domains Про задачу Діріхле для рівнянь Бельтрамі з джерелами в однозв’язаних областях Article published earlier |
| spellingShingle | On the Dirichlet problem for Beltrami equations with sources in simply connected domains Gutlyanskiĭ, V.Ya. Nesmelova, O.V. Ryazanov, V.I. Yakubov, E. Математика |
| title | On the Dirichlet problem for Beltrami equations with sources in simply connected domains |
| title_alt | Про задачу Діріхле для рівнянь Бельтрамі з джерелами в однозв’язаних областях |
| title_full | On the Dirichlet problem for Beltrami equations with sources in simply connected domains |
| title_fullStr | On the Dirichlet problem for Beltrami equations with sources in simply connected domains |
| title_full_unstemmed | On the Dirichlet problem for Beltrami equations with sources in simply connected domains |
| title_short | On the Dirichlet problem for Beltrami equations with sources in simply connected domains |
| title_sort | on the dirichlet problem for beltrami equations with sources in simply connected domains |
| topic | Математика |
| topic_facet | Математика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/202282 |
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