On the boundary-value problems for quasiconformal functions in the plane
Generalized solvability of the classical boundary value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the...
Збережено в:
| Дата: | 2015 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2015
|
| Назва видання: | Український математичний вісник |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/140873 |
| 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 boundary-value problems for quasiconformal functions in the plane / V.Y. Gutlyanskii, V.I. Ryazanov, A.S. Yefimushkin // Український математичний вісник. — 2015. — Т. 12, № 3. — С. 363-389. — Бібліогр.: 29 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-140873 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1408732018-07-18T01:23:27Z On the boundary-value problems for quasiconformal functions in the plane Gutlyanskii, V.Y. Ryazanov, V.I. Yefimushkin, A.S. Generalized solvability of the classical boundary value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. Finally, some applications to the boundary value problems for A-harmonic functions are given. 2015 Article On the boundary-value problems for quasiconformal functions in the plane / V.Y. Gutlyanskii, V.I. Ryazanov, A.S. Yefimushkin // Український математичний вісник. — 2015. — Т. 12, № 3. — С. 363-389. — Бібліогр.: 29 назв. — англ. 1810-3200 2010 MSC: 31A05, 31A20, 31A25, 31B25, 35Q15, 30E25, 31C05, 35F45 http://dspace.nbuv.gov.ua/handle/123456789/140873 en Український математичний вісник 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 |
Generalized solvability of the classical boundary value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. Finally, some applications to the boundary value problems for A-harmonic functions are given. |
| format |
Article |
| author |
Gutlyanskii, V.Y. Ryazanov, V.I. Yefimushkin, A.S. |
| spellingShingle |
Gutlyanskii, V.Y. Ryazanov, V.I. Yefimushkin, A.S. On the boundary-value problems for quasiconformal functions in the plane Український математичний вісник |
| author_facet |
Gutlyanskii, V.Y. Ryazanov, V.I. Yefimushkin, A.S. |
| author_sort |
Gutlyanskii, V.Y. |
| title |
On the boundary-value problems for quasiconformal functions in the plane |
| title_short |
On the boundary-value problems for quasiconformal functions in the plane |
| title_full |
On the boundary-value problems for quasiconformal functions in the plane |
| title_fullStr |
On the boundary-value problems for quasiconformal functions in the plane |
| title_full_unstemmed |
On the boundary-value problems for quasiconformal functions in the plane |
| title_sort |
on the boundary-value problems for quasiconformal functions in the plane |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140873 |
| citation_txt |
On the boundary-value problems for quasiconformal functions in the plane / V.Y. Gutlyanskii, V.I. Ryazanov, A.S. Yefimushkin // Український математичний вісник. — 2015. — Т. 12, № 3. — С. 363-389. — Бібліогр.: 29 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT gutlyanskiivy ontheboundaryvalueproblemsforquasiconformalfunctionsintheplane AT ryazanovvi ontheboundaryvalueproblemsforquasiconformalfunctionsintheplane AT yefimushkinas ontheboundaryvalueproblemsforquasiconformalfunctionsintheplane |
| first_indexed |
2025-07-10T11:26:56Z |
| last_indexed |
2025-07-10T11:26:56Z |
| _version_ |
1837259104292175872 |
| fulltext |
Український математичний вiсник
Том 12 (2015), № 3, 363 – 389
On the boundary value problems for
quasiconformal functions in the plane
Vladimir Gutlyanskii, Vladimir Ryazanov,
Artem Yefimushkin
Abstract. Generalized solvability of the classical boundary value prob-
lems for analytic and quasiconformal functions in arbitrary Jordan do-
mains with boundary data that are measurable with respect to the log-
arithmic capacity is established. Moreover, it is shown that the spaces
of the found solutions have the infinite dimension. Finally, some appli-
cations to the boundary value problems for A-harmonic functions are
given.
2010 MSC. 31A05, 31A20, 31A25, 31B25, 35Q15, 30E25, 31C05, 35F45.
Key words and phrases. Dirichlet, Hilbert, Riemann, Neumann,
Poincare problems, analytic functions, quasiconformal functions, Bel-
trami equations, A-harmonic functions, nonlinear problems.
1. Introduction
The classical boundary value problems of the theory of analytic func-
tions, such as the Dirichlet, Hilbert, Riemann, Neumann and Poincare
problems, play a fundamental role in contemporary analysis and its ap-
plications to actual problems of mathematical physics. Note that these
boundary value problems are closely interconnected, see e.g. mono-
graphs [9, 21,29], and also recent works [6, 25–28].
In this paper we continue the development of the theory of the above
boundary value problems for analytic functions in arbitrary Jordan do-
mains with measurable boundary data and extend the theory to the more
general class of quasiconformal functions. The latter class of functions is
generated by the regular solutions of the well-known Beltrami equation,
see e.g. [18]. Recall that quasiconformal functions form just the class
of mappings with bounded distortion by Reshetnyak, see e.g. [24], or,
equivalently, the class of quasiregular mappings, see e.g. [19].
Received 02.09.2015
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
364 On the boundary value problems...
Let D be a domain in the complex plane C and µ : D → C be a
measurable function with |µ(z)| < 1 a.e. Recall that a partial differential
equation
fz̄ = µ(z) · fz (1.1)
where fz̄ = ∂̄f = (fx + ify)/2, fz = ∂f = (fx − ify)/2, z = x + iy, fx
and fy are partial derivatives of the function f in x and y, respectively,
is said to be a Beltrami equation. The Beltrami equation (1.1) is said to
be nondegenerate if ||µ||∞ < 1.
Note that analytic functions satisfy (1.1) with µ(z) ≡ 0. Recall also
that a regular solution of the Beltrami equation is a continuous, discrete
and open mapping f : D → C of the Sobolev class W 1,1
loc satisfying (1.1)
a.e. Note that, in the case of nondegenerate Beltrami equations, a reg-
ular solution f belongs to class W 1,p
loc for some p > 2 and, moreover, its
Jacobian Jf (z) ̸= 0 for almost all z ∈ D, and it is called a quasicon-
formal function, see e.g. Chapter VI in [18]. Moreover, f is called a
quasiconformal mapping if in addition it is a homeomorphism.
It is clear that if we start to consider boundary value problems with
measurable boundary data, then the requests on the existence of the
limits at all points ζ ∈ ∂D and along all paths terminating in ζ (as
well as the conception of the index) lose any sense. Thus, the notion
of solutions of such boundary value problems should be extended. The
nontangential limits from the function theory of a complex variable were
a suitable tool, see e.g. [6] and [25]. In [28], it was proposed an alternative
approach admitting tangential limits for analytic functions. At present
paper, we use this approach to the Beltrami equations.
One of the relevant problems concerns to the measurement of sets on
boundaries of domains. In this connection, note that the sets of the length
measure zero as well as of the harmonic measure zero are invariant under
conformal mappings, however, they are not invariant under quasiconfor-
mal mappings. The latter follows, e.g., from the famous Ahlfors–Beurling
example of quasisymmetric mappings of the real axis that are not abso-
lutely continuous, see [2]. Hence we are forced to apply here the so-called
absolute harmonic measure by Nevanlinna, in other words, logarithmic
capacity whose zero sets are invariant under quasiconformal mappings,
see e.g. [22].
2. Definitions and preliminary remarks
Let D = {z ∈ C : |z| < 1} be the unit disk in the complex plane C.
By the well–known Priwalow uniqueness theorem analytic functions in D
coincide if they have the equal boundary values along all nontangential
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 365
paths to a set E of points in ∂D of a positive measure, see e.g. Theorem
IV.2.5 in [23]. The theorem is valid also for analytic functions in Jordan
domains with rectifiable boundaries, see e.g. Section IV.2.6 in [23].
However, examples of Lusin and Priwalow show that there exist non-
trivial analytic functions in D whose radial boundary values are equal to
zero on sets E ⊆ ∂D of a positive measure, see e.g. Section IV.5 in [23].
Simultaneously, by Theorem IV.6.2 in [23] of Lusin and Priwalow the
uniqueness result is valid if E is of the second category. Theorem 1 in [4]
demonstrates that the latter condition is necessary.
Recall Baire’s terminology for categories of sets and functions. Name-
ly, given a topological space X, a set E ⊆ X is of first category if it can
be written as a countable union of nowhere dense sets, and is of second
category if E is not of first category. Also, given topological spaces X
and X∗, f : X → X∗ is said to be a function of Baire’s class 1 if f−1(U)
for every open set U in X∗ is an Fσ set in X where an Fσ set is the union
of a sequence of closed sets.
Given a Jordan curve C in C, we say that a family of Jordan arcs
{Jζ}ζ∈C is of class BS (of the Bagemihl–Seidel class), cf. [4], 740–741,
if all Jζ lie in a ring R generated by C and a Jordan curve C∗ in C,
C∗ ∩ C = Ø, Jζ is joining C∗ and ζ ∈ C, every z ∈ R belongs to a single
arc Jζ , and for a sequence of mutually disjoint Jordan curves Cn in R
such that Cn → C as n→ ∞, Jζ ∩ Cn consists of a single point for each
ζ ∈ C and n = 1, 2, . . ..
In particular, a family of Jordan arcs {Jζ}ζ∈C is of class BS if Jζ
are induced by an isotopy of C. For instance, every curvilinear ring R
one of whose boundary component is C can be mapped with a conformal
mapping g onto a circular ring R and the inverse mapping g−1 : R → R
maps radial lines in R onto suitable Jordan arcs Jζ and centered circles
in R onto Jordan curves giving the corresponding isotopy of C to other
boundary component of R.
Now, if Ω ⊂ C is an open set bounded by a finite collection of mutually
disjoint Jordan curves, then we say that a family of Jordan arcs {Jζ}ζ∈∂Ω
is of class BS if its restriction to each component of ∂Ω is so.
Theorem 1 in [4] can be written in the following way, see footnote 9
in [4].
Proposition 1. Let D be a bounded domain in C whose boundary con-
sists of a finite number of mutually disjoint Jordan curves and let
{γζ}ζ∈∂D be a family of Jordan arcs of class BS in D.
SupposeM is an Fσ set of first category on ∂D and Φ(ζ) is a complex-
valued function of Baire’s class 1 on M . Then there is a nonconstant
single-valued analytic function f : D → C such that, for all ζ ∈ M ,
366 On the boundary value problems...
along γζ
lim
z→ζ
f(z) = Φ(ζ) . (2.1)
On this basis, in the case of domains D whose boundaries consist
of rectifiable Jordan curves, it was formulated Theorem 2 in [4] on the
existence of analytic functions f : D → C such that (2.1) holds a.e. on
∂D with respect to the natural parameter for each prescribed measurable
function Φ : ∂D → C.
3. On the Dirichlet problem for analytic functions
We need for our goals the following statement on analytic functions
with prescribed boundary values that is similar to Theorem 2 in [4] but
formulated in terms of logarithmic capacity instead of the natural param-
eter, see definitions, notations and comments in [6].
Theorem 1. Let D be a bounded domain in C whose boundary consists
of a finite number of mutually disjoint Jordan curves and let a function
Φ : ∂D → C be measurable with respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there is a nonconstant single-valued analytic function f : D →
C such that (2.1) holds along γζ for a.e. ζ ∈ ∂D with respect to the
logarithmic capacity.
Proof. Note first of all that C := C(∂D) < ∞ because ∂D is bounded
and Borel, even compact, and show that there is a sigma-compact set S
in ∂D of first category such that C(S) = C. More precisely, S will be
the union of a sequence of sets Sm in ∂D of the Cantor type that are
nowhere dense in ∂D.
Namely, Sm is constructed in the following way. First we remove
an open arc A1 in ∂D of the logarithmic capacity 2−mC and one more
open arc A2 in ∂D \ A1 of the logarithmic capacity 2−(m+1)C such that
∂D \ (A1 ∪A2) consists of 2 segments of ∂D with the equal logarithmic
capacity. Then we remove a union A3 of 2 open arcs in each these
segments of the total logarithmic capacity 2−(m+2)C such that new 4
segments in ∂D \ (A1 ∪ A2 ∪ A3) have the equal logarithmic capacity.
Repeating by induction this construction, we obtain the compact sets
Sm = ∂D \
∞∪
m=1
Am with the logarithmic capacity (1− 2−(m−1)) · C → C
as m→ ∞.
Note also that the logarithmic capacity is Borel’s regular measure
as well as Radon’s measure in the sense of points 2.2.3 and 2.2.5 in [7],
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 367
correspondingly, see Section 2 in [6]. Hence the Lusin theorem holds for
the logarithmic capacity on C, see Theorem 2.3.5 in [7].
By the Lusin theorem one can find a sequence of compacta Kn in S
with C(S \Kn) < 2−n such that Φ|Kn is continuous for each n = 1, 2, . . .,
i.e., for every open set U ⊆ C,Wn := Φ|−1
Kn
(U) = Vn∩S for some open set
Vn in C, andWn is sigma-compact because Vn and S are sigma-compact.
Consequently, W := Φ|−1
K (U) =
∞∪
n=1
Wn is also sigma-compact where
K =
∞∪
n=1
Kn. Hence the restriction of Φ on the set K is a function of
Baire’s class 1. Finally, note that by the construction C(Z) = 0 where
Z = ∂D \K = (∂D \ S) ∪ (S \K) and
K =
∞∪
m,n=1
(Sm ∩Kn)
where each set Em,n := Sm ∩ Kn, m,n = 1, 2, . . ., is nowhere dense in
∂D.
Thus, the conclusion of Theorem 1 follows from Proposition 1.
Corollary 1. Let D be a bounded domain in C whose boundary consists
of a finite number of mutually disjoint Jordan curves and let a function
φ : ∂D → R be measurable with respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there is a harmonic function u : D → R such that
lim
z→ζ
u(z) = φ(ζ) (3.1)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Corollary 2. Let D be a Jordan domain in C and a function Φ : ∂D → C
be measurable with respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there is a harmonic function u : D → R such that
lim
z→ζ
∇u(z) = Φ(ζ) (3.2)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Here we use the complex notation for the gradient ∇u := ux + i · uy.
Proof. Indeed, by Theorem 1 there is a single-valued analytic function
f : D → R such that
lim
z→ζ
f(z) = Φ(ζ) (3.3)
368 On the boundary value problems...
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity. Then
any indefinite integral F of f is also a single-valued analytic function in
the simply connected domain D and the harmonic functions u = Re F
and v = Im F satisfy the Cauchy–Riemann system vx = −uy and vy =
ux. Hence
f = F ′ = Fx = ux + i · vx = ux − i · uy = ∇u .
Thus, (3.2) follows from (3.3) and, consequently, u is the required func-
tion.
Corollary 3. Let a function Φ : ∂D → C be measurable with respect to
the logarithmic capacity. Then there is a harmonic function u : D → R
such that (3.2) holds along radial lines for a.e. ζ ∈ ∂D with respect to
the logarithmic capacity.
4. On the Dirichlet problem for quasiconformal functions
We show that quasiconformal functions are able to take arbitrary
measurable values along prescribed families of Jordan arcs terminating
at the boundary.
Theorem 2. Let D be a bounded domain in C whose boundary consists
of a finite number of mutually disjoint Jordan curves, µ : D → C be
a Lebesgue measurable function with ||µ||∞ < 1, and let a function Φ :
∂D → C be measurable with respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then the Beltrami equation (1.1) has a regular solution f : D → C such
that (2.1) holds along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic
capacity.
Proof. Extending µ by zero everywhere outside of D, we obtain the exis-
tence of a quasiconformal mapping h : C → C a.e. satisfying the Beltrami
equation (1.1) with the given µ, see e.g. Theorem V.B.3 in [1]. Setting
D∗ = h(D) and Γξ = h(γh−1(ξ)), ξ ∈ ∂D∗, we see that {Γξ}ξ∈∂D∗ is a
family of Jordan arcs of class BS in D∗.
The logarithmic capacity of a set coincides with its transfinite diame-
ter, see e.g. [8] and the point 110 in [22]. Hence the mappings h and h−1
transform sets of logarithmic capacity zero on ∂D into sets of logarith-
mic capacity zero on ∂D∗ and vice versa because these quasiconformal
mappings are continuous by Hölder on ∂D and ∂D∗ correspondingly, see
e.g. Theorem II.4.3 in [18].
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 369
Further, the function φ = Φ ◦ h−1 is measurable with respect to log-
arithmic capacity. Indeed, under this mapping measurable sets with re-
spect to logarithmic capacity are transformed into measurable sets with
respect to logarithmic capacity because such a set can be represented
as the union of a sigma-compactum and a set of logarithmic capacity
zero. On the other hand, compacta under continuous mappings are trans-
formed into compacta and compacta are measurable sets with respect to
logarithmic capacity.
By Theorem 1 there is an analytic function F : D∗ → C such that
lim
w→ξ
F (w) = φ(ξ) (4.1)
holds along Γξ for a.e. ξ ∈ ∂D∗ with respect to logarithmic capacity.
Thus, by the arguments given above f := F ◦ h is the desired solution of
(1.1).
5. On Riemann–Hilbert problem for analytic functions
Recall that the classical setting of the Hilbert (Riemann–Hilbert)
boundary value problem is to find analytic functions f in a domain D ⊂ C
bounded by a rectifiable Jordan curve with the boundary condition
lim
z→ζ
Re {λ(ζ) · f(z)} = φ(ζ) ∀ ζ ∈ ∂D (5.1)
where functions λ and φ were continuously differentiable with respect to
the natural parameter s on ∂D and, moreover, |λ| ≠ 0 everywhere on
∂D. Hence without loss of generality one can assume that |λ| ≡ 1 on ∂D.
Theorem 3. Let D be a bounded domain in C whose boundary consists of
a finite number of mutually disjoint Jordan curves, and let λ : ∂D → C,
|λ(ζ)| ≡ 1, φ : ∂D → R and ψ : ∂D → R be measurable functions with
respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there is a nonconstant single-valued analytic function f : D → C
such that
lim
z→ζ
Re {λ(ζ) · f(z)} = φ(ζ) , (5.2)
lim
z→ζ
Im {λ(ζ) · f(z)} = ψ(ζ) (5.3)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
370 On the boundary value problems...
Remark 1. Thus, the space of all solutions f of the Hilbert problem (5.2)
in the given sense has the infinite dimension for any such prescribed φ,
λ and {γζ}ζ∈D because the space of all functions ψ : ∂D → R which
are measurable with respect to the logarithmic capacity has the infinite
dimension.
The latter is valid even for its subspace of continuous functions ψ :
∂D → R. Indeed, by the Riemann theorem every Jordan domain G can
be mapped with a conformal mapping g onto the unit disk D and by
the Caratheodory theorem g can be extended to a homeomorphism of G
onto D. By the Fourier theory, the space of all continuous functions ψ̃ :
∂D → R, equivalently, the space of all continuous 2π-periodic functions
ψ∗ : R → R, has the infinite dimension.
Proof. Indeed, set Ψ(ζ) = φ(ζ) + i · ψ(ζ) and Φ(ζ) = λ(ζ) · Ψ(ζ) for all
ζ ∈ ∂D. Then by Theorem 1 there is a single-valued analytic function f
such that
lim
z→ζ
f(z) = Φ(ζ) (5.4)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity. Then
also
lim
z→ζ
λ(ζ) · f(z) = Ψ(ζ) (5.5)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
6. On Riemann–Hilbert problem for Beltrami equations
Theorem 4. Let D be a bounded domain in C whose boundary consists
of a finite number of mutually disjoint Jordan curves, µ : D → C be
a Lebesgue measurable function with ||µ||∞ < 1, and let λ : ∂D → C,
|λ(ζ)| ≡ 1, φ : ∂D → R and ψ : ∂D → R be functions that are measurable
with respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then the Beltrami equation (1.1) has a regular solution f : D → C such
that
lim
z→ζ
Re {λ(ζ) · f(z)} = φ(ζ) , (6.1)
lim
z→ζ
Im {λ(ζ) · f(z)} = ψ(ζ) (6.2)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Proof. Indeed, set Ψ(ζ) = φ(ζ) + i · ψ(ζ) and Φ(ζ) = λ(ζ) · Ψ(ζ) for all
ζ ∈ ∂D. Then by Theorem 2 the Beltrami equation (1.1) has a regular
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 371
solution f : D → C such that
lim
z→ζ
f(z) = Φ(ζ) (6.3)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity. Then
also
lim
z→ζ
λ(ζ) · f(z) = Ψ(ζ) (6.4)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Remark 2. Thus, the space of all solutions f of the Hilbert problem
(6.1) for the Beltrami equation (1.1) in the given sense has the infinite
dimension for any such prescribed φ, λ and {γζ}ζ∈D because the space
of all functions ψ : ∂D → R which are measurable with respect to the
logarithmic capacity has the infinite dimension, see Remark 1.
7. Neumann and Poincare problems for analytic functions
As known the Neumann problem has no classical solutions generally
speaking even for harmonic functions and smooth boundary data, see e.g.
[20]. Here we study the Neumann problem and more general boundary
value problems for analytic functions with measurable boundary data in
a generalized form.
First of all recall a more general problem on directional derivatives for
the harmonic functions in the unit disk D = {z ∈ C : |z| < 1}, z = x+ iy.
The classic setting of the latter problem is to find a function u : D → R
that is twice continuously differentiable, admits a continuous extension
to the boundary of D together with its first partial derivatives, satisfies
the Laplace equation
∆u :=
∂2u
∂x2
+
∂2u
∂y2
= 0 ∀ z ∈ D (7.1)
and the boundary condition with a prescribed continuous date φ : ∂D →
R:
∂u
∂ν
= φ(ζ) ∀ ζ ∈ ∂D (7.2)
where ∂u
∂ν denotes the derivative of u at ζ in a direction ν = ν(ζ),
|ν(ζ)| = 1:
∂u
∂ν
:= lim
t→0
u(ζ + t · ν)− u(ζ)
t
. (7.3)
372 On the boundary value problems...
The Neumann problem is a special case of the above problem on di-
rectional derivatives with the boundary condition
∂u
∂n
= φ(ζ) ∀ ζ ∈ ∂D (7.4)
where n denotes the unit interior normal to ∂D at the point ζ.
In turn, the above problem on directional derivatives is a special case
of the Poincare problem with the boundary condition
a · u + b · ∂u
∂ν
= φ(ζ) ∀ ζ ∈ ∂D (7.5)
where a = a(ζ) and b = b(ζ) are real-valued functions given on ∂D.
Recall also that twice continuously differentiable solutions of the Lap-
lace equation are called harmonic functions. As known, such functions
are infinitely differentiable as well as they are real and imaginary parts
of analytic functions. The above boundary value problems for analytic
functions are formulated in a similar way. Let us start from the problem
on directional derivatives.
Theorem 5. Let D be a Jordan domain in C with a rectifiable boundary,
ν : ∂D → C, |ν(ζ)| ≡ 1, and Φ : ∂D → C be measurable functions with
respect to the natural parameter.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there is a nonconstant single-valued analytic function f : D → C
such that
lim
z→ζ
∂f
∂ν
(z) = Φ(ζ) (7.6)
along γζ for a.e. ζ ∈ ∂D with respect to the natural parameter.
Proof. Indeed, by Theorem 5 in [4] there is a harmonic function u : D →
R such that
lim
z→ζ
∇u(z) = Ψ(ζ) := ν(ζ) · Φ(ζ) (7.7)
along γζ for a.e. ζ ∈ ∂D with respect to the natural parameter where we
use the complex notation for the gradient ∇u = ux + i · uy .
As known, there is a conjugate harmonic function v such that f :=
u+i·v is a single-valued analytic function in the simply connected domain
D, see e.g. point I.A in [15]. The functions u and v satisfy the Cauchy–
Riemann system vx = −uy and vy = ux, i.e., ∇v = i ·∇u. Note also that
the derivatives of u and v in the direction ν are projections of gradients
of u and v onto ν. Hence
∂f
∂ν
=
∂u
∂ν
+ i · ∂v
∂ν
= Re ν · ∇u + i · Re ν · ∇v = ν · ∇u .
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 373
Thus, (7.6) follows from (7.7) and, consequently, f is the required func-
tion.
Remark 3. We are able to say more in the case Re n · ν > 0 where
n = n(ζ) is the unit interior normal with a tangent to ∂D at a point
ζ ∈ ∂D. In view of (7.6), since the limit Φ(ζ) is finite, there is a finite
limit f(ζ) of f(z) as z → ζ in D along the straight line passing through
the point ζ and being parallel to the vector ν(ζ) because along this line,
for z and z0 that are close enough to ζ,
f(z) = f(z0) −
1∫
0
∂f
∂ν
(z0 + τ(z − z0)) dτ . (7.8)
Thus, at each point with the condition (7.6), there is the directional
derivative
∂f
∂ν
(ζ) := lim
t→0
f(ζ + t · ν)− f(ζ)
t
= Φ(ζ) . (7.9)
In particular, the following statement on the solvability of the Neu-
mann problem is a consequence of Theorem 5 and Remark 3.
Corollary 4. For each measurable function Φ : ∂D → C, one can find
a nonconstant single-valued analytic function f : D → C such that, for
a.e. point ζ ∈ ∂D, there exist:
1) the finite radial limit
f(ζ) := lim
r→1
f(rζ) (7.10)
2) the normal derivative
∂f
∂n
(ζ) := lim
t→0
f(ζ + t · n)− f(ζ)
t
= Φ(ζ) (7.11)
3) the radial limit
lim
r→1
∂f
∂n
(rζ) =
∂f
∂n
(ζ) (7.12)
where n = n(ζ) denotes the unit interior normal to ∂D at the point ζ.
Finally, arguing as in the proof of Theorem 5 and using Corollary 2
instead of Theorem 5 in [4], we obtain the following theorem that solves
the problem on directional derivatives for analytic functions in arbitrary
Jordan domains.
374 On the boundary value problems...
Theorem 6. Let D be a Jordan domain in C, ν : ∂D → C, |ν(ζ)| ≡ 1,
and Φ : ∂D → C be measurable functions with respect to the logarithmic
capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there is a nonconstant single-valued analytic function f : D → C
such that
lim
z→ζ
∂f
∂ν
(z) = Φ(ζ) (7.13)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Remark 4. As it follows from the given scheme of the proof, Theorems
5 and 6 are valid for multiply connected domains bounded by a finite
collection of mutually disjoint Jordan curves, however, the corresponding
analytic function f can be multivalent.
8. Neumann, Poincare problems for Beltrami equations
Theorem 7. Let D be a Jordan domain in C, µ : D → C be a function
of the Hölder class Cα with α ∈ (0, 1) and |µ(z)| ≤ k < 1, z ∈ D, and
let ν : ∂D → C, |ν(ζ)| ≡ 1, and Φ : ∂D → C be measurable with respect
to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then the Beltrami equation (1.1) has a regular solution f : D → C of
the class C1+α such that
lim
z→ζ
∂f
∂ν
(z) = Φ(ζ) (8.1)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Lemma 1. Let S be a subspace of a metric space (X, d) and let h : S → R
be a Hölder continuous function, namely, for some C > 0 and α ∈ (0, 1),
|h(x1)− h(x2)| ≤ C · dα(x1, x2) ∀ x1, x2 ∈ S . (8.2)
Then there is an extension H : X → R of h to X such that
|H(x1)−H(x2)| ≤ C · dα(x1, x2) ∀ x1, x2 ∈ X . (8.3)
The same is valid for the functions h : S → C as well as h : S → Rn.
Proof. Let us prove that d∗(x1, x2) := dα(x1, x2), (x1, x2) ∈ (X ×X), is
a distance on X. Indeed, by the triangle inequality for d we have that
d∗(x1, x2) ≤ (d(x1, x3) + d(x3, x2))
α ∀ x1, x2, x3 ∈ X .
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 375
Hence it remains to show that
φ(t, τ) := tα + τα − (t + τ)α ≥ 0 ∀ t, τ ∈ R+, α ∈ (0, 1) .
It sufficient for this to note that φ(t, 0) ≡ 0 and
∂φ
∂τ
(t, τ) = α ·
[
1
τ1−α
− 1
(t + τ)1−α
]
≥ 0 .
Thus, the conclusion of Lemma 1 follows from point 2.10.44 in [7] on
the Lipschitz functions.
Lemma 2. Let f : D → C be a smooth quasiconformal mapping satisfy-
ing the Beltrami equation (1.1) with a continuous complex coefficient µ
in a domain D ⊆ C. Then
C(Zf ) = 0 = C(f(Zf )) , (8.4)
where C(E) is the logarithmic capacity of E ⊆ C, Jf is the Jacobian of
f , and
Zf = { z ∈ C : Jf (z) = 0 } . (8.5)
Proof. First of all, note that f satisfies (1.1) for all z ∈ D because the
functions µ(z), fz and fz̄ are continuous, and by (1.1) the maximal dis-
tortion Mf (z) = |fz| + |fz̄| under f at z is equal to 0 at all points of
Zf . Note also that, in view of countable subadditivity of logarithmic
capacity, with no loss of generality we may assume that D is bounded
and then C(D) <∞.
Let us take an arbitrary ε > 0. In view of continuity of Mf (z), for
every z∗ ∈ Zf , there is a disk Bz∗ centered at z∗ with a small enough ra-
dius such that Bz∗ ⊂ D and such that sup
z∈Bz∗
Mf (z) ≤ ε. The union of all
these disks gives an open neighborhood U of Zf such that sup
z∈U
Mf (z) ≤ ε.
The logarithmic capacity of a set can be characterized as its transfinite
diameter, see e.g. [8] and the point 110 in [22]. Hence
C(f(Zf )) ≤ C(f(U)) ≤ ε · C(U) ≤ ε · C(D) . (8.6)
Thus, by arbitrariness of ε > 0 we obtain the right equality in (8.4).
Note that the function Jf (z) = |fz|2 − |fz̄|2 is continuous and, conse-
quently, the set Zf is closed in D, i.e. Zf and f(Zf ) are sigma-compacta
in D and D∗ = f(D), correspondingly. Thus, the equality C(f(Zf )) = 0
implies the equality C(Zf ) = 0 by the same characterization of loga-
rithmic capacity because the inverse mapping f−1 is quasiconformal and
hence it is Hölder continuous on every compact set in the domain D∗,
see e.g. Theorem II.4.3 in [18].
376 On the boundary value problems...
Proof of Theorem 7. By Lemma 1 µ is extended to a Hölder continuous
function µ∗ : C → C. Hence also, for every k∗ ∈ (k, 1), there is a
neighborhood U of the closure of the domain D such that |µ(z)| < k∗.
Let D∗ be a connected component of U containing D.
Now, there is a quasiconformal mapping h : D∗ → C a.e. satisfying
the Beltrami equation (1.1) with the complex coefficient µ∗ := µ∗|D∗ in
D∗, see e.g. Theorem V.B.3 in [1]. Note that the mapping h has the
Hölder continuous first partial derivatives in D∗ with the same order of
the Hölder continuity as µ, see e.g. [11] and also [12]. Thus, by Lemma 2
C(Zh) = 0 = C(h(Zh)) , (8.7)
where C(E) is the logarithmic capacity of E ⊆ C, Jh is the Jacobian of
h, and
Zh = { z ∈ C : Jh(z) = 0 } . (8.8)
Moreover, setting D∗ = h(D) and Γξ = h(γh−1(ξ)), ξ ∈ ∂D∗, we see that
{Γξ}ξ∈∂D∗ is a family of Jordan arcs of class BS in D∗.
Next, setting
hν(z) := lim
t→0
h(z + t · ν)− h(z)
t · ν
, (8.9)
we see that the function hν(z) is continuous and hence measurable and
by (8.7) also hν(z) ̸= 0 a.e. in z ∈ D∗ with respect to the logarithmic
capacity.
The logarithmic capacity of a set coincides with its transfinite diame-
ter, see e.g. [8] and the point 110 in [22]. Hence the mappings h and h−1
transform sets of logarithmic capacity zero on ∂D into sets of logarith-
mic capacity zero on ∂D∗ and vice versa because these quasiconformal
mappings are Hölder continuous on ∂D and ∂D∗ correspondingly, see e.g.
Theorem II.4.3 in [18].
Further, the functions N := ν ◦h−1 and φ := (Φ/hν)◦h−1 are measu-
rable with respect to logarithmic capacity. Indeed, measurable sets with
respect to logarithmic capacity are transformed under the mappings h
and h−1 into measurable sets with respect to logarithmic capacity because
such a set can be represented as the union of a sigma-compactum and a
set of logarithmic capacity zero and compacta under continuous mappings
are transformed into compacta and compacta are measurable with respect
to logarithmic capacity.
Finally, by Theorem 6 there is a nonconstant single-valued analytic
function F : D∗ → C such that
lim
w→ξ
∂F
∂N
(w) = φ(ξ) (8.10)
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 377
holds along Γξ for a.e. ξ ∈ ∂D∗ with respect to the logarithmic capacity.
It remains to note that f := F ◦ h is the desired solution of (1.1)
because
∂f
∂ν
= F ′ ◦ h · ∂h
∂ν
= F ′ ◦ h · νhν =
∂F
∂N
◦ h · hν ,
i.e. (8.1) holds along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic
capacity.
Remark 5. Again, we are able to say more in the case Re n · ν > 0
where n = n(ζ) is the unit interior normal with a tangent to ∂D at a
point ζ ∈ ∂D. In view of (8.1), since the limit Φ(ζ) is finite, there is a
finite limit f(ζ) of f(z) as z → ζ in D along the straight line passing
through the point ζ and being parallel to the vector ν(ζ) because along
this line, for z and z0 that are close enough to ζ,
f(z) = f(z0) −
1∫
0
∂f
∂ν
(z0 + τ(z − z0)) dτ . (8.11)
Thus, at each point with the condition (8.1), there is the directional
derivative
∂f
∂ν
(ζ) := lim
t→0
f(ζ + t · ν)− f(ζ)
t
= Φ(ζ) . (8.12)
In particular, we obtain on the basis of Theorem 7 and Remark 5 the
following statement on the Neumann problem.
Corollary 5. Let µ : D → C be in Cα, α ∈ (0, 1), and |µ(z)| ≤ k < 1,
and let Φ : ∂D → C be measurable with respect to the logarithmic capacity.
Then the Beltrami equation (1.1) has a regular solution f : D → C of
the class C1+α such that, for a.e. point ζ ∈ ∂D, there exist:
1) the finite radial limit
f(ζ) := lim
r→1
f(rζ) (8.13)
2) the normal derivative
∂f
∂n
(ζ) := lim
t→0
f(ζ + t · n)− f(ζ)
t
= Φ(ζ) (8.14)
3) the radial limit
lim
r→1
∂f
∂n
(rζ) =
∂f
∂n
(ζ) (8.15)
where n = n(ζ) denotes the unit interior normal to ∂D at the point ζ.
378 On the boundary value problems...
Remark 6. As it follows from Remark 4, Theorems 7 is valid for multiply
connected domains D bounded by a finite collection of mutually disjoint
Jordan curves, however, then f can be multivalent in the spirit of the
theory of multivalent analytic functions.
Namely, in finitely connected domains D in C, we say that a contin-
uous discrete open mapping f : B(z0, ε0) → C, where B(z0, ε0) ⊆ D,
is a local regular solution of (1.1) if f ∈ W 1,1
loc , Jf (z) ̸= 0 and f satis-
fies (1.1) a.e. in B(z0, ε0). Local regular solutions f0 : B(z0, ε0) → C
and f∗ : B(z∗, ε∗) → C of (1.1) are called the extension of each to
other if there is a finite chain of such solutions fi : B(zi, εi) → C,
i = 1, . . . ,m, 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 multi-valued solution
of the Beltrami equation (1.1) in D if the disks B(zj , εj) cover the whole
domain D and fj are extensions of each to other through the collection
and this collection is maximal by inclusion, cf. e.g. [5] and [16].
9. On the Riemann problem for analytic functions
Recall that the classical setting of the Riemann problem in a smooth
Jordan domain D of the complex plane C is to find analytic functions
f+ : D → C and f− : C \ D → C that admit continuous extensions to
∂D and satisfy the boundary condition
f+(ζ) = A(ζ) · f−(ζ) + B(ζ) ∀ ζ ∈ ∂D (9.1)
with prescribed Hölder continuous functions A : ∂D → C and B : ∂D →
C.
Recall also that the Riemann problem with shift inD is to find analytic
functions f+ : D → C and f− : C \D → C satisfying the condition
f+(α(ζ)) = A(ζ) · f−(ζ) + B(ζ) ∀ ζ ∈ ∂D (9.2)
where α : ∂D → ∂D was a one-to-one sense preserving correspondence
having the non-vanishing Hölder continuous derivative with respect to
the natural parameter on ∂D. The function α is called a shift function.
The special case A ≡ 1 gives the so-called jump problem and B ≡ 0 gives
the problem on gluing of analytic functions.
Theorem 8. Let D be a domain in C whose boundary consists of a finite
number of mutually disjoint Jordan curves, A : ∂D → C and B : ∂D → C
be functions that are measurable with respect to the logarithmic capacity.
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 379
Suppose that {γ+ζ }ζ∈∂D and {γ−ζ }ζ∈∂D are families of Jordan arcs of class
BS in D and C \D, correspondingly.
Then there exist single-valued analytic functions f+ : D → C and
f− : C \ D → C that satisfy (9.1) for a.e. ζ ∈ ∂D with respect to
the logarithmic capacity where f+(ζ) and f−(ζ) are limits of f+(z) and
f−(z) az z → ζ along γ+ζ and γ−ζ , correspondingly.
Furthermore, the space of all such couples (f+, f−) has the infinite
dimension for every couple (A,B) and any collections γ+ζ and γ−ζ , ζ ∈
∂D.
Theorem 8 is a special case of the following lemma on the generalized
Riemann problem with shift that may have of independent interest.
Lemma 3. Under the hypotheses of Theorem 8, let in addition α : ∂D →
∂D be a homeomorphism keeping components of ∂D such that α and α−1
have the (N)−property of Lusin with respect to the logarithmic capacity.
Then there exist single-valued analytic functions f+ : D → C and
f− : C \ D → C that satisfy (9.2) for a.e. ζ ∈ ∂D with respect to
the logarithmic capacity where f+(ζ) and f−(ζ) are limits of f+(z) and
f−(z) az z → ζ along γ+ζ and γ−ζ , correspondingly.
Furthermore, the space of all such couples (f+, f−) has the infinite
dimension for every couple (A,B) and any collections γ+ζ and γ−ζ , ζ ∈
∂D.
Proof. For brevity, we write here “C-measurable” instead of the expres-
sion “measurable with respect to the logarithmic capacity”.
First, let D be bounded and let g− : ∂D → C be a C-measurable
function. Then the function
g+ := {A · g− +B} ◦ α−1 (9.3)
is also C−measurable. Indeed, E := {A · g− + B}−1(Ω) is a subset of
∂D that is C-measurable for every open set Ω ⊆ C because the function
A · g− + B is C−measurable by the hypotheses. Hence the set E is
the union of a sigma-compact set and a set of the logarithmic capacity
zero, see e.g. Section 2 in [6]. However, continuous mappings transform
compact sets into compact sets and, thus, by (N)−property α(E) =
α ◦ {A · g− + B}−1(Ω) = (g+)−1(Ω) is a C-measurable set, i.e. the
function g+ is really C-measurable.
Then by Theorem 1 there is a single-valued analytic function f+ :
D → C such that
lim
z→ξ
f+(z) = g+(ξ) (9.4)
380 On the boundary value problems...
along γ+ξ for a.e. ξ ∈ ∂D with respect to the logarithmic capacity. Note
that g+(α(ζ)) is determined by the given limit for a.e. ζ ∈ ∂D because
α−1 also has the (N)-property of Lusin.
Note that C \ D consists of a finite number of (simply connected)
Jordan domains D0, D1, . . . , Dm in the extended complex plane C =
C ∪ {∞}. Let ∞ ∈ D0. Then again by Theorem 1 there exist single-
valued analytic functions f−l : Dl → C, l = 1, . . . ,m, such that
lim
z→ζ
f−l (z) = g−l (ζ) , g−l := g−|∂Dl
, (9.5)
along γ−ζ for a.e. ζ ∈ ∂Dl with respect to the logarithmic capacity.
Now, let S be a circle that contains D and let j be the inversion of
C with respect to S. Set
D∗ = j(D0), g∗ = g0 ◦ j, g−0 := g−|∂D0 , γ
∗
ξ = j
(
γ−j(ξ)
)
, ξ ∈ ∂D∗ .
Then by Theorem 1 there is a single-valued analytic function f∗ : D∗ → C
such that
lim
w→ξ
f∗(w) = g∗(ξ) (9.6)
along γ∗ξ for a.e. ξ ∈ ∂D∗ with respect to the logarithmic capacity.
Note that f−0 := g∗ ◦ j is a single-valued analytic function in D0 and by
construction
lim
z→ζ
f−0 (z) = g−0 (ζ) , g−0 := g−|∂D0 , (9.7)
along γ−ζ for a.e. ζ ∈ ∂D0 with respect to the logarithmic capacity.
Thus, the functions f−l , l = 0, 1, . . . ,m, form an analytic function
f− : C \ D → C satisfying (9.2) for a.e. ζ ∈ ∂D with respect to the
logarithmic capacity.
The space of all such couples (f+, f−) has the infinite dimension for
every couple (A,B) and any collections γ+ζ and γ−ζ , ζ ∈ ∂D, in view of
the above construction because of the space of all measurable functions
g− : ∂D → C has the infinite dimension.
The case of unboundedD is reduced to the case of boundedD through
the complex conjugation and the inversion of C with respect to a circle
S in some of the components of C \D arguing as above.
Remark 7. Some investigations were devoted also to the nonlinear Rie-
mann problems with boundary conditions of the form
Φ( ζ, f+(ζ), f−(ζ) ) = 0 ∀ ζ ∈ ∂D . (9.8)
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 381
It is natural as above to weaken such conditions to the following
Φ( ζ, f+(ζ), f−(ζ) ) = 0 for a.e. ζ ∈ ∂D . (9.9)
It is easy to see that the proposed approach makes possible also to reduce
such problems to the algebraic and measurable solvability of the relations
Φ( ζ, v, w ) = 0 (9.10)
with respect to complex-valued functions v(ζ) and w(ζ), cf. e.g. [10].
Example 1. For instance, correspondingly to the scheme given
above, special nonlinear problems of the form
f+(ζ) = φ( ζ, f−(ζ) ) for a.e. ζ ∈ ∂D (9.11)
are always solved if the function φ : ∂D × C → C satisfies the Cara-
theodory conditions with respect to the logarithmic capacity: φ(ζ, w) is
continuous in the variable w ∈ C for a.e. ζ ∈ ∂D with respect to the
logarithmic capacity and it is C-measurable in the variable ζ ∈ ∂D for
all w ∈ C.
Furthermore, the spaces of solutions of such problems always have
the infinite dimension. Indeed, by the Egorov theorem, see e.g. Theorem
2.3.7 in [7], see also Section 17.1 in [17], the function φ(ζ, ψ(ζ)) is C-
measurable in ζ ∈ ∂D for every C−measurable function ψ : ∂D → C if
the function φ satisfies the Caratheodory conditions, and the space of all
C-measurable functions ψ : ∂D → C has the infinite dimension, see e.g.
Remark 1.
10. On the Riemann problem for the Beltrami equations
Theorem 9. Let D be a domain in C whose boundary consists of a finite
number of mutually disjoint Jordan curves, µ : C → C be a measurable
(by Lebesgue) function with ||µ||∞ < 1, A : ∂D → C and B : ∂D → C
be functions that are measurable with respect to the logarithmic capacity.
Suppose that {γ+ζ }ζ∈∂D and {γ−ζ }ζ∈∂D are families of Jordan arcs of class
BS in D and C \D, correspondingly.
Then the Beltrami equation (1.1) has regular solutions f+ : D → C
and f− : C \ D → C that satisfy (9.1) for a.e. ζ ∈ ∂D with respect to
the logarithmic capacity where f+(ζ) and f−(ζ) are limits of f+(z) and
f−(z) az z → ζ along γ+ζ and γ−ζ , correspondingly.
Furthermore, the space of all such couples (f+, f−) has the infinite
dimension for every couple (A,B) and any collections γ+ζ and γ−ζ , ζ ∈
∂D.
382 On the boundary value problems...
Theorem 9 is a special case of the following lemma on the Riemann
problem with shift for the Beltrami equation.
Lemma 4. Under the hypotheses of Theorem 9, let in addition α : ∂D →
∂D be a homeomorphism keeping components of ∂D such that α and α−1
have the (N)-property of Lusin with respect to the logarithmic capacity.
Then the Beltrami equation (1.1) has regular solutions f+ : D → C
and f− : C \ D → C that satisfy (9.2) for a.e. ζ ∈ ∂D with respect to
the logarithmic capacity where f+(ζ) and f−(ζ) are limits of f+(z) and
f−(z) az z → ζ along γ+ζ and γ−ζ , correspondingly.
Furthermore, the space of all such couples (f+, f−) has the infinite
dimension for every couple (A,B) and any collections γ+ζ and γ−ζ , ζ ∈
∂D.
Proof. First of all, there is a quasiconformal mapping h : C → C that
is a regular homeomorphic solution of the Beltrami equation (1.1) with
the given complex coefficient µ, see e.g. Theorem V.B.3 in [1]. Set D∗ =
h(D) and α∗ = h ◦α ◦ h−1. Then α∗ : ∂D∗ → ∂D∗ is a homeomorphism.
Recall that the logarithmic capacity of a set coincides with its trans-
finite diameter, see [8] and the point 110 in [22]. Hence the mappings h
and h−1 transform sets of logarithmic capacity zero on ∂D into sets of
logarithmic capacity zero on ∂D∗ and vice versa because quasiconformal
mappings are continuous by Hölder on ∂D and ∂D∗ correspondingly, see
e.g. Theorem II.4.3 in [18]. Thus, α∗ and α−1
∗ have the (N)-property
with respect to the logarithmic capacity.
Moreover, arguing similarly to the proof of Theorem 7, it is easy to
show that the functions A∗ := A◦h−1 and B∗ := B ◦h−1 are measurable
with respect to the logarithmic capacity. Setting Γ+
ξ = h(γ+
h−1(ξ)
) and
Γ−
ξ = h(γ−
h−1(ξ)
), ξ ∈ ∂D∗, we see also that {Γ+
ξ }ξ∈∂D∗ and {Γ−
ξ }ξ∈∂D∗
are families of Jordan arcs of class BS in D∗ and C\D∗, correspondingly.
Hence by Lemma 3 there exist single-valued analytic functions f+∗ :
D∗ → C and f−∗ : C \D∗ → C such that
f+∗ (α∗(ξ)) = A∗(ξ) · f−∗ (ξ) + B∗(ξ) ∀ ξ ∈ ∂D∗ (10.1)
for a.e. ξ ∈ ∂D∗ with respect to the logarithmic capacity where f+∗ (ξ)
and f−∗ (ξ) are limits of f+∗ (w) and f−∗ (w) as w → ξ along Γ+
ξ and Γ−
ξ .
Furthermore, the space of all such couples (f+∗ , f
−
∗ ) has the infinite di-
mension.
Finally, f+ := f+∗ ◦h and f− := f−∗ ◦h are the desired regular solutions
of the Beltrami equation (1.1) in D and C \D, correspondingly.
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 383
Remark 8. As it follows from the construction of solutions in the given
proof, Remark 6 on the nonlinear Riemann problems is valid for the
Beltrami equations, see also Example 1 in the last section.
11. On mixed boundary value problems
In order to demonstrate the potentiality of our approach, we consider
here some nonlinear boundary value problems.
Theorem 10. Let D be a domain in C whose boundary consists of a
finite number of mutually disjoint Jordan curves, µ : C → C be Lebesgue
measurable, µ ∈ Cα(C \D) and ||µ||∞ < 1, φ : ∂D × C → C satisfy the
Carathéodory conditions and ν : ∂D → C, |ν(ζ)| ≡ 1, be measurable with
respect to the logarithmic capacity.
Suppose that {γ+ζ }ζ∈∂D and {γ−ζ }ζ∈∂D are families of Jordan arcs of
class BS in D and C \D. Then the Beltrami equation (1.1) has regular
solutions f+ : D → C and f− : C \ D → C of class C1+α(C \ D) such
that
f+(ζ) = φ
(
ζ,
[
∂f
∂ν
]−
(ζ)
)
(11.1)
for a.e. ζ ∈ ∂D with respect to the logarithmic capacity where f+(ζ) and[
∂f
∂ν
]−
(ζ) are limits of the functions f+(z) and ∂f−
∂ ν (z) as z → ζ along
γ+ζ and γ−ζ , correspondingly.
Furthermore, the space of all such couples (f+, f−) has the infinite
dimension for any such prescribed functions µ, φ, ν and collections γ+ζ
and γ−ζ , ζ ∈ ∂D.
Theorem 10 is a special case of the following lemma on the nonlinear
Riemann problem with shift.
Lemma 5. Under the hypotheses of Theorem 10, let in addition β :
∂D → ∂D be a homeomorphism keeping components of ∂D such that β
and β−1 have the (N)-property of Lusin with respect to the logarithmic
capacity.
Then the Beltrami equation (1.1) has regular solutions f+ : D → C
and f− : C \D → C of class C1+α(C \D) such that
f+(β(ζ)) = φ
(
ζ,
[
∂f
∂ν
]−
(ζ)
)
(11.2)
384 On the boundary value problems...
for a.e. ζ ∈ ∂D with respect to the logarithmic capacity where f+(ζ) and[
∂f
∂ν
]−
(ζ) are limits of the functions f+(z) and ∂f−
∂ ν (z) as z → ζ along
γ+ζ and γ−ζ , correspondingly.
Furthermore, the space of all such couples (f+, f−) has the infinite
dimension for any such prescribed µ, φ, ν, β and collections {γ+ζ }ζ∈∂D
and {γ−ζ }ζ∈∂D.
Proof. Indeed, let ψ : ∂D → C be a function that is measurable with
respect to the logarithmic capacity. Then by Theorem 7 the Beltrami
equation (1.1) has a regular solution f− : C \D → C of class C1+α such
that
lim
z→ζ
∂f−
∂ ν
(z) = ψ(ζ) (11.3)
along γ−ζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Now, the function Ψ(ζ) := φ(ζ, ψ(ζ)) is measurable with respect
to the logarithmic capacity, see Example 1 after Remark 6. Then the
function Φ = Ψ ◦ α−1 is also measurable with respect to the logarith-
mic capacity because the homeomorphism α has the (N)-property, cf.
arguments in the proof of Lemma 4.
Thus, by Theorem 2 the Beltrami equation (1.1) has a regular solution
f+ : D → C such that f+(z) → Ψ(ζ) as z → ζ along γζ for a.e. ζ ∈ ∂D
with respect to the logarithmic capacity. Finally, f+ and f− are the
desired functions because α−1 also has the (N)-property.
It remains also to note that the space of all such couples (f+, f−)
has the infinite dimension because the space of all functions ψ : ∂D → C
which are measurable with respect to the logarithmic capacity has the
infinite dimension, see Remark 1.
12. On applications to equations of the divergence type
The partial differential equations in the divergence form below take a
significant part in many problems of mathematical physics, in particular,
in anisotropic inhomogeneous media, see e.g. [3, 13] and [14].
In this connection, note that if f = u+ i ·v is a regular solution of the
Beltrami equation (1.1), then the function u is a continuous generalized
solution of the divergence-type equation
divA(z)∇u = 0 , (12.1)
called A-harmonic function, i.e. u ∈ C ∩W 1,1 and∫
D
⟨A(z)∇u,∇φ⟩ = 0 ∀ φ ∈ C∞
0 (D) , (12.2)
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 385
where A(z) is the matrix function:
A =
( |1−µ|2
1−|µ|2
−2Imµ
1−|µ|2
−2Imµ
1−|µ|2
|1+µ|2
1−|µ|2
)
. (12.3)
Moreover, we have for the stream function v = Im f the following relation
∇ v = J A(z)∇u , (12.4)
where
J =
(
0 −1
1 0
)
. (12.5)
As we see in (12.3), the matrix A(z) is symmetric and its entries aij =
aij(z) are dominated by the quantity
Kµ(z) =
1 + |µ(z)|
1 − |µ(z)|
, (12.6)
and, thus, they are bounded if the Beltrami equation (1.1) is not degen-
erate.
Vice verse, uniformly elliptic equations (12.1) with symmetric A(z)
and detA(z) ≡ 1 just correspond to nondegenerate Beltrami equations
(1.1). Let all such matrix functions A(z) form the class B.
Recall that the equation (12.1) is called uniformly elliptic if aij ∈ L∞
and there is ε > 0 such that ⟨A(z)η, η⟩ ≥ ε|η|2 for all η ∈ R2. A matrix
function A(z) with the latter property is called uniformly positive definite.
The following statement on a potential boundary behavior of A-
harmonic functions of the Dirichlet type is a direct consequence of The-
orem 2.
Corollary 6. Let D be a bounded domain in C whose boundary consists
of a finite number of mutually disjoint Jordan curves, A(z), z ∈ D, be a
matrix function of class B and let a function φ : ∂D → R be measurable
with respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there exist A-harmonic functions u : D → R such that
lim
z→ζ
u(z) = φ(ζ) (12.7)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Furthermore, the space of all such A-harmonic functions u has the
infinite dimension for any such prescribed A, φ and {γζ}ζ∈∂D.
386 On the boundary value problems...
The next conclusion in the particular case of the Poincare problem
on directional derivatives follows directly from Theorem 7.
Corollary 7. Let D be a Jordan domain in C, A(z), z ∈ D, be a matrix
function of class B ∩Cα, α ∈ (0, 1), and let ν : ∂D → C, |ν(ζ)| ≡ 1, and
φ : ∂D → R be measurable with respect to the logarithmic capacity.
Suppose that {γζ}ζ∈∂D is a family of Jordan arcs of class BS in D.
Then there exist A-harmonic functions u : D → R of the class C1+α such
that
lim
z→ζ
∂u
∂ν
(z) = φ(ζ) (12.8)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Furthermore, the space of all such A-harmonic functions u has the
infinite dimension for any such prescribed A, φ, ν and {γζ}ζ∈∂D.
Now, the following statement concerning to the Neumann problem
for A-harmonic functions is a special significant case of Corollary 7.
Corollary 8. Let A(z), z ∈ D, be a matrix function of class B∩Cα, α ∈
(0, 1), and let φ : ∂D → R be measurable with respect to the logarithmic
capacity.
Then there exist A-harmonic functions u : D → R of the class C1+α
such that, for a.e. point ζ ∈ ∂D with respect to the logarithmic capacity,
there exist:
1) the finite radial limit
u(ζ) := lim
r→1
u(rζ), (12.9)
2) the normal derivative
∂u
∂n
(ζ) := lim
t→0
u(ζ + t · n)− u(ζ)
t
= φ(ζ), (12.10)
3) the radial limit
lim
r→1
∂u
∂n
(rζ) =
∂u
∂n
(ζ) (12.11)
where n = n(ζ) denotes the unit interior normal to ∂D at the point ζ.
Furthermore, the space of all such A-harmonic functions u has the
infinite dimension for any such prescribed A and φ.
The following result on the Riemann problem with shift for A-har-
monic functions is not a direct consequence of Lemma 4 but of its proof
where we are able to choose f−(ζ) such that Imf−(ζ) = 0 for a.e. ζ ∈ ∂D
with respect to the logarithmic capacity.
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 387
Corollary 9. Let D be a domain in C whose boundary consists of a finite
number of mutually disjoint Jordan curves, A(z), z ∈ D, be a matrix
function of class B, B : ∂D → R and C : ∂D → R be functions that are
measurable with respect to the logarithmic capacity and let α : ∂D → ∂D
be a homeomorphism keeping components of ∂D such that α and α−1
have the (N)-property of Lusin with respect to the logarithmic capacity.
Suppose that {γ+ζ }ζ∈∂D and {γ−ζ }ζ∈∂D are families of Jordan arcs of
class BS in D and C \D, correspondingly. Then there exist A-harmonic
functions u+ : D → R and u− : C \D → R such that
u+(α(ζ)) = B(ζ) · u−(ζ) + C(ζ) (12.12)
for a.e. ζ ∈ ∂D with respect to the logarithmic capacity where u+(ζ)
and u−(ζ) are limits of u+(z) and u−(z) az z → ζ along γ+ζ and γ−ζ ,
correspondingly.
Furthermore, the space of all such couples (u+, u−) has the infinite
dimension for any such prescribed A, B, C, α and collections {γ+ζ }ζ∈∂D
and {γ−ζ }ζ∈∂D.
Similarly, the following result on the mixed nonlinear Riemann prob-
lem with shifts for A-harmonic functions is not a direct consequence
of Lemma 5 but of its proof where we may choose f−(ζ) such that
Im
[
∂f
∂ν
]−
(ζ) = 0 for a.e. ζ ∈ ∂D with respect to the logarithmic ca-
pacity.
Corollary 10. Let D be a domain in C whose boundary consists of a
finite number of mutually disjoint Jordan curves, A(z), z ∈ C, be a
matrix function of class B∩Cα(C\D), α ∈ (0, 1), ν : ∂D → C, |ν(ζ)| ≡ 1,
be a measurable function, β : ∂D → ∂D be a homeomorphism such that
β and β−1 have the (N)-property of Lusin and φ : ∂D × R → R satisfy
the Caratheodory conditions with respect to the logarithmic capacity.
Suppose that {γ+ζ }ζ∈∂D and {γ−ζ }ζ∈∂D are families of Jordan arcs of
class BS in D and C \D, correspondingly. Then there exist A-harmonic
functions u+ : D → R and u− : C \ D → R of class C1+α(C \ D) such
that
u+(β(ζ)) = φ
(
ζ,
[
∂u
∂ν
]−
(ζ)
)
(12.13)
for a.e. ζ ∈ ∂D with respect to the logarithmic capacity where u+(ζ) and[
∂u
∂ν
]−
(ζ) are limits of the functions u+(z) and ∂u−
∂ ν (z) as z → ζ along
γ+ζ and γ−ζ , correspondingly.
Furthermore, the space of all such couples (u+, u−) has the infinite
dimension for any such prescribed A, ν, β, φ and collections γ+ζ and γ−ζ ,
ζ ∈ ∂D.
388 On the boundary value problems...
In particular, we are able to obtain from the last corollary solutions
of the problem on gluing of the Dirichlet problem in the unit disk D and
the Neumann problem outside of D in the class of A-harmonic functions.
References
[1] L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York,
1966.
[2] L. Ahlfors, A. Beurling, The boundary correspondence under quasiconformal
mappings // Acta Math., 96 (1956), 125-–142.
[3] K. Astala, T. Iwaniec, G. Martin, Elliptic partial differential equations and qua-
siconformal mappings in the plane // Princeton Math. Ser., 48, Princeton Univ.
Press, Princeton, NJ, 2009.
[4] F. Bagemihl, W. Seidel, Regular functions with prescribed measurable boundary
values almost everywhere // Proc. Nat. Acad. Sci. U. S. A. 41 (1955), 740–743.
[5] B. Bojarski, V. Gutlyanskii, V. Ryazanov, On existence and representation of
solutions for general degenerate Beltrami equations // Complex Variables and
Elliptic Equations, 59 (2014), No. 1, 67-–75.
[6] A. Efimushkin, V. Ryazanov, On the Riemann–Hilbert problem for the Beltrami
equations in quasidisks // Ukr. Math. Vis. 12 (2015), No. 2, 190–209; see also
arXiv.org: 1404.1111v4 [math.CV] 22 Sep 2015, 1–25.
[7] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.
[8] M. Fékete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichun-
gen mit ganzzahligen Koeffizienten // Math. Z., 17 (1923), 228–249.
[9] F. D. Gakhov, Boundary value problems, Dover Publications. Inc., New York,
1990.
[10] M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer
Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-
Verlag, Berlin, 1986.
[11] T. Iwaniec, Regularity of solutions of certain degenerate elliptic systems of equa-
tions that realize quasiconformal mappings in n-dimensional space // Differen-
tial and integral equations. Boundary value problems. Tbilisi: Tbilis. Gos. Univ.,
1979, 97–111 (in Russian).
[12] T. Iwaniec, Regularity theorems for solutions of partial differential equations
for quasiconformal mappings in several dimensions // Dissertationes Math.
(Rozprawy Mat.) 198 (1982), 45.
[13] T. Iwaniec, p-harmonic tensors and quasiregular mappings // Ann. of Math.,
(2) 136 (1992), No. 3, 589—624.
[14] T. Iwaniec, C. Sbordone, Quasiharmonic fields // Ann. Inst. H. Poincare Anal.
Non Lineaire, 18, (2001), No. 5, 519—572.
[15] P. Koosis, Introduction to Hp spaces // Cambridge Tracts in Mathematics, 115,
Cambridge Univ. Press, Cambridge, 1998.
[16] D. Kovtonyuk, I. Petkov, V. Ryazanov, R. Salimov, On the Dirichlet problem
for the Beltrami equation // J. Anal. Math., 122 (2014), No. 4, 113–141.
[17] M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, P. E. Sobolevskii, Integral
operators in spaces of summable functions // Monographs and Textbooks on
Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International
Publishing, Leiden, 1976.
V. Gutlyanskii, V. Ryazanov, A. Yefimushkin 389
[18] O. Lehto, K. J. Virtanen, Quasiconformal mappings in the plane, Springer-
Verlag, Berlin-Heidelberg, 1973.
[19] O. Martio, S. Rickman, J. Vaisala, Definitions for quasiregular mappings //
Ann. Acad. Sci. Fenn. Ser. A1. Math., 448 (1969), 1–40.
[20] S. G. Mikhlin, Partielle Differentialgleichungen in der mathematischen Physik //
Math. Lehrbücher und Monographien, 30, Akademie-Verlag, Berlin, 1978.
[21] N. I. Muskhelishvili, Singular integral equations. Boundary problems of function
theory and their application to mathematical physics, Dover Publications. Inc.,
New York, 1992.
[22] R. Nevanlinna, Eindeutige analytische Funktionen, Ann Arbor, Michigan, 1944.
[23] I. I. Priwalow, Randeigenschaften analytischer Funktionen. VEB Deutscher Ver-
lag der Wissenschaften, Berlin, 1956.
[24] Yu. G. Reshetnyak, Space mappings with bounded distortion, Nauka, Novosi-
birsk, 1982; Transl. of Mathematical Monographs, 73, Amer. Math. Soc., Prov-
idence, RI, 1988.
[25] V. Ryazanov, On the Riemann-Hilbert Problem without Index // Ann. Univ.
Bucharest, Ser. Math. 5 (LXIII) (2014), No. 1, 169–178.
[26] V. Ryazanov, Infinite dimension of solutions of the Dirichlet problem // Open
Math. (the former Central European J. Math.), 13 (2015), No. 1, 348–350.
[27] V. Ryazanov, On Neumann and Poincare problems for Laplace equation //
arXiv.org: 1510.00733v5 [math.CV] 26 Oct 2015, 1–5.
[28] V. Ryazanov, On Hilbert and Riemann problems. An alternative approach //
arXiv.org: 1510.06380v3 [math.CV] 28 Oct 2015, 1–8.
[29] I. N. Vekua, Generalized analytic functions, Pergamon Press, London etc., 1962.
Contact information
Vladimir
Gutlyanskii
Vladimir Ryazanov
Artem Yefimushkin
Institute of Applied Mathematics
and Mechanics, NAS of Ukraine
E-Mail: vladimirgut@mail.ru
vl.ryazanov1@gmail.com
a.yefimushkin@gmail.com
|