On recent advances in boundary value problems in the plane
The survey is devoted to recent advances in nonclassical solutions of the main boundary value problems such as the well–known Dirichlet, Hilbert, Neumann, Poincare and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical...
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irk-123456789-1409002018-07-18T01:23:53Z On recent advances in boundary value problems in the plane Gutlyanskii, V.Y. Ryazanov, V.I. The survey is devoted to recent advances in nonclassical solutions of the main boundary value problems such as the well–known Dirichlet, Hilbert, Neumann, Poincare and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical physics and based on the nonstandard point of view of the geometrical function theory with a clear visual sense. The traditional approach of the latter is the meaning of the boundary values of functions in the sense of the so–called angular limits or limits along certain classes of curves terminated at the boundary. This become necessary if we start to consider boundary data that are only measurable, and it is turned out to be useful under the study of problems in the field of mathematical physics, too. Thus, we essentially widen the notion of solutions and, furthermore, obtain spaces of solutions of the infinite dimension for all the given boundary value problems. The latter concerns to the Laplace equation as well as to its counterparts in the potential theory for inhomogeneous and anisotropic media. 2016 Article On recent advances in boundary value problems in the plane / V.Y. Gutlyanskii, V.I. Ryazanov // Український математичний вісник. — 2016. — Т. 13, № 2. — С. 167-212. — Бібліогр.: 76 назв. — рос. 1810-3200 2010 MSC: 30C62, 30D40, 37E30 http://dspace.nbuv.gov.ua/handle/123456789/140900 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The survey is devoted to recent advances in nonclassical solutions of the main boundary value problems such as the well–known Dirichlet, Hilbert, Neumann, Poincare and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical physics and based on the nonstandard point of view of the geometrical function theory with a clear visual sense. The traditional approach of the latter is the meaning of the boundary values of functions in the sense of the so–called angular limits or limits along certain classes of curves terminated at the boundary. This become necessary if we start to consider boundary data that are only measurable, and it is turned out to be useful under the study of problems in the field of mathematical physics, too. Thus, we essentially widen the notion of solutions and, furthermore, obtain spaces of solutions of the infinite dimension for all the given boundary value problems. The latter concerns to the Laplace equation as well as to its counterparts in the potential theory for inhomogeneous and anisotropic media. |
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On recent advances in boundary value problems in the plane |
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On recent advances in boundary value problems in the plane / V.Y. Gutlyanskii, V.I. Ryazanov // Український математичний вісник. — 2016. — Т. 13, № 2. — С. 167-212. — Бібліогр.: 76 назв. — рос. |
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Український математичний вiсник
Том 13 (2016), № 2, 167 – 212
On recent advances in boundary value
problems in the plane
Vladimir Gutlyanskii and Vladimir Ryazanov
Abstract. The survey is devoted to recent advances in nonclassical
solutions of the main boundary value problems such as the well–known
Dirichlet, Hilbert, Neumann, Poincare and Riemann problems in the
plane. Such solutions are essentially different from the variational solu-
tions of the classical mathematical physics and based on the nonstandard
point of view of the geometrical function theory with a clear visual sense.
The traditional approach of the latter is the meaning of the boundary
values of functions in the sense of the so-called angular limits or limits
along certain classes of curves terminated at the boundary. This become
necessary if we start to consider boundary data that are only measur-
able, and it is turned out to be useful under the study of problems in
the field of mathematical physics, too. Thus, we essentially widen the
notion of solutions and, furthermore, obtain spaces of solutions of the
infinite dimension for all the given boundary value problems. The latter
concerns to the Laplace equation as well as to its counterparts in the
potential theory for inhomogeneous and anisotropic media.
2010 MSC. 30C62, 30D40, 37E30.
Key words and phrases. Dirichlet, Hilbert (Riemann–Hilbert), Neu-
mann, Poincare and Riemann boundary value problems, Laplace equa-
tion, Beltrami equations, harmonic and analytic functions, quasiconfor-
mal and A−harmonic functions.
1. Introduction
The Dirichlet, Hilbert (Riemann–Hilbert), Neumann, Poincare and
Riemann boundary value problems are basic in the theory of analytic
functions and they are closely interconnected, see e.g. the monographs
[24,56] and [72] for the history, and also the recent papers [21,34,61–69]
and [74].
Received 26.06.2016
ISSN 1810 – 3200. c© Iнститут математики НАН України
168 On recent advances in boundary value problems
Recall that twice continuously differentiable solutions of the Laplace
equation
∆u :=
∂2u
∂x2
+
∂2u
∂y2
= 0 ∀ z ∈ D (1.1)
are called harmonic functions. As well known, they are infinitely dif-
ferentiable. The classic Dirichlet problem in the unit disk D = {z ∈
C : |z| < 1}, z = x + iy, is the problem on the existence of harmonic
functions u : D → R such that
lim
z→ζ
u(z) = ϕ(ζ) ∀ ζ ∈ ∂D (1.2)
for a prescribed continuous function ϕ : ∂D → R.
The request (1.2) is too strong and has no sense if the boundary func-
tion ϕ is only measurable. However, Luzin has shown in his dissertation
that, for every measurable function ϕ : ∂D → R, there exists a harmonic
function in D such that (1.2) holds for a.e. ζ ∈ ∂D along any nontan-
gential path, see, e.g., [51]. F. Gehring in [26] has rediscovered this fact
in a similar way on the basis of another deep Luzin result, see Section 2.
Furthermore, it was proved in [61] that the space of such solutions has
the infinite dimension, see also [63].
Moreover, it was demonstrated in [34, 61–69] and [74] that all other
boundary value problems mentioned above for harmonic and analytic
functions as well as their generalizations in the extended sense are suc-
cessively reduced to the first boundary value problem. In particular, it
is well–known that the Neumann problem has no classical solutions gen-
erally speaking even for smooth boundary data, see e.g. [54]. The main
goal of the short note [65] was to show that the problem has nonclassical
solutions for arbitrary measurable data. The result was based on a re-
duction of this problem to the Hilbert boundary value problem recently
solved for arbitrary measurable coefficients and for arbitrary measurable
boundary data in [61].
Let us start from a more general problem on directional deriva-
tives. The classic setting of the latter problem is to find a harmonic
function u : D → R that admits a continuous extension to the boundary
of D together with its first partial derivatives and satisfies the boundary
condition
∂u
∂ν
= ϕ(ζ) ∀ ζ ∈ ∂D (1.3)
with a prescribed continuous date ϕ : ∂D → R 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
. (1.4)
V. Gutlyanskii, V. Ryazanov 169
The Neumann problem is a special case of the above problem on
directional derivatives with the boundary condition
∂u
∂n
= ϕ(ζ) ∀ ζ ∈ ∂D (1.5)
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 (1.6)
where a = a(ζ) and b = b(ζ) are real-valued functions given on ∂D.
Recall that the classical setting of the Riemann problem in a
smooth Jordan domain D of the complex plane C was on finding an-
alytic 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 (1.7)
with prescribed Hölder continuous functions A : ∂D → C and B : ∂D →
C.
Recall also that the Riemann problem with shift in D was on
finding such functions f+ : D → C and f− : C \ D → C satisfying the
condition
f+(α(ζ)) = A(ζ) · f−(ζ) + B(ζ) ∀ ζ ∈ ∂D (1.8)
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.
The classical setting of the Hilbert (Riemann–Hilbert) bound-
ary value problem was on finding analytic functions f in a domain
D ⊂ C bounded by a rectifiable Jordan curve with the boundary condi-
tion
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) ∀ ζ ∈ ∂D (1.9)
with functions λ and ϕ that are continuously differentiable with respect
to the natural parameter s on ∂D and, moreover, |λ| 6= 0 everywhere on
∂D. Hence without loss of generality one can assume that |λ| ≡ 1 on ∂D.
It is clear that if we start to consider the Hilbert and Riemann prob-
lems with measurable boundary data, the requests on the existence of
the limits at all points ζ ∈ ∂D and along all paths terminating in ζ lose
170 On recent advances in boundary value problems
any sense (as well as the conception of the index). Thus, the notion of
solutions of the Hilbert and Riemann problems should be widened in this
case. The nontangential limits were a suitable tool from the function the-
ory of one complex variable, see e.g. [61–69] and [74]. In [34] and [62], it
was proposed an alternative approach based on the use of special families
of curves terminating at the boundary, see [7], and admitting tangential
curves.
Moreover, we extend the results on the boundary value problems to
the case of quasiconformal functions (the Beltrami equations) as well as
to A-harmonic functions that leads to problems of mathematical physics
in inhomogeneous and anisotropic media, see [34,69] and [74]. The latter
demands the application of the so-called logarithmic capacity zero that
is invariant under quasiconformal mappings.
2. Angular limits in Dirichlet problem for Laplace equa-
tion
The following deep (non–trivial) result of Luzin was one the main
theorems of his (1915) dissertation, see e.g. [51, p. 78].
Theorem A. For any measurable function ϕ : [0, 1] → R, there is
a continuous function Φ : [0, 1] → R such that Φ′ = ϕ a.e.
Just on the basis of Theorem A, Luzin proved the next significant
result of his dissertation, see e.g. [51, p. 80], that is key for our goals.
Theorem B. Let ϕ(ϑ) be real, measurable, almost everywhere finite
and have the period 2π. Then there exists a harmonic function u in the
unit disk D such that u(z) → ϕ(ϑ) for a.e. ϑ as z → eiϑ along any
nontangential path.
Here a path in D terminating at a point ζ = eiϑ ∈ ∂D is called
nontangential if its part in a neighborhood of ζ lies inside of an angle
in D with the vertex at ζ. Hence such limits are called also angular
limits. The latter is a traditional tool of the geometric function theory,
see e.g. monographs [19,44,51,59] and [60].
Note that the Luzin dissertation was published only in Russian in
the book [51] prepared by his pupils Barri and Men’shov after his death
but Theorem A was published with a complete proof in English in the
book [70, p. 217], as Theorem VII(2.3). Hence Frederick Gehring in [26]
has rediscovered Theorem B and his proof on the basis of Theorem A has
in fact coincided with the original proof of Luzin. Since the proof is very
short and nice and has a common interest, we give it for completeness
here.
V. Gutlyanskii, V. Ryazanov 171
Proof. By Theorem A we can find a continuous function Φ(ϑ) such that
Φ′(ϑ) = ϕ(ϑ) for a.e. ϑ. Considering the Poisson integral
U(reiϑ) =
1
2π
2π∫
0
1− r2
1− 2r cos(ϑ− t) + r2
Φ(t) dt
for 0 < r < 1, U(0) := 0, we see by the Fatou result, see e.g. 3.441
in [76], p. 53, that ∂
∂ϑ U(z) → Φ′(ϑ) as z → eiϑ along any nontangential
path whenever Φ′(ϑ) exists. Thus, the conclusion follows for the function
u(z) = ∂
∂ϑ U(z).
Remark 2.1. Note that the given function u is harmonic in the punc-
tured unit disk D \ {0} because the function U is harmonic in D and
the differential operator ∂
∂ϑ is commutative with the Laplace operator
∆. Setting u(0) = 0, we see that
u(reiϑ) = − r
π
2π∫
0
(1− r2) sin(ϑ− t)
(1− 2r cos(ϑ− t) + r2)2
Φ(t) dt → 0 as r → 0 ,
i.e. u(z) → u(0) as z → 0, and, moreover, the integral of u over each circle
|z| = r, 0 < r < 1, is equal to zero. Thus, by the criterion for a harmonic
function on the averages over circles we have that u is harmonic in D.
The alternative argument for the latter is the removability of isolated
singularities for harmonic functions, see e.g. [57].
Corollary 5.1 to Theorem 5.1 in [61] has strengthened Theorem B as
the next.
Theorem C. For each (Lebesgue) measurable function ϕ : ∂D → R,
the space of all harmonic functions u : D → R with the angular limits
ϕ(ζ) for a.e. ζ ∈ ∂D has the infinite dimension.
Theorem C is the direct consequence of Theorem B and Theorem 5.1
in [61] :
Theorem 2.2. The space of all harmonic functions in D with angular
limit 0 at a.e. point of ∂D has the infinite dimension.
We give its complete proof here in view of its importance because we
will successively reduce all other boundary value problems to Theorem
C.
172 On recent advances in boundary value problems
Proof. Indeed, let Φ : [0, 2π] → R be integrable and differentiable a.e.
with Φ′(t) = 0. Then the function
U(z) : =
1
2π
2π∫
0
1− r2
1− 2r cos(ϑ− t) + r2
Φ(t) dt , z = reiϑ, r < 1 ,
is harmonic on D with U(z) → Φ(Θ) as z → eiΘ, see e.g. Theorem 1.3
in [25] or Theorem IX.1.1 in [28], and ∂
∂ϑ U(z) → Φ′(Θ) as z → eiΘ
along any nontangential path whenever Φ′(Θ) exists, see e.g. 3.441 in
[76], p. 53, or Theorem IX.1.2 in [28]. Thus, the harmonic function
u(z) = ∂
∂ϑ U(z) has nontangential limit 0 at a.e. point of ∂D.
Let us give a subspace of such functions u with an infinite basis.
Namely, let ϕ : [0, 1] → [0, 1] be the Cantor function, see e.g. 8.15
in [27], and let ϕn : [0, 2π] → [0, 1] be equal to ϕ((t− an−1)/(an− an−1))
on [an−1, an) where a0 = 0 and an = 2π(2−1 + . . . + 2−n), n = 1, 2, . . .
and 0 outside of [an−1, an). Denote by Un and un the harmonic functions
corresponding to ϕn as in the first item.
By the construction the supports of the functions ϕn are mutually
disjoint and, thus, the series
∞∑
n=1
γnϕn is well defined for every sequence
γn ∈ R, n = 1, 2, . . .. If in addition we restrict ourselves to the sequences
γ = {γn} in the space l with the norm ‖γ‖ =
∞∑
n=1
|γn|, then the series is
a suitable function Φ for the first item.
Denote by U and u the harmonic functions corresponding to the
function Φ as in the first item and by H0 the class of all such u. Note
that un, n = 1, 2, . . ., form a basis in the space H0 with the locally
uniform convergence in D which is metrizable.
Firstly,
∞∑
n=1
γnϕn 6= 0 if γ 6= 0. Really, let us assume that γn 6= 0 for
some n = 1, 2, . . .. Then u 6= 0 because the limits lim
z→ζ
U(z) exist for all
ζ = eiϑ with ϑ ∈ (an−1, an) and can be arbitrarily close to 0 as well as
to γn.
Secondly, u∗m =
m∑
n=1
γnϕn → u locally uniformly in D as m → ∞.
Indeed, elementary calculations give the following estimate of the re-
mainder term
|u(z)− u∗m(z)| ≤ 2r(1 + r)
(1− r)3
·
∞∑
n=m+1
|γn| → 0 as m→ ∞
in every disk D(r) = {z ∈ C : |z| ≤ r}, r < 1.
V. Gutlyanskii, V. Ryazanov 173
Remark 2.3. In Section 5, one can find our more refined results which
are counterparts of Theorem A, B, C as well as 2.2 in terms of logarith-
mic capacity that makes possible to extend the theory of boundary value
problems to the so-called A−harmonic functions corresponding to gen-
eralizations of the Laplace equation in inhomogeneous and anisotropic
media.
By the well–known Lindelöf maximum principle, see e.g. Lemma 1.1
in [25], it follows the uniqueness theorem for the Dirichlet problem in
the class of bounded harmonic functions u on the unit disk D = {z ∈
C : |z| < 1}. In general there is no uniqueness theorem in the Dirichlet
problem for the Laplace equation even under zero boundary data. In
comparison with the examples in the proof of Theorem 2.2, here we give
more elementary examples and constructions of solutions from [63].
Many such nontrivial solutions u for the Laplace equation can be
given by the Poisson–Stiltjes integral
u(z) =
1
2π
2π∫
0
Pr(ϑ− t) dΦ(t) , z = reiϑ, r < 1 , (2.1)
with an arbitrary singular function Φ : [0, 2π] → R, i.e., where Φ is
of bounded variation and Φ′ = 0 a.e., and where we use the standard
notation for the Poisson kernel
Pr(Θ) =
1− r2
1− 2r cosΘ + r2
, r < 1 . (2.2)
Indeed, u in (2.1) is harmonic for every function Φ : [0, 2π] → R of
bounded variation and by the Fatou theorem, see e.g. Theorem I.D.3.1
in [44], u(z) → Φ′(Θ) as z → eiΘ along any nontangential path whenever
Φ′(Θ) exists. Thus, u(z) → 0 as z → eiΘ for a.e. Θ ∈ [0, 2π] along any
nontangential paths for every singular function Φ.
Example 2.4. The first natural example is given by the formula (2.1)
with Φ(t) = ϕ(t/2π) where ϕ : [0, 1] → [0, 1] is the well–known Cantor
function, see e.g. [18] and further references therein.
Example 2.5. However, the simplest example of such a kind is given
by nondecreasing step-like data Φϑ0 with values 0 and 2π and with the
jump at ϑ0 ∈ (0, 2π):
u(z) = Pr(ϑ− ϑ0) =
1− r2
1− 2r cos(ϑ− ϑ0) + r2
, z = reiϑ, r < 1 .
174 On recent advances in boundary value problems
We see that u(z) → 0 as z → eiΘ for all Θ ∈ (0, 2π) except Θ = ϑ0.
Note that the function u is harmonic in the unit disk D because
u(z) = Re
ζ0 + z
ζ0 − z
=
1− |z|2
1− 2Re zζ0 + |z|2
, ζ0 = eiϑ0 , z ∈ D ,
where the function w = g(z) = gζ0(z) : = (ζ0 + z)/(ζ0 − z) is analytic
(conformal) in D and maps D onto half-plane Rew > 0, g(0) = 1, g(ζ0) =
∞.
Remark 2.6. On the basis of the latter examples, it was given an al-
ternative proof of Theorem 2.2 and its strengthening for the class of all
harmonic functions in D with the angular limit 0 at every point of ∂D
except a countable collection of points, see Theorem 2.1 in [63]. In par-
ticular, that makes possible to formulate the corresponding more refined
results in terms of the so-called logarithmic capacity.
Note also that the harmonic functions u given in the proof of Theorem
2.2 and in (2.1) cannot be represented in the form of the Poisson integral
with any integrable function ϕ : [0, 2π] → R because such integral would
have nontangential limits ϕ a.e. in ∂D, see e.g. Corollary IX.1.1 in [28].
Consequently, u do not belong to the classes hp for any p > 1, see e.g.
Theorem IX.2.3 in [28]. However, the functions u in (2.1) belong to the
class h1, see e.g. Theorem IX.2.2 in [28].
Recall that hp, p ∈ (0,∞), denotes the class of harmonic functions u
in D with
sup
r∈(0,1)
2π∫
0
|u(reiϑ)|p dϑ
1
p
< ∞
and functions in h1 have angular limits a.e. in ∂D, see e.g. Corollary
IX.2.2 in [28].
3. Angular limits in Hilbert problem for analytic func-
tions
Boundary value problems for analytic functions are due to the well-
known Riemann dissertation (1851) contained a general setting of a prob-
lem on finding analytic functions with a connection between its real and
imaginary parts on the boundary. However, it has contained no concrete
boundary value problems.
The first concrete problem of such a type has been proposed by Hilbert
(1904) and called by the Hilbert problem or the Riemann–Hilbert prob-
lem. Recall that this problem consists in finding an analytic function
V. Gutlyanskii, V. Ryazanov 175
f in a domain bounded by a rectifiable Jordan curve C with the linear
boundary condition
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) ∀ ζ ∈ C (3.1)
where it was assumed that the functions λ and ϕ are continuously dif-
ferentiable with respect to the natural parameter s on C and, moreover,
|λ| 6= 0 everywhere on C. Hence without loss of generality we may assume
that |λ| ≡ 1 on C.
The first way for solving this problem based on the theory of singular
integral equations was given by Hilbert (1904), see [35]. This attempt was
not quite successful because of the theory of singular integral equations
has been not yet enough developed at that time. However, just that way
became the main approach in this research direction with important con-
tributions of Georgian and Russian mathematicians and mechanicians,
see e.g. [24, 56] and [72]. In particular, the existence of solutions to this
problem was in that way proved for Hölder continuous λ and ϕ. But sub-
sequent weakening conditions on λ and ϕ led to strengthening conditions
on the contour C, say to the Lyapunov curves or the Radon condition of
bounded rotation or even to smooth curves.
However, Hilbert (1905) has solved his problem with the above set-
tings to (3.1) in the second way based on the reduction it to solving
the corresponding two Dirichlet problems, see e.g. [36]. The main goal
of the paper [61] was to show that this approach is more simple and
leads to perfectly general results in the problem for the arbitrary recti-
fiable Jordan domains with coefficients λ and boundary data ϕ that are
only measurable with respect to the natural parameter. The key was the
Gehring–Luzin result, see Theorem B in Section 2, on the Dirichlet prob-
lem for harmonic functions. But the way of the reduction of the Hilbert
problem to the corresponding 2 Dirichlet problems was original in [61].
First we have the result in the unit disk D, see Theorems 2.1 and 5.2
in [61].
Theorem 3.1. Let λ : ∂D → C, |λ(ζ)| ≡ 1, and ϕ : ∂D → R be
measurable functions. Then there exist analytic functions f : D → C
such that along any nontangential path
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) for a.e. ζ ∈ ∂D (3.2)
and the space of such analytic functions has the infinite dimension.
The proof is short and, in view of its importance, we give it here.
176 On recent advances in boundary value problems
Proof. First, consider the function α(ζ) = arg λ(ζ) where argω is the
principal value of the argument of ω ∈ C with |ω| = 1, i.e., the unique
number α ∈ (−π, π] such that ω = eiα. Note that the function argω is
continuous on ∂D \ {−1} and the sets λ−1(∂D \ {−1}) and λ−1(−1) are
measurable because the function λ(ζ) is measurable. Thus, the function
α(ζ) is measurable on ∂D. Furthermore, α ∈ L∞(∂D) because |α(ζ)| ≤ π
for all ζ ∈ ∂D. Hence
g(z) : =
1
2πi
∫
∂D
α(ζ)
z + ζ
z − ζ
dζ
ζ
, z ∈ D , (3.3)
is an analytic function in D with u(z) = Re g(z) → α(ζ) as z → ζ along
any nontangential path in D for a.e. ζ ∈ ∂D, see e.g. Corollary IX.1.1
in [28] and Theorem I.E.1 in [44]. Denote A(z) = exp{ig(z)} that is an
analytic function.
Since α ∈ L∞(∂D), we have that u ∈ hp for all p ≥ 1, see e.g.
Theorem IX.2.3 in [28], and then also v = Im g ∈ hp for all p ≥ 1 by
the theorem of M. Riesz (1927), see e.g. Theorem IX.2.4 in [28]. Hence
there exists a function β : ∂D → R, β ∈ Lp, for all p ≥ 1 such that
v(z) → β(ζ) as z → ζ for a.e. ζ ∈ ∂D along any nontangential path,
see e.g. Theorem IX.2.3 and Corollary IX.2.2 in [28]. Thus, by Theorem
B there exists an analytic function B : D → C such that Re B(z) →
B(ζ) : = ϕ(ζ) · exp{β(ζ)} as z → ζ along any nontangential path for
a.e. ζ ∈ ∂D. Finally, elementary calculations show that one of the desired
analytic functions in (3.7) is f = A · B.
Let U : D → R be a harmonic function with angular limit 0 at
a.e. point of ∂D from Theorem 2.2. Then there is the unique harmonic
function V : D → R with V (0) = 0 such that C = U + iV is an analytic
function. Thus, setting in the last item f = A(B+C) instead of f = A·B,
we obtain by Theorem 2.2 the space of analytic functions of the infinite
dimension satisfying (3.7).
Remark 3.2. As it follows from formula (3.3), the first analytic function
A in the proof is calculated in the explicit form. The function β : ∂D → R
in the proof can also explicitly be calculated by the following formula,
see e.g. Theorem I.E.4.1 in [44], for a.e. ζ ∈ ∂D
β(ζ) : = lim
ε→+0
1
π
π∫
ε
α(ζe−it)− α(ζeit)
2 tg t
2
dt . (3.4)
The second analytic function B in the proof is equal to ∂
∂ϑ G(z),
V. Gutlyanskii, V. Ryazanov 177
z = reiϑ, with
G(z) : =
1
2πi
∫
∂D
Φ(ζ)
z + ζ
z − ζ
dζ
ζ
, z ∈ D , (3.5)
where Φ : ∂D → R is a continuous function such that ∂
∂ϑ Φ(ζ) = B(ζ),
ζ = eiϑ, for a.e. ϑ ∈ [0, 2π], see the nontrivial construction of Theorem
VII(2.3) in [70].
The case of arbitrary rectifiable Jordan curves is reduced to the case
of the unit circle as in Theorem 3.1 and Remark 5.2 in [61].
Theorem 3.3. Let D be a Jordan domain in C with a rectifiable bound-
ary and let λ : ∂D → C, |λ(ζ)| ≡ 1, and ϕ : ∂D → R be measurable
functions with respect to the natural parameter on ∂D. Then there exist
analytic functions f : D → C such that along any nontangential path
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) for a.e. ζ ∈ ∂D (3.6)
with respect to the natural parameter on ∂D and the space of such analytic
functions has the infinite dimension.
Proof. This case is reduced to the case of the unit disk D in the following
way. First, by the Riemann theorem, see e.g. Theorem II.2.1 in [28], there
exists a conformal mapping ω of any Jordan domain D onto D. By the
Caratheodory (1912) theorem ω can be extended to a homeomorphisms
of D onto D and, if ∂D is rectifiable, then by the theorem of F. and M.
Riesz (1916) length ω−1(E) = 0 whenever E ⊂ ∂D with |E| = 0, see
e.g. Theorem II.C.1 and Theorems II.D.2 in [44]. Conversely, by the
Lavrentiev (1936) theorem |ω(E)| = 0 whenever E ⊂ ∂D and length E =
0, see [49], see also the point III.1.5 in [60].
Hence ω and ω−1 transform measurable sets into measurable sets.
Indeed, every measurable set is the union of a sigma-compact set and
a set of measure zero, see e.g. Theorem III(6.6) in [70], and continuous
mappings transform compact sets into compact sets. Thus, a function
ϕ : ∂D → R is measurable with respect to the natural parameter on ∂D
if and only if the function Φ = ϕ ◦ ω−1 : ∂D → R is measurable with
respect to the linear measure on ∂D.
By the Lindelöf (1917) theorem, see e.g. Theorem II.C.2 in [44], if ∂D
has a tangent at a point ζ, then arg [ω(ζ)− ω(z)]− arg [ζ − z] → const
as z → ζ. In other words, the conformal images of sectors in D with
a vertex at ζ is asymptotically the same as sectors in D with a vertex
at w = ω(ζ). Thus, nontangential paths in D are transformed under ω
178 On recent advances in boundary value problems
into nontangential paths in D. Finally, a rectifiable Jordan curve has a
tangent a.e. with respect to the natural parameter and, thus, Theorem
3.3 follows from Theorem 3.1.
Remark 3.4. The conceptions of a harmonic measure introduced by R.
Nevanlinna in [57] and a principal asymptotic value based on one nice
result of F. Bagemihl [6] make possible with a great simplicity and gen-
erality to formulate the existence theorems for the Dirichlet and Hilbert
problems in arbitrary Jordan domains, see Theorem 4.1 and Remark 5.2
in [61].
In view of the theorems of Riemann and Caratheodory, this approach
makes possible also to formulate the corresponding theorems for arbitra-
ry simply connected domains D in C having at least 2 boundary points.
The only difference is that the functions λ and ϕ should be given as
functions of prime ends of D but not of points of ∂D and harmonic
measures of sets of prime ends are given through the natural one-to-one
correspondence between the prime ends of D and the boundary points
of D under Riemann mappings ω : D → D, see e.g. [17].
Moreover, in [64] it was proved the existence of multivalent solutions
with the infinite number of branches for the Hilbert problem in the gen-
eral settings of finitely connected domains bounded by mutually disjoint
Jordan curves, measurable coefficients and measurable boundary data.
The general theorem is formulated in terms of harmonic measure and
principal asymptotic values. It is also given the corresponding reinforced
criterion for domains with rectifiable boundaries stated in terms of the
natural parameter and nontangential limits. Furthermore, it is shown
that the dimension of the spaces of these solutions is infinite.
Let us start from the simplest kind of multiply connected domains.
Recall that a domain D in C = C∪{∞} is called circular if its boundary
consists of finite number of mutually disjoint circles and points. We call
such a domain nondegenerate if its boundary consists only of circles.
The following statement was first proved as Theorem 2.1 in [64].
Theorem 3.5. Let D∗ be a bounded nondegenerate circular multiply con-
nected domain and let λ : ∂D∗ → C, |λ(ζ)| ≡ 1, and ϕ : ∂D∗ → R be
measurable functions. Then there exist multivalent analytic functions
f : D∗ → C with the infinite number of branches such that
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) (3.7)
along any nontangential path to a.e. ζ ∈ ∂D∗.
V. Gutlyanskii, V. Ryazanov 179
Proof. Indeed, by the Poincare theorem, see e.g. Theorem VI.1 in [28],
there is a locally conformal mapping g of the unit disc D = {z ∈ C :
|z| < 1} onto D∗. Let h : D∗ → D be the corresponding multivalent
analytic function that is inverse to g. D∗ without a finite number of cuts
is simply connected and hence h has there only single-valued branches
that are extended to the boundary by the Caratheodory theorem.
By Section VI.2 in [28], ∂D without a countable set of its points con-
sists of a countable collection of arcs every of which is a one-to-one image
of a circle in ∂D∗ without its one point under every extended branch of
h. Note that by the reflection principle g is conformally extended into
a neighborhood of every such arc and, thus, nontangential paths to its
points go into nontangential paths to the corresponding points of circles
in ∂D∗ and inversely.
Setting Λ = λ ◦ g and Φ = ϕ ◦ g with the extended g on the given
arcs of ∂D we obtain measurable functions on ∂D. Thus, by Theorem 2.1
in [61], see Theorem 3.1 above, there exist analytic functions F : D → C
such that
lim
w→η
Re {Λ(η) · F (w)} = Φ(η) (3.8)
along any nontangential path to a.e. η ∈ ∂D. By the above arguments,
we see that f = F ◦ h are desired multivalent analytic solutions of (3.7).
To solve the Riemann–Hilbert problem in the case of domains boun-
ded by a finite number of rectifiable Jordan curves we should extend
to this case the known results of Caratheodory (1912), Lindelöf (1917),
F. and M. Riesz (1916) and Lavrentiev (1936) for Jordan’s domains.
Namely, it was proved as Lemma 3.1 in [64] the following statements.
Lemma 3.6. Let D be a bounded domain in C whose boundary compo-
nents are Jordan curves, D∗ be a bounded nondegenerate circular domain
in C and let ω : D → D∗ be a conformal mapping. Then
(i) ω can be extended to a homeomorphism of D onto D∗;
(ii) arg [ω(ζ) − ω(z)] − arg [ζ − z] → const as z → ζ whenever ∂D
has a tangent at ζ ∈ ∂D;
(iii) for rectifiable ∂D, length ω−1(E) = 0 whenever |E| = 0, E ⊂
∂D∗;
(iv) for rectifiable ∂D, |ω(E)| = 0 whenever length E = 0, E ⊂ ∂D.
Proof. (i) Indeed, we are able to transform D∗ into a simply connected
domain D∗ through a finite sequence of cuts. Thus, we come to the
desired conclusion applying the Caratheodory theorems to simply con-
nected domains D∗ and D∗ := ω−1(D∗), see e.g. Theorem 9.4 in [17] and
Theorem II.C.1 in [44].
180 On recent advances in boundary value problems
(ii) In the construction from the previous item, we may assume that
the point ζ is not the end of the cuts in D generated by the cuts in D∗ un-
der the extended mapping ω−1. Thus, we come to the desired conclusion
twice applying the Caratheodory theorems, the reflection principle for
conformal mappings and the Lindelöf theorem for the Jordan domains,
see e.g. Theorem II.C.2 in [44].
Points (iii) and (iv) are proved similarly to the last item on the basis of
the corresponding results of F. and M. Riesz and Lavrentiev for Jordan
domains with rectifiable boundaries, see e.g. Theorem II.D.2 in [44],
and [49], see also the point III.1.5 in [60].
Lemma 3.6 makes possible to reduce the case of domains with rectifi-
able boundaries to the case of circular domains, see Theorem 3.1 in [64].
Theorem 3.7. Let D be a bounded multiply connected domain in C
whose boundary components are rectifiable Jordan curves and λ : ∂D →
C, |λ(ζ)| ≡ 1, and ϕ : ∂D → R be measurable functions with respect
to the natural parameter on ∂D. Then there exist multivalent analytic
functions f : D → C with the infinite number of branches such that along
any nontangential path
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) for a.e. ζ ∈ ∂D (3.9)
with respect to the natural parameters of the boundary components of D.
Proof. This case is reduced to the case of a bounded nondegenerate circu-
lar domain D∗ in the following way. First, there is a conformal mapping
ω of D onto a circular domain D∗, see e.g. Theorem V.6.2 in [28]. Note
that D∗ is not degenerate because isolated singularities of conformal map-
pings are removable that is due to the well-known Weierstrass theorem,
see e.g. Theorem 1.2 in [17]. Without loss of generality, we may assume
that D∗ is bounded.
By point (i) in Lemma 3.6 ω can be extended to a homeomorphisms
of D onto D∗. If ∂D is rectifiable, then by point (iii) in Lemma 3.6
length ω−1(E) = 0 whenever E ⊂ ∂D∗ with |E| = 0, and by (iv) in
Lemma 3.6, conversely, |ω(E)| = 0 whenever E ⊂ ∂D with length E = 0.
In the last case ω and ω−1 transform measurable sets into measur-
able sets. Indeed, every measurable set is the union of a sigma-compact
set and a set of measure zero, see e.g. Theorem III(6.6) in [70], and
continuous mappings transform compact sets into compact sets. Thus,
a function ϕ : ∂D → R is measurable with respect to the natural pa-
rameter on ∂D if and only if the function Φ = ϕ ◦ ω−1 : ∂D∗ → R is
measurable with respect to the natural parameter on ∂D∗.
V. Gutlyanskii, V. Ryazanov 181
By point (ii) in Lemma 3.6, if ∂D has a tangent at a point ζ ∈ ∂D,
then arg [ω(ζ)−ω(z)]−arg [ζ−z] → const as z → ζ. In other words, the
conformal images of sectors in D with a vertex at ζ is asymptotically the
same as sectors in D∗ with a vertex at w = ω(ζ). Thus, nontangential
paths in D are transformed under ω into nontangential paths in D∗ and
inversely. Finally, a rectifiable Jordan curve has a tangent a.e. with
respect to the natural parameter and, thus, Theorem 3.3 follows from
Theorem 3.1.
Theorem 4.1 in [64] is formulated for finitely connected domains boun-
ded by arbitrary Jordan curves in terms of harmonic measure and the
so-called principal asymptotic values. Finally, Theorems 5.1 in [64] says
on the dimension of these spaces of solutions:
Theorem 3.8. The spaces of solutions of the Hilbert boundary value
problem in Theorems 3.5 and 3.7 have the infinite dimension.
Proof. By Theorem 5.1 in [61], see Theorem 3.1 above, the space of
solutions of the problem (3.8) has the infinite dimension. Thus, the
conclusion follows by the construction of these solutions in the given
theorems through the successive reduction to (3.8).
Remark 3.9. Of course, results concerning to the infinite dimension of
the space of solutions are not in some sense new and treated in terms of
the infinite index of the Hilbert boundary value problem, see e.g. [29] and
[55]. By the general theory of boundary value problems, each additional
singularities, including singularities on the boundary contour, increases
the index. Hence the above results can be also interpreted as the case of
the infinite index.
Note that the considered situations admit the boundary functions
with uncountable singularities. Such examples are given by Poisson-
Stiltjes integral with the Cantor type functions under its differential, see
e.g. [63] or Section 2 above, and the corresponding examples of analytic
functions in the simplest case of the Hilbert problem under λ = 1 and
ϕ = 0 a.e.
Indeed, a Cantor type set C is perfect, i.e. it is closed and without
isolated points. Hence C is of the continuum cardinality by the well–
known W.H. Young theorem, see [75]. The corresponding Cantor type
function has the symmetric Lebesgue derivative +∞ at every point in
C except ends, see e.g. the survey [18]. Then by the Fatou theorem,
see e.g. Theorem I.D.3.2 in [44], the corresponding harmonic function
has the radial limit +∞ on the set of the continuum (maximal possible)
cardinality.
182 On recent advances in boundary value problems
In this connection, it would be also interesting to study the prob-
lem on a maximal possible cardinality of the dimension of the spaces of
solutions for the Hilbert problem.
4. Angular limits in Neumann problem for Laplace
equation
It is well–known that the Neumann problem has no classical solutions
generally speaking even for smooth boundary data, see e.g. [54]. On the
basis of solving of Hilbert boundary value problem in [61], see the last
section, it was proved in [65] the existence of nonclassical solutions of
the Neumann problem for the harmonic functions in the Jordan rectifi-
able domains with arbitrary measurable boundary distributions of normal
derivatives. The same is stated for a special case of the Poincare problem
on directional derivatives. Moreover, it is shown that the spaces of the
found solutions have the infinite dimension.
Let us start from the more general problem on directional derivatives,
see Theorem 1 and Remark 1 in [65].
Theorem 4.1. Let ν : ∂D → C, |ν(ζ)| ≡ 1, and ϕ : ∂D → R be
measurable functions. Then there exist harmonic functions u : D → R
such that
lim
z→ζ
∂u
∂ν
(z) = ϕ(ζ) (4.1)
along any nontangential paths to a.e. point ζ ∈ ∂D.
Remark 4.2. We are able to say more in the case of Re n(ζ)ν(ζ) > 0.
Indeed, the latter magnitude is a scalar product of n = n(ζ) and ν = ν(ζ)
interpreted as vectors in R2 and it has the geometric sense of projection
of the vector ν onto the inner normal n to ∂D at the point ζ. In view of
(4.1), since the limit ϕ(ζ) is finite, there is a finite limit u(ζ) of u(z) as
z → ζ in D along the straight line passing through the point ζ and being
parallel to the vector ν because along this line
u(z) = u(z0) −
1∫
0
∂u
∂ν
(z0 + τ(z − z0)) dτ . (4.2)
Thus, at each point with condition (4.1), there is the directional deriva-
tive
∂u
∂ν
(ζ) := lim
t→0
u(ζ + t · ν)− u(ζ)
t
= ϕ(ζ) . (4.3)
V. Gutlyanskii, V. Ryazanov 183
In particular, in the case of the Neumann problem, Re n(ζ)ν(ζ) ≡ 1
and we have by Theorem 4.1 and Remark 4.2 the following result, see
Theorem 2 in [65].
Theorem 4.3. For each measurable function ϕ : ∂D → R, one can find
harmonic functions u : D → R such that, at a.e. point ζ ∈ ∂D, there
exist:
1) the finite radial limit
u(ζ) := lim
r→1
u(rζ) (4.4)
2) the normal derivative
∂u
∂n
(ζ) := lim
t→0
u(ζ + t · n)− u(ζ)
t
= ϕ(ζ) (4.5)
3) the nontangential limit
lim
z→ζ
∂u
∂n
(z) =
∂u
∂n
(ζ) (4.6)
where n = n(ζ) denotes the unit interior normal to ∂D at the point ζ.
Proof. To prove Theorem 4.1, let us show that the problem on directional
derivatives is equivalent to the corresponding Riemann–Hilbert problem.
Indeed, let u be a harmonic function u : D → R satisfying the bound-
ary condition (4.1). Then the functions U = ux and V = −uy satisfy
the system of Cauchy-Riemann: Uy = −Vx and Ux = Vy in view of
(1.1). Thus, the function f = U + iV is analytic in D and along any
nontangential path to a.e. ζ ∈ ∂D
lim
z→ζ
Re ν(ζ) · f(z) = ϕ(ζ) (4.7)
that is equivalent to (4.1). Inversely, let f : D → C be an analytic
function satisfying the boundary condition (4.7). Then any indefinite
integral F of f is also a single-valued analytic function in D and u = Re F
is a harmonic function satisfying the boundary condition (4.1) because
the directional derivative
∂u
∂ν
= Re ν · ∇u = Re ν · ∇u = (ν,∇u) (4.8)
is the scalar product of ν and the gradient ∇u interpreted as vectors in
R2.
Thus, Theorem 4.1 is a direct consequence of Theorem 2.1 in [61], see
Theorem 3.1 above on the Hilbert boundary value problem with λ(ζ) =
ν(ζ), ζ ∈ ∂D.
184 On recent advances in boundary value problems
The proof of the following result in domains bounded by rectifiable
Jordan curves, Theorem 3 in [65], is perfectly similar to the proof of
Theorem 4.1 above but it is based on more general Theorem 3.1 in [61],
see Theorem 3.3 in Section 3.
Theorem 4.4. Let D be a domain in C bounded by a rectifiable Jordan
curve, ν : ∂D → C, |ν(ζ)| ≡ 1, and ϕ : ∂D → R be measurable func-
tions with respect to the natural parameter. Then there exist harmonic
functions u : D → R such that along any nontangential paths
lim
z→ζ
∂u
∂ν
(z) = ϕ(ζ) (4.9)
for a.e. point ζ ∈ ∂D with respect to the natural parameter.
Remark 4.5. Again we are able to say more in the case with Re n·ν > 0
where n = n(ζ) is the unit inner normal at a point ζ ∈ ∂D with a tangent
to ∂D. In view of (4.9), since the limit ϕ(ζ) is finite, there is a finite
limit u(ζ) of u(z) as z → ζ in D along the straight line passing through
the point ζ and being parallel to the vector ν because along this line, for
z and z0 that are close enough to ζ,
u(z) = u(z0) −
1∫
0
∂u
∂ν
(z0 + τ(z − z0)) dτ . (4.10)
Thus, at each point with the condition (4.9), there is the directional
derivative
∂u
∂ν
(ζ) := lim
t→0
u(ζ + t · ν)− u(ζ)
t
= ϕ(ζ) . (4.11)
In particular, in the case of the Neumann problem, Re n(ζ) · ν(ζ) ≡
1 > 0 and we have by Theorem 4.6 and Remark 4.5 the following signif-
icant result, Theorem 4 in [65]. Here we also apply the well-known fact
that any rectifiable curve has a tangent a.e. with respect to the natural
parameter.
Theorem 4.6. Let D be a domain in C bounded by a rectifiable Jordan
curve and ϕ : ∂D → R be a measurable function with respect to the
natural parameter. Then one can find harmonic functions u : D → R
such that, at a.e. point ζ ∈ ∂D with respect to the natural parameter,
there exist:
1) the finite normal limit
u(ζ) := lim
z→ζ
u(z) (4.12)
V. Gutlyanskii, V. Ryazanov 185
2) the normal derivative
∂u
∂n
(ζ) := lim
t→0
u(ζ + t · n)− u(ζ)
t
= ϕ(ζ) (4.13)
3) the nontangential limit
lim
z→ζ
∂u
∂n
(z) =
∂u
∂n
(ζ) (4.14)
where n = n(ζ) denotes the unit interior normal to ∂D at the point ζ.
Note that here the tangent τ(s) to ∂D is measurable with respect
to the natural parameter s as the derivative dζ(s)/ds and, thus, the
inner normal n(s) to ∂D is also measurable with respect to the natural
parameter.
Finally, we have the following significant result, Theorem 5 in [65].
Theorem 4.7. The spaces of harmonic functions in Theorems 4.1, 4.3,
4.4 and 4.6, being nonclassical solutions of the problem on directional
derivatives and the Neumann problem, correspondingly, have the infinite
dimension for any prescribed measurable boundary data.
Proof. In view of the equivalence of the problem on the directional deriva-
tives to the corresponding Hilbert boundary value problem established
under the proof of Theorem 4.1, the conclusion of Theorem 4.7 follows
directly from Theorem 5.2 and Remark 5.2 in [61], see Theorems 3.1 and
3.3 in Section 3 above.
5. Logarithmic capacity in Hilbert problem for analytic
functions
Here we give more refined results for harmonic and analytic func-
tions in terms of the so-called logarithmic capacity that makes possi-
ble to extend the theory of boundary value problems to quasiconformal
functions (Beltrami equations) and to A−harmonic functions (generaliza-
tions of the Laplace equation in inhomogeneous and anisotropic media),
see [21,34,69,74] and Sections 6–7 and 9–10 further.
Recall some notions and facts which are relevant to logarithmic ca-
pacity, see e.g. [16,57] and [58]. First of all, given a bounded Borel set E
in the plane C, a mass distribution on E is a nonnegative completely
additive function of a set ν defined on its Borel subsets with ν(E) = 1.
The function
Uν(z) :=
∫
E
log
∣∣∣∣
1
z − ζ
∣∣∣∣ dν(ζ) (5.1)
186 On recent advances in boundary value problems
is called a logarithmic potential of the mass distribution ν at a point
z ∈ C. A logarithmic capacity C(E) of the Borel set E is the quantity
C(E) = e−V , V = inf
ν
Vν(E) , Vν(E) = sup
z
Uν(z) . (5.2)
Note that it is sufficient to take the supremum in (5.2) over the set E
only. If V = ∞, then C(E) = 0. It is known that 0 ≤ C(E) < ∞,
C(E1) ≤ C(E2) if E1 ⊆ E2, C(E) = 0 if E =
∞⋃
n=1
En, with C(En) = 0,
n = 1, 2, . . ., see e.g. Lemma III.4 in [16].
It is well-known the following geometric characterization of the loga-
rithmic capacity, see e.g. the point 110 in [57]:
C(E) = τ(E) := lim
n→∞
V
2
n(n−1)
n (5.3)
where Vn denotes the supremum (really, maximum) of the product
V (z1, . . . , zn) =
l=1,...,n∏
k<l
|zk − zl| (5.4)
taken over all collections of points z1, . . . , zn in the set E. Following
Fékete, see [23], the quantity τ(E) is called the transfinite diameter
of the set E. By the geometric interpretation of the logarithmic capacity
as the transfinite diameter we immediately see that if C(E) = 0, then
C(f(E)) = 0 for an arbitrary mapping f that is continuous by Hölder
and, in particular, for conformal and quasiconformal mappings on the
compact sets, see e.g. Theorem II.4.3 in [50].
In order to introduce sets that are measurable with respect to log-
arithmic capacity, we define, following [16], inner C∗ and outer C∗
capacities:
C∗(E) : = sup
F⊆E
C(E) (5.5)
where supremum is taken over all compact sets F ⊂ C, and
C∗(E) : = inf
E⊆O
C(O) (5.6)
where infimum is taken over all open sets O ⊂ C. Further, a bounded
set E ⊂ C is called measurable with respect to the logarithmic
capacity if
C∗(E) = C∗(E) , (5.7)
V. Gutlyanskii, V. Ryazanov 187
and the common value of C∗(E) and C∗(E) is still denoted by C(E).
Note, see e.g. Lemma III.5 in [16], that the outer capacity is semiadditive,
i.e.,
C∗
( ∞⋃
n=1
En
)
≤
∞∑
n=1
C∗(En) . (5.8)
A function ϕ : E → C defined on a bounded set E ⊂ C is called
measurable with respect to logarithmic capacity if, for all open
sets O ⊆ C, the sets
Ω = {z ∈ E : ϕ(z) ∈ O} (5.9)
are measurable with respect to logarithmic capacity. It is clear from the
definition that the set E is itself measurable with respect to logarithmic
capacity.
Note also that sets of logarithmic capacity zero coincide with sets of
the so-called absolute harmonic measure zero introduced by Nevan-
linna, see Chapter V in [57]. Hence a set E is of (Hausdorff) length zero if
C(E) = 0, see Theorem V.6.2 in [57]. However, there exist sets of length
zero having a positive logarithmic capacity, see e.g. Theorem IV.5 in [16].
Remark 5.1. It is known that Borel sets and, in particular, compact and
open sets are measurable with respect to logarithmic capacity, see e.g.
Lemma I.1 and Theorem III.7 in [16]. Moreover, as it follows from the
definition, any set E ⊂ C of finite logarithmic capacity can be represented
as a union of the sigma-compactum (union of countable collection of
compact sets) and the set of logarithmic capacity zero. It is also known
that the Borel sets and, in particular, compact sets are measurable with
respect to all Hausdorff’s measures and, in particular, with respect to
measure of length, see e.g. theorem II(7.4) in [70]. Consequently, any
set E ⊂ C of finite logarithmic capacity is measurable with respect to
measure of length. Thus, on such a set any function ϕ : E → C being
measurable with respect to logarithmic capacity is also measurable with
respect to measure of length on E. However, there exist functions that
are measurable with respect to measure of length but not measurable
with respect to logarithmic capacity, see e.g. Theorem IV.5 in [16].
We are especially interested by functions ϕ : ∂D → C defined on
the unit circle ∂D = {z ∈ C : |z| = 1}. However, in view of (5.3),
it suffices to examine the corresponding problems on the segments of
the real axis because any closed arc on ∂D admits a bi-Lipschitz (even
infinitely smooth, the so-called stereographic) mapping g onto such a
segment.
188 On recent advances in boundary value problems
In this connection, recall that a mapping g : X → X ′ between metric
spaces (X, d) and (X ′, d′) is said to be Lipschitz if d′(g(x1), g(x2)) 6
C ·d(x1, x2) for any x1, x2 ∈ X and for a finite constant C. If, in addition,
d(x1, x2) 6 c ·d′(g(x1), g(x2)) for any x1, x2 ∈ X and for a finite constant
c, then mapping g is called bi-Lipschitz.
First of all, it was proved the following analog of the Luzin theorem,
see Theorem 3.1 in [69], cf. Theorem A in Section 2.
Theorem 5.2. Let ϕ : [a, b] → R be a measurable function with respect to
logarithmic capacity. Then there is a continuous function Φ : [a, b] → R
such that Φ′(x) = ϕ(x) a.e. on (a, b) with respect to logarithmic capacity.
Furthermore, the function Φ can be choosen such that Φ(a) = Φ(b) = 0
and |Φ(x)| ≤ ε for a prescribed ε > 0 and all x ∈ [a, b].
Then, on this basis, it was obtain the following analog of the Gehring–
Luzin theorem, see Theorem 4.1 in [69], cf. Theorem B in Section 2.
Theorem 5.3. Let ϕ : R → R be 2π-periodic, measurable and finite a.e.
with respect to logarithmic capacity. Then there is a harmonic function
u(z), z ∈ D, such that u(z) → ϕ(ϑ) as z → eiϑ along any nontangential
path for all ϑ ∈ R except a set of logarithmic capacity zero.
We call λ : ∂D → C a function of bounded variation, write
λ ∈ BV(∂D), if
Vλ(∂D) : = sup
j=k∑
j=1
|λ(ζj+1)− λ(ζj)| < ∞ (5.10)
where the supremum is taken over all finite collections of points ζj ∈ ∂D,
j = 1, . . . , k, with the cyclic order meaning that ζj lies between ζj+1 and
ζj−1 for every j = 1, . . . , k. Here we assume that ζk+1 = ζ1 = ζ0. The
quantity Vλ(∂D) is called the variation of the function λ.
Remark 5.4. Note that by the definition Vλ(∂D) = Vλ◦h(∂D), i.e., the
variation is invariant under every homeomorphism h : ∂D → ∂D and,
thus, the definition can be extended in a natural way to an arbitrary
Jordan curve in C because a Jordan curve is a continuous one-to-one
image of the unit circle in C.
It was established in [69], see Theorem 5.1, the following interesting
fact.
Theorem 5.5. Let α : ∂D → R be a function of bounded variation and
let f : D → C be an analytic function such that
lim
z→ζ
Re f(z) = α(ζ) for a.e. ζ ∈ ∂D (5.11)
V. Gutlyanskii, V. Ryazanov 189
with respect to logarithmic capacity along any nontangential path. Then
lim
z→ζ
Im f(z) = β(ζ) for a.e. ζ ∈ ∂D (5.12)
with respect to logarithmic capacity along any nontangential path where
β : ∂D → R is a function that is finite a.e. and measurable with respect
to logarithmic capacity.
Remark 5.6. Recall a subtle fact due to N. Luzin: the harmonic func-
tions in a unit disk with continuous (even absolutely continuous!) bound-
ary values can have conjugate harmonic functions whose boundary val-
ues are not continuous functions. Moreover, they are not even essentially
bounded in a neighborhood of any point of the unit circle, see e.g. [8].
Thus, the interconnection between the boundary values of conjugate har-
monic functions is a quite complicated item, see also I.E in [44].
Moreover, correspondingly to Proposition 5.1 in [69], we have the
following:
Lemma 5.7. For every function λ : ∂D → ∂D of the class BV(∂D)
there is a function αλ : ∂D → R of the class BV(∂D) such that λ(ζ) =
exp{iαλ(ζ)}, ζ ∈ ∂D.
Finally, on the basis of Theorems 5.3 and 5.5, Lemma 5.7, see also
Remark 2.3, it was derived similarly to the proof of Theorem 3.1 the
following significant result on the Hilbert boundary value problem, see
Theorems 6.1 and 8.1 in [69].
Theorem 5.8. Let λ : ∂D → ∂D be of bounded variation and ϕ : ∂D → R
be measurable with respect to logarithmic capacity. Then the space of all
analytic functions f : D → C such that along any nontangential path
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) for a.e. ζ ∈ ∂D (5.13)
with respect to logarithmic capacity has the infinite dimension.
Note that, in view of Remark 5.1, we strengthen in Theorem 5.8 in
comparison with Theorem 3.1 its hypothesis as well as its conclusion.
Thus, Theorem 5.8 is not a consequence of Theorem 3.1.
6. Angular limits in Hilbert problem for quasiconformal
functions
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. The equation of the form
fz̄ = µ(z) · fz (6.1)
190 On recent advances in boundary value problems
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 (6.1) is said
to be nondegenerate if ||µ||∞ < 1.
Note that there were recently established a great number of new
theorems on the existence and on the boundary behavior of homeo-
morphic solutions and, on this basis, on the Dirichlet problem for the
Beltrami equations with essentially unbounded distortion quotients
Kµ(z) = (1 + |µ(z)|)/(1− |µ(z)|), see e.g. papers [9–13,33,45–47,65–67]
and monographs [5,32,52] with many references therein. However, under
the study of the Hilbert boundary value problem for (6.1) we restrict
ourselves with the nondegenerate case because this research leads to a
very delicate Lusin’s problem on interconnections of the boundary data
of conjugate harmonic functions, see Remark 5.6, and to the difficult
problem on the distortion of boundary measures under mappings which
are more general than quasiconformal mappings.
Recall that homeomorphic solutions with distributional derivatives
of the nondegenerate Beltrami equations (6.1) are called quasiconfor-
mal mappings, see e.g. [3, 14] and [50]. The images of the unit disk
D = {z ∈ C : |z| < 1} under the quasiconformal mappings C onto itself
are called quasidisks and their boundaries are called quasicircles or
quasiconformal curves. It is known that every smooth (or Lipschitz)
Jordan curve is a quasiconformal curve and, at the same time, quasicon-
formal curves can be nonrectifiable as it follows from the known examples,
see e.g. the point II.8.10 in [50].
Note also that a Jordan curve generally speaking has no tangents.
Hence we need a replacement for the notion of a nontangential limit
usually applied. In this connection, recall the Bagemihl theorem in [6],
see also Theorem III.1.8 in [58], stated that, for any function Ω : D → C,
except at most countable set of points ζ ∈ ∂D, for all pairs of arcs γ1 and
γ2 in D terminating at ζ ∈ ∂D,
C(Ω, γ1) ∩ C(Ω, γ2) 6= ∅ , (6.2)
where C(Ω, γ) denotes the cluster set of Ω at ζ along γ, i.e.,
C(Ω, γ) = {w ∈ C : Ω(zn) → w, zn → ζ, zn ∈ γ} .
Immediately by the theorems of Riemann and Caratheodory, this result
is extended to an arbitrary Jordan domain D in C. Given a function
Ω : D → C and ζ ∈ ∂D, denote by P (Ω, ζ) the intersection of all cluster
sets C(Ω, γ) for arcs γ in D terminating at ζ. Later on, we call the points
of the set P (Ω, ζ) principal asymptotic values of Ω at ζ. Note that if
V. Gutlyanskii, V. Ryazanov 191
Ω has a limit along at least one arc in D terminating at a point ζ ∈ ∂D
with the property (6.2), then the principal asymptotic value is unique.
Recall that a mapping f : D → C is called discrete if the pre-image
f−1(y) consists of isolated points for every y ∈ C, and open if f maps
every open set U ⊆ D onto an open set in C.
The regular solution of a Beltrami equation (6.1) is a continuous,
discrete and open mapping f : D → C with distributional derivatives
satisfying (6.1) a.e. Note that, in the case of nondegenerate Beltrami
equations (6.1), a regular solution f belongs to class W 1,p
loc for some p > 2
and, moreover, its Jacobian Jf (z) 6= 0 for almost all z ∈ D, and it is
called a quasiconformal function, see e.g. Chapter VI in [50], or, in
the modern manner, a quasiregular mapping, see [53].
Next, considering the corresponding generalization of the Hilbert boun-
dary value problem for the Beltrami equations, we are able to formulate
the following result, see Theorem 7.1 in [69].
Theorem 6.1. Let D be a Jordan domain in C bounded by a quasicon-
formal curve, µ : D → C be a measurable (by Lebesgue) function with
||µ||∞ < 1, λ : ∂D → C, |λ(ζ)| ≡ 1 be a function of bounded variation
and let ϕ : ∂D → R be a measurable function with respect to logarithmic
capacity.
Then the Beltrami equation (6.1) has a regular solution f such that
in the sense of the unique principal asymptotic value
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) for a.e. ζ ∈ ∂D (6.3)
with respect to logarithmic capacity. If in addition ∂D is rectifiable, then
the limit in (6.3) holds a.e. with respect to the natural parameter along
any nontangential path. Finally, the space of all such solutions has the
infinite dimension.
In particular, the latter conclusion of Theorem 6.1 holds in the case
of smooth and, more generally, Lipschitz boundaries. We give here the
complete proof of Theorem 6.1 in view its significance and to demonstrate
the way of its reduction to Theorem 5.8.
Proof. Without loss of generality we may assume that 0 ∈ D and 1 ∈
∂D. Extending µ by zero everywhere outside of D, we obtain the exis-
tence of a quasiconformal mapping ω : C → C with the normalization
ω(0) = 0, ω(1) = 1 and ω(∞) = ∞ satisfying the Beltrami equation
(6.1) with the given µ, see e.g. Theorem V.B.3 in [3]. By the theorems
of Riemann and Caratheodory, the Jordan domain ω(D) can be mapped
by a conformal mapping g with the normalization g(0) = 0 and g(1) = 1
192 On recent advances in boundary value problems
onto the unit disk D. It is clear that h := g ◦ ω is a quasiconformal
homeomorphism with normalization h(0) = 0 and h(1) = 1 satisfying
the same Beltrami equation.
By the reflection principle for quasiconformal mappings, using the
conformal reflection (inversion) with respect to the unit circle in the
image and quasiconformal reflection with respect to ∂D in the preimage,
we can extend h to a quasiconformal mapping H : C → C with the
normalization H(0) = 0, H(1) = 1 and H(∞) = ∞, see e.g. I.8.4, II.8.2
and II.8.3 in [50]. Note that Λ = λ ◦ H−1 is a function of bounded
variation, VΛ(∂D) = Vλ(∂D).
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 [50].
Further, the function Φ = ϕ ◦ H−1 is measurable with respect to
logarithmic capacity. Indeed, under this mapping measurable sets with
respect 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
and compacta under continuous mappings are transformed into compacta
and compacta are measurable sets with respect to logarithmic capacity.
Thus, the Hilbert boundary value problem (6.3) for the Beltrami
equation (6.1) is reduced to the corresponding Hilbert problem for ana-
lytic functions F in the unit disk:
lim
z→ζ
Re Λ(ζ) · F (z) = Φ(ζ) (6.4)
and by Theorem 5.8 there is an analytic function F : D → C for which
this boundary condition holds for a.e. ζ ∈ ∂D with respect to logarithmic
capacity along any nontangential path.
So, the desired solution of the original Hilbert boundary value prob-
lem (6.3) for the Beltrami equation (6.1) exists and can be represented
as f = F ◦H.
Finally, since the distortion of angles under the quasiconformal map-
ping is bounded, see e.g. [1, 2] and [71], then in the case of a rectifiable
boundary of D condition (6.3) can be understood along any nontangen-
tial path a.e. with respect to the natural parameter.
V. Gutlyanskii, V. Ryazanov 193
7. Angular limits in Neumann problem for A−harmonic
functions
The Neumann problem with respect to angular limits for A-harmonic
functions in the case of boundary data that are measurable with respect to
logarithmic capacity was first solved in the paper [74]. The corresponding
partial differential equations in the divergence form below take a sig-
nificant part in many problems of mathematical physics, in particular,
in anisotropic and inhomogeneous media. These equations are closely
interconnected with Beltrami equations, see e.g. [5, 39] and [42].
In this connection, note that if f = u+ i ·v is a regular solution of the
Beltrami equation (6.1), then the function u is a continuous generalized
solution of the divergence-type equation
divA(z)∇u = 0 , (7.1)
called A−harmonic function, i.e. u ∈ C ∩W 1,1 and
∫
D
〈A(z)∇u,∇ϕ〉 = 0 ∀ ϕ ∈ C∞
0 (D) , (7.2)
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
)
. (7.3)
As we see in (7.3), 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)| , (7.4)
and, thus, they are bounded if the Beltrami equation (6.1) is not degen-
erate.
Vice verse, uniformly elliptic equations (7.1) with symmetric A(z)
and detA(z) ≡ 1 just correspond to nondegenerate Beltrami equations
(6.1) with coefficient
µ =
1
det (I +A)
(a22 − a11 − 2ia21) =
a22 − a11 − 2ia21
1 + TrA + detA
(7.5)
where I denotes identity 2× 2 matrix, TrA = a22+ a11, see e.g. theorem
16.1.6 in [5]. Following [34], call all such matrix functions A(z) of the
194 On recent advances in boundary value problems
class B. Recall that equation (7.1) is said to be uniformly elliptic, if
aij ∈ L∞ and 〈A(z)η, η〉 ≥ ε|η|2 for some ε > 0 and for all η ∈ R2.
First, similarly to the paper [65], see Section 4 above, it is proved
theorems on Poincare and Neumann problems for harmonic functions in
the unit disk D but in terms of logarithmic capacity on the unit circle
∂D, see Theorems 1 and 2 in [74]:
Theorem 7.1. Let ν : ∂D → C, |ν(ζ)| ≡ 1 be a function of bounded
variation, and let ϕ : ∂D → R be a measurable function with respect to
logarithmic capacity. Then there exist harmonic functions u : D → R
such that
lim
z→ζ
∂u
∂ν
= ϕ(ζ) (7.6)
along any nontangential paths for a.e. ζ ∈ ∂D with respect to logarithmic
capacity.
Proof. Indeed, by Theorem 6.1 in [69], see Theorem 6.1 above, there
exist analytic function f : D → C such that
lim
z→ζ
Re ν(ζ) · f(z) = lim
z→ζ
Re ν(ζ) · f(z) = ϕ(z) (7.7)
along any nontangential paths for a.e. ζ ∈ ∂D with respect to logarithmic
capacity. Note that an indefinite integral F of f in D is also an analytic
function and, correspondingly, the harmonic functions u = ReF and
v = ImF satisfy the system of Cauchy–Riemann vx = −uy and vy = ux.
Hence
f = F ′ = Fx = ux + i · vx = ux − i · uy = ∇u
where ∇u = ux + i · uy is the gradient of the function u in complex
form. Thus, (7.6) follows from (7.7), i.e. u is one of the desired harmonic
functions because its directional derivative
∂u
∂ν
= Re ν · ∇u = Re ν · ∇u = 〈ν,∇u〉
is the scalar product of ν and the gradient ∇u interpreted as vectors in
R2.
Remark 7.2. We can say more in the case Re n · ν > 0 where n = n(ζ)
is the unit interior normal with a tangent to ∂D at the point ζ ∈ ∂D. In
view of (7.6), since the limit ϕ(ζ) is finite, there is a finite limit u(ζ) of
u(z) as z → ζ in D along the straight line passing through the point ζ
V. Gutlyanskii, V. Ryazanov 195
and being parallel to the vector ν(ζ) because along this line, for z and
z0 that are close enough to ζ,
u(z) = u(z0) −
1∫
0
∂u
∂ν
(z0 + τ(z − z0)) dτ .
Thus, at each point with the condition (7.6), there is the directional
derivative
∂u
∂ν
(ζ) := lim
t→0
u(ζ + t · ν)− u(ζ)
t
= ϕ(ζ) .
In particular, in the case of the Neumann problem, we have by The-
orem 7.1 and Remark 7.2 the following significant result.
Theorem 7.3. For each function ϕ : ∂D → R that is measurable with
respect to logarithmic capacity, one can find harmonic functions u : D →
C such that, for a.e. point ζ ∈ ∂D with respect to logarithmic capacity,
there exist:
1) the finite radial limit
u(ζ) := lim
r→1
u(rζ)
2) the normal derivative
∂u
∂n
(ζ) := lim
t→0
u(ζ + t · n)− u(ζ)
t
= ϕ(ζ)
3) the nontangential limit
lim
z→ζ
∂u
∂n
(z) =
∂f
∂n
(ζ)
where n = n(ζ) denotes the unit interior normal to ∂D at the point ζ.
Then these results are extended to smooth enough Jordan domains
by reduction to the case of the unit disk, see Theorem 3 and 4 in [74].
Theorem 7.4. Let D be an almost smooth Jordan domain in the complex
plane C, ν : ∂D → C, |ν(ζ)| ≡ 1 be a function of bounded variation
and let ϕ : ∂D → R be a function that is measurable with respect to
logarithmic capacity. Then there exist harmonic functions u : D → R
such that along any nontangential paths
lim
z→ζ
∂u
∂ν
= ϕ(ζ) (7.8)
for a.e. point ζ ∈ ∂D with respect to logarithmic capacity.
196 On recent advances in boundary value problems
Recall that a Jordan domain is called almost smooth if its bound-
ary is Lipschitzian and has tangent to almost all points with respect to
logarithmic capacity.
Theorem 7.5. Let D be an almost smooth Jordan domain in the complex
plane C and let a function ϕ : ∂D → R be measurable with respect to
logarithmic capacity. Then one can find harmonic functions u : D → C
such that for a.e. ζ ∈ ∂D with respect to logarithmic capacity, there
exist:
1) the finite normal limit
u(ζ) := lim
z→ζ
u(z)
2) the normal derivative
∂u
∂n
(ζ) := lim
t→0
u(ζ + t · n) − u(ζ)
t
= ϕ(ζ)
3) the nontangential limit
lim
z→ζ
∂u
∂n
(z) =
∂u
∂n
(ζ) .
And only then the author prove the corresponding results on Poincare
and Neumann problems for A−harmonic functions, see Theorems 5 and
6 in [74]:
Theorem 7.6. Let D be an almost smooth Jordan domain in C, A(z),
z ∈ D, be a matrix function of the class B∩Cα, α ∈ (0, 1), ν : ∂D → C,
|ν(ζ)| ≡ 1, be a function of bounded variation and let a function ϕ :
∂D → R be measurable with respect to logarithmic capacity. Then there
exist A-harmonic functions u : D → R of the class C1+α such that
lim
z→ζ
∂u
∂ν
(z) = ϕ(ζ) (7.9)
along any nontangential paths for a.e. ζ ∈ ∂D with respect to logarithmic
capacity.
Note that the last and the next theorem are based on the well–known
fact that the homeomorphic solutions of the Beltrami equations with
complex coefficients in the class Cα, α ∈ (0, 1), belong to the class C1+α,
see e.g. [37] and [38].
V. Gutlyanskii, V. Ryazanov 197
Theorem 7.7. Let D be an almost smooth Jordan domain in C, the
interior unit normal n = n(ζ) to ∂D has bounded variation, A(z), z ∈
D, be a matrix function of class B ∩ Cα, α ∈ (0, 1) and let a function
ϕ : ∂D → R be measurable with respect to logarithmic capacity. Then
there exist A-harmonic function u : D → R of class C1+α such that for
a.e. ζ ∈ ∂D with respect to logarithmic capacity there exist:
1) the finite normal limit
u(ζ) := lim
z→ζ
u(z)
2) the normal derivative
∂u
∂n
(ζ) := lim
t→0
u(ζ + t · n) − u(ζ)
t
= ϕ(ζ)
3) the nontangential limit
lim
z→ζ
∂u
∂n
(z) =
∂u
∂n
(ζ) .
Finally, it was established by Theorem 7 in [74] that all the spaces of
solutions in Theorems 7.1–7.7 have the infinite dimension.
8. Tangent limits in boundary value problems for analytic
functions
In this section, we demonstrate an alternative approach making pos-
sible to obtain new results with tangent limits in the Hilbert boundary
value problem, see [62]. In particular, it is shown that the spaces of
the found solutions have the infinite dimension for prescribed collections
of Jordan arcs terminating in almost every boundary point. Moreover,
similar results are proved for the Riemann boundary value problem.
Jordan arcs {Jζ}ζ∈C is of class BS (of Bagemihl–Seidel class),
cf. [7], 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
is generated 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
198 On recent advances in boundary value problems
in R onto Jordan curves giving the corresponding isotopy of C to other
boundary component of R. We may also to choose in R a curve which is
tangent to its boundary components and which intersects every centered
circle in R only one time and to obtain the rest lines by its rotation.
Finally, 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.
In these terms, it is easy to prove the following results on the Hilbert
boundary value problem, see Theorem 1 and Remark 1 in [62].
Theorem 8.1. Let D be a bounded domain in C whose boundary consists
of a finite number of mutually disjoint rectifiable Jordan curves, and let
λ : ∂D → C, |λ(ζ)| ≡ 1, ϕ : ∂D → R and ψ : ∂D → R 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 exist single-valued analytic functions f : D → C such that
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) , (8.1)
lim
z→ζ
Im {λ(ζ) · f(z)} = ψ(ζ) (8.2)
along γζ for a.e. ζ ∈ ∂D with respect to the natural parameter.
Remark 8.2. Thus, the space of all solutions f of the Hilbert problem
(8.1) in the given sense has the infinite dimension for any prescribed ϕ, λ
and {γζ}ζ∈D because the space of all measurable functions ψ : ∂D → R
has the infinite dimension.
Proof. Indeed, set Ψ(ζ) = ϕ(ζ) + iψ(ζ) and Φ(ζ) = λ(ζ) · Ψ(ζ) for all
ζ ∈ ∂D. Then by Theorem 2 in [7] there is a single-valued analytic
function f such that
lim
z→ζ
f(z) = Φ(ζ) (8.3)
along γζ for a.e. ζ ∈ ∂D with respect to the natural parameter. Then
also
lim
z→ζ
λ(ζ) · f(z) = Ψ(ζ) (8.4)
along γζ for a.e. ζ ∈ ∂D with respect to the natural parameter.
Similar results can be formulated for arbitrary Jordan domains with
respect to the harmonic measure, see Theorem 2 and Remark 2 in [62].
V. Gutlyanskii, V. Ryazanov 199
Theorem 8.3. 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 harmonic measure. Suppose that {γζ}ζ∈∂D is a family
of Jordan arcs of class BS in D.
Then there exist single-valued analytic functions f : D → C such that
lim
z→ζ
Re {λ(ζ) · f(z)} = ϕ(ζ) , (8.5)
lim
z→ζ
Im {λ(ζ) · f(z)} = ψ(ζ) (8.6)
along γζ for a.e. ζ ∈ ∂D with respect to the harmonic measure.
Remark 8.4. Again, the space of all solutions f of the Riemann–Hilbert
problem (8.5) in the given sense has the infinite dimension for any pre-
scribed ϕ, λ and {γζ}ζ∈D because the space of all functions ψ : ∂D → R
that are measurable with respect to the harmonic measure has the infinite
dimension.
Proof. Theorem 8.3 is reduced to Theorem 8.1 in the following way.
First, there is a conformal mapping ω of D onto a circular domain
D∗ whose boundary consists of a finite number of circles and points, see
e.g. Theorem V.6.2 in [28]. Note that D∗ cannot be degenerate because
isolated singularities of conformal mappings are removable that is due
to the well-known Weierstrass theorem, see e.g. Theorem 1.2 in [17].
Applying in the case of need the inversion with respect to a boundary
circle of D∗, we may assume that D∗ is bounded.
Remark that ω is extended to a homeomorphism ω∗ of D onto D∗,
see e.g. point (i) of Lemma 3.1 in [64]. Set Λ = λ ◦ Ω, Φ = ϕ ◦ Ω and
Ψ = ψ ◦ Ω where Ω : ∂D∗ → ∂D is the restriction of Ω∗ := ω−1
∗ to
∂D∗. Let us show that these functions are measurable with respect to
the natural parameter on ∂D∗.
For this goal, note first of all that the sets of the harmonic measure
zero are invariant under conformal mappings between multiply connected
Jordan domains because a composition of a harmonic function with a con-
formal mapping is again a harmonic function. Moreover, a set E ⊂ ∂D∗
has the harmonic measure zero if and only if it has the length zero, say
in view of the integral representation of the harmonic measure through
the Green function of the domain D∗, see e.g. Section II.4 in [57].
Hence Ω and Ω−1 transform measurable sets into measurable sets
because every measurable set is the union of a sigma-compact set and
a set of measure zero, see e.g. Theorem III(6.6) in [70], and continuous
mappings transform compact sets into compact sets. Thus, the functions
200 On recent advances in boundary value problems
λ, ϕ and ψ are measurable with respect to the harmonic measure on ∂D
if and only if the functions Λ, Φ and Ψ are measurable with respect to
the natural parameter on ∂D∗.
Then by Theorem 8.1 there exist single-valued analytic functions F :
D → C such that
lim
w→ξ
Re {Λ(ξ) · F (w)} = Φ(ξ) , (8.7)
lim
w→ξ
Im {Λ(ξ) · F (w)} = Ψ(ξ) (8.8)
along Γξ = ω(γΩ(ξ)) for a.e. ξ ∈ ∂D∗ with respect to the natural param-
eter.
Thus, by the construction the functions f = F ◦ ω are the desired
analytic functions f : D → C satisfying the boundary conditions (8.5)
and (8.6) along γζ for a.e. ζ ∈ ∂D with respect to the harmonic measure.
Remark 8.5. Many investigations were devoted to the nonlinear Hilbert
(Riemann–Hilbert) boundary value problems with conditions of the type
Φ( ζ, f(ζ) ) = 0 ∀ ζ ∈ ∂D , (8.9)
see e.g. [20, 43] and [73]. It is natural also to weaken such conditions to
Φ( ζ, f(ζ) ) = 0 for a.e. ζ ∈ ∂D . (8.10)
It is easy to see that the proposed approach makes possible also to reduce
such problems to the algebraic and measurable solvability of the relation
Φ( ζ, v ) = 0 (8.11)
with respect to a complex-valued function v(ζ), cf. e.g. [30].
Through suitable modifications of Φ under the corresponding map-
pings of Jordan boundary curves onto the unit circle S = {ζ ∈ C : |ζ| =
1}, we may assume that ζ belongs to S.
The following results on the Riemann boundary value problem can be
found as Theorem 3 and Lemma 1 in [62].
Theorem 8.6. Let D be a domain in C whose boundary consists of a
finite number of mutually disjoint rectifiable Jordan curves, A : ∂D → C
and B : ∂D → C be measurable functions with respect to the natural
parameter. 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.
V. Gutlyanskii, V. Ryazanov 201
Then there exist single-valued analytic functions f+ : D → C and
f− : C \ D → C that satisfy (1.7) for a.e. ζ ∈ ∂D with respect to the
natural parameter 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 8.6 is a special case of the following lemma on the generalized
Riemann problem with shifts that can be useful for other goals, too.
Lemma 8.7. Under the hypotheses of Theorem 8.6, 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 natural
parameter.
Then there exist single-valued analytic functions f+ : D → C and
f− : C \ D → C that satisfy (1.8) for a.e. ζ ∈ ∂D with respect to the
natural parameter 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, let D be bounded and let g− : ∂D → C be a measurable
function. Note that the function
g+ := {A · g− +B} ◦ α−1 (8.12)
is measurable. Indeed, E := {A · g− + B}−1(Ω) is a measurable subset
of ∂D for every open set Ω ⊆ C because the function A · g− + B is
measurable by the hypotheses. Hence the set E is the union of a sigma-
compact set and a set of measure zero, see e.g. Theorem III(6.6) in [70].
However, continuous mappings transform compact sets into compact sets
and, thus, α(E) = α ◦ {A · g− + B}−1(Ω) = (g+)−1(Ω) is a measurable
set, i.e. the function g+ is really measurable.
Then by Theorem 2 in [7] there is a single-valued analytic function
f+ : D → C such that
lim
z→ξ
f+(z) = g+(ξ) (8.13)
along γ+ξ for a.e. ξ ∈ ∂D with respect to the natural parameter. 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.
202 On recent advances in boundary value problems
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 2 in [7] 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
, (8.14)
along γ−ζ for a.e. ζ ∈ ∂Dl with respect to the natural parameter.
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 2 in [7] there is a single-valued analytic function f∗ :
D∗ → C such that
lim
w→ξ
f∗(w) = g∗(ξ) (8.15)
along γ∗ξ for a.e. ξ ∈ ∂D∗ with respect to the natural parameter. Note
that f−0 := g∗ ◦ j is a single-valued analytic function in D0 and by con-
struction
lim
z→ζ
f−0 (z) = g−0 (ζ) , g−0 := g−|∂D0 , (8.16)
along γ−ζ for a.e. ζ ∈ ∂D0 with respect to the natural parameter.
Thus, the functions f−l , l = 0, 1, . . . ,m, form an analytic function
f− : C \ D → C satisfying (1.8) for a.e. ζ ∈ ∂D with respect to the
natural parameter.
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 8.8. Some investigations were devoted also to the nonlinear
Riemann problems with boundary conditions of the form
Φ( ζ, f+(ζ), f−(ζ) ) = 0 ∀ ζ ∈ ∂D . (8.17)
It is natural as above to weaken such conditions to the following
Φ( ζ, f+(ζ), f−(ζ) ) = 0 for a.e. ζ ∈ ∂D . (8.18)
V. Gutlyanskii, V. Ryazanov 203
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 (8.19)
with respect to complex-valued functions v(ζ) and w(ζ), cf. e.g. [30].
Through suitable modifications of Φ under the corresponding map-
pings of Jordan boundary curves onto the unit circle S = {ζ ∈ C : |ζ| =
1}, we may assume that ζ belongs to S.
Example 8.9. 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 (8.20)
in the unit disk D = {z ∈ C : |z| < 1} are always solved if the function
ϕ : S × C → C satisfies the Caratheodory conditions: ϕ(ζ, w) is
continuous in the variable w ∈ C for a.e. ζ ∈ S and it is measurable in
the variable ζ ∈ S for all w ∈ C.
Furthermore, the spaces of solutions of such problems always have
the infinite dimension. Indeed, the function ϕ(ζ, ψ(ζ)) is measurable in
ζ ∈ S for every measurable function ψ : S → C if the function ϕ satisfies
the Caratheodory conditions, see e.g. Section 17.1 in [48], and the space
of all measurable functions ψ : S → C has the infinite dimension.
Problems. Finally, it is necessary to point out the open problems on
solvability of Hilbert and Riemann problems along any prescribed families
of arcs terminating in the boundary but not only along families of the
Bagemihl–Seidel class and, more generally, along any prescribed families
of paths to a.e. boundary point.
9. Tangent limits in boundary value problems for quasi-
conformal functions
Here we restrict ourselves by a history survey and short comments
of results in [34] extending the approach from the last section to the
boundary value problems for the Beltrami equations, see (6.1).
The first relevant problem is the measurement of sets on boundaries of
domains. Recall 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 quasiconformal mappings as it follows from
the famous Ahlfors-Beurling example of quasisymmetric mappings of the
real axis that are not absolutely continuous, see [4]. Hence we are forced
204 On recent advances in boundary value problems
to apply here instead of them the so-called absolute harmonic measure
by Nevanlinna, in other words, logarithmic capacity, see e.g. [57], or Sec-
tion 5, whose zero sets are invariant under quasiconformal mappings, see
comments after (5.3)–(5.4).
By the well–known Priwalow uniqueness theorem analytic functions
in the unit disk D = {z ∈ C : |z| < 1} coincide if they have the equal
boundary values along all nontangential paths to a set E of points in ∂D
of a positive length, see e.g. Theorem IV.2.5 in [60]. The theorem is valid
also for analytic functions in Jordan domains with rectifiable boundaries,
see e.g. Section IV.2.6 in [60].
However, examples of Luzin 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 [60].
Simultaneously, by Theorem IV.6.2 in [60] of Luzin and Priwalow the
uniqueness result is valid if E is of the second category. Theorem 1 in [7]
demonstrates that the latter condition is necessary.
Theorem 1 in [7] can be formulated in the following way, see footnote
9 there. For the definition of class BS, see Section 8.
Theorem D. Let D be a bounded domain in C whose boundary
consists 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 class 1 on M . Then there is a nonconstant
single-valued analytic function f : D → C such that, for all ζ ∈ M ,
along γζ
lim
z→ζ
f(z) = Φ(ζ) . (9.1)
Recall Baire’s terminology for categories of sets and functions. Na-
mely, 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 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.
On the basis of Theorem D, in the case of domains D whose bound-
aries consist of rectifiable Jordan curves, it was formulated Theorem 2
in [7] on the existence of analytic functions f : D → C such that (9.1)
holds a.e. on ∂D with respect to the natural parameter for each pre-
scribed measurable function Φ : ∂D → C.
The following statement is similar to Theorem 2 in [7] but formulated
in terms of logarithmic capacity instead of the natural parameter, see
V. Gutlyanskii, V. Ryazanov 205
Theorem 1 in [34]. It is necessary to stress that this theorem does not
assume that boundary Jordan curves are rectifiable and that this fact is
key.
Theorem 9.1. 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 (9.1) holds along γζ for a.e. ζ ∈ ∂D with respect to the
logarithmic capacity.
This result has made possible to solve the corresponding Dirichlet,
Hilbert, Riemann, Neumann, Poincare and mixed boundary value prob-
lems for analytic functions as well as for quasiconformal mappings with
an arbitrary prescribed complex dilatation µ, see Theorems 2–10 in [34].
Hence we give here only its proof.
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 [22],
correspondingly. Hence the classic Luzin theorem holds for the logarith-
mic capacity on C, see e.g. Theorem 2.3.5 in [22].
By the Luzin 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
206 On recent advances in boundary value problems
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 9.1 follows from Theorem D.
We give, for example, only one theorem on the directional derivative
problem for the Beltrami equation, see Theorem 7 in [34], because, firstly,
many of the rest theorems are similar to the corresponding theorems in
Section 8 and, secondly, they can be found in the given paper.
Theorem 9.2. 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 (6.1) has a regular solution f : D → C of
the class C1+α such that
lim
z→ζ
∂f
∂ν
(z) = Φ(ζ) (9.2)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
10. Tangent limits in boundary value problems for
A-harmonic functions
Here we give short comments to results in [34] extending the approach
from the last two sections to the boundary value problems for the Laplace
equation and its generalizations corresponding to problems of mathemat-
ical physics in inhomogeneous and anisotropic media. For simplicity, we
first give the corresponding results for harmonic functions that are simple
consequences of Theorem 9.1.
Corollary 10.1. Let D be a bounded domain in C whose boundary con-
sists of a finite number of mutually disjoint Jordan curves and let a
function ϕ : ∂D → R be measurable with respect to the logarithmic ca-
pacity.
V. Gutlyanskii, V. Ryazanov 207
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) (10.1)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Corollary 10.2. 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) (10.2)
along γζ for a.e. ζ ∈ ∂D with respect to the logarithmic capacity.
Here we use the complex writing for the gradient ∇u := ux + i · uy.
Proof. Indeed, by Theorem 9.1 there is a single-valued analytic function
f : D → R such that
lim
z→ζ
f(z) = Φ(ζ) (10.3)
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, (10.2) follows from (10.3) and, consequently, u is the desired func-
tion.
The following statement on the directional derivative problem for
A−harmonic functions can be derived from Theorem 9.2, see Corollary 7
in [34].
Corollary 10.3. 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) (10.4)
208 On recent advances in boundary value problems
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.
We refer the reader to the paper [34] for a great number of other
consequences on boundary value problems for A−harmonic functions.
Finally, we recall that it was recently obtained a great number of the
existence theorems for the degenerate Beltrami equations, see e.g. the
papers [9–11, 15, 41, 67, 68] and the monographs [32, 40, 52]. The authors
conjecture that the main part of results on boundary value problems can
be extended to the corresponding degenerate equations but in terms of the
so-called logarithmic measure that is a more refined measurement than
logarithmic capacity, see e.g. [57]. Moreover, they should have suitable
nonlinear and spatial analogues.
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Contact information
Vladimir
Gutlyanskii,
Vladimir Ryazanov
Institute of Applied Mathematics
and Mechanics,
National Academy of Sciences of Ukraine,
Ukraine, Slavyansk
E-Mail: vladimirgut@mail.ru,
vl ryazanov@mail.ru
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