Dirichlet problem for Poisson equations in Jordan domains
We study the Dirichlet problem for the Poisson equations △u(z) = g(z) with g ∈ Lp, p > 1, and continuous boundary data φ : ∂D → ℝ in arbitrary Jordan domains D in ℂ and prove the existence of continuous solutions u of the problem. Мы изучаем задачу Дирихле для уравнений Пуассона △u(z) = g(z) с g...
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| Cite this: | Dirichlet problem for Poisson equations in Jordan domains / V. Gutlyanskii, V. Ryazanov, E. Yakubov // Праці Інституту прикладної математики і механіки НАН України. — Слов’янськ: ІПММ НАН України, 2018. — Т. 32. — С. 30-41. — Бібліогр.: 27 назв. — англ. |
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Gutlyanskii, V. Ryazanov, V. Yakubov, E. 2020-06-06T11:48:29Z 2020-06-06T11:48:29Z 2018 Dirichlet problem for Poisson equations in Jordan domains / V. Gutlyanskii, V. Ryazanov, E. Yakubov // Праці Інституту прикладної математики і механіки НАН України. — Слов’янськ: ІПММ НАН України, 2018. — Т. 32. — С. 30-41. — Бібліогр.: 27 назв. — англ. DOI: 10.37069/1683-4720-2018-32-4 1683-4720 MSC: Primary 30C62,31A05, 31A20, 31A25, 31B25, 35J61. Secondary 30E25, 31C05, 34M50, 35F45, 35Q15. https://nasplib.isofts.kiev.ua/handle/123456789/169122 517.5 We study the Dirichlet problem for the Poisson equations △u(z) = g(z) with g ∈ Lp, p > 1, and continuous boundary data φ : ∂D → ℝ in arbitrary Jordan domains D in ℂ and prove the existence of continuous solutions u of the problem. Мы изучаем задачу Дирихле для уравнений Пуассона △u(z) = g(z) с g ∈ Lp, p > 1, и непрерывными граничными данными φ : ∂D → ℝ в произвольных жордановых областях D ⊂ ℂ и доказываем существование непрерывных решений u этой задачи. Ми вивчаємо задачу Дiрихле для рiвнянь Пуасона △u(z) = g(z) с g ∈ Lp, p > 1, та неперервними граничними даними φ : ∂D → ℝ в довiльних жорданових областях D ⊂ ℂ та доводимо iснування неперервних рiшень u цiєї задачi en Інститут прикладної математики і механіки НАН України Праці Інституту прикладної математики і механіки НАН України Dirichlet problem for Poisson equations in Jordan domains Задача Дирихле для уравнений Пуассона в жордановых областях Задача Дiрихле для рiвнянь Пуасона у жорданових областях Article published earlier |
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Dirichlet problem for Poisson equations in Jordan domains |
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Dirichlet problem for Poisson equations in Jordan domains Gutlyanskii, V. Ryazanov, V. Yakubov, E. |
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Dirichlet problem for Poisson equations in Jordan domains |
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Dirichlet problem for Poisson equations in Jordan domains |
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Dirichlet problem for Poisson equations in Jordan domains |
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Dirichlet problem for Poisson equations in Jordan domains |
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dirichlet problem for poisson equations in jordan domains |
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Gutlyanskii, V. Ryazanov, V. Yakubov, E. |
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Gutlyanskii, V. Ryazanov, V. Yakubov, E. |
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Праці Інституту прикладної математики і механіки НАН України |
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Задача Дирихле для уравнений Пуассона в жордановых областях Задача Дiрихле для рiвнянь Пуасона у жорданових областях |
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We study the Dirichlet problem for the Poisson equations △u(z) = g(z) with g ∈ Lp, p > 1, and continuous boundary data φ : ∂D → ℝ in arbitrary Jordan domains D in ℂ and prove the existence of continuous solutions u of the problem.
Мы изучаем задачу Дирихле для уравнений Пуассона △u(z) = g(z) с g ∈ Lp, p > 1, и непрерывными граничными данными φ : ∂D → ℝ в произвольных жордановых областях D ⊂ ℂ и доказываем существование непрерывных решений u этой задачи.
Ми вивчаємо задачу Дiрихле для рiвнянь Пуасона △u(z) = g(z) с g ∈ Lp, p > 1, та неперервними граничними даними φ : ∂D → ℝ в довiльних жорданових областях D ⊂ ℂ та доводимо iснування неперервних рiшень u цiєї задачi
|
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DOI: 10.37069/1683-4720-2018-32-4 |
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1683-4720 |
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https://nasplib.isofts.kiev.ua/handle/123456789/169122 |
| citation_txt |
Dirichlet problem for Poisson equations in Jordan domains / V. Gutlyanskii, V. Ryazanov, E. Yakubov // Праці Інституту прикладної математики і механіки НАН України. — Слов’янськ: ІПММ НАН України, 2018. — Т. 32. — С. 30-41. — Бібліогр.: 27 назв. — англ. |
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2025-11-26T13:28:23Z |
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2025-11-26T13:28:23Z |
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| fulltext |
ISSN 1683-4720 Працi IПММ НАН України. 2018. Том 32
UDC 517.5
DOI: 10.37069/1683-4720-2018-32-4
c⃝2018. V. Gutlyanskĭı, V. Ryazanov, E. Yakubov
DIRICHLET PROBLEM FOR POISSON EQUATIONS
IN JORDAN DOMAINS
First, we study the Dirichlet problem for the Poisson equations △u(z) = g(z) with g ∈ Lp, p > 1, and
continuous boundary data φ : ∂D → R in arbitrary Jordan domains D in C and prove the existence
of continuous solutions u of the problem in the class W 2,p
loc . Moreover, u ∈ W 1,q
loc for some q > 2 and u
is locally Hölder continuous. Furthermore, u ∈ C1,α
loc with α = (p − 2)/p if p > 2. Then, on this basis
and applying the Leray–Schauder approach, we obtain the similar results for the Dirichlet problem
with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form
△u(z) = h(z) · f(u(z)) with the same assumptions on h as for g above and continuous functions
f : R → R, either bounded or with nondecreasing |f | of |t| such that f(t)/t → 0 as t → ∞. We
also give here applications to mathematical physics that are relevant to problems of diffusion with
absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson
equations in the unit disk D ⊂ C with arbitrary boundary data φ : ∂D → R that are measurable with
respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions u
of the problem in terms of the angular limits in D a.e. on ∂D with respect to logarithmic capacity with
the same local properties as above. Finally, we extend these results to almost smooth Jordan domains
with qusihyperbolic boundary condition by Gehring–Martio.
MSC: Primary 30C62,31A05, 31A20, 31A25, 31B25, 35J61. Secondary 30E25, 31C05, 34M50, 35F45,
35Q15.
Keywords: Dirichlet problem, quasilinear Poisson equation, logarithmic potential, logarithmic capacity,
angular limits.
1. Introduction.
First of all, recall that the Poisson kernel is the 2π−periodic function
Pr(Θ) :=
1− r2
1− 2r cosΘ + r2
, r < 1 , Θ ∈ R . (1)
Here we will apply the notation of the Poisson integral in the unit disk D :
Pφ(z) :=
1
2π
π∫
−π
Pr(ϑ− t) φ(eit) dt , z = reiϑ, r < 1 , ϑ ∈ R (2)
for arbitrary continuous functions φ : ∂D → R. As known, Pφ is a harmonic function
in D that is extended by continuity to D with φ as its boundary data, see e.g. I.D.2
in [18].
Similarly, given a Jordan domain D in C and a continuous boundary function
φ : ∂D → R, let us denote by Dφ the harmonic function in D that has the continuous
extension to D with φ as its boundary data. As known, by the Lindelöf maximum
principle, see e.g. Lemma 1.1 in [10], we have the uniqueness theorem for the bounded
30
Dirichlet problem for Poisson equations in Jordan domains
harmonic functions with continuous boundary data. By the Riemann theorem, see e.g.
Theorem II.2.1 in [14], there is a conformal mapping f : D → D that is extended to
a homeomorphism f̃ : D → D by the Caratheodory theorem, see e.g. Theorem II.3.4
in [14]. Thus, the Dirichlet operator Dφ has the following useful representation
Dφ(z) = Pφ◦f−1
∗
(f(z)), z ∈ D, where f∗ = f̃ |∂D . (3)
It is also known, see e.g. Corollary 1 in [16], that the Newtonian potential
Ng(z) :=
1
2π
∫
C
log |z − w| g(w) dm(w) (4)
of integrable functions g : C → R with compact support satisfies the Poisson equation
△Ng = g (5)
in the distributional sense, i.e.,∫
C
Ng(z)△ψ(z) dm(z) =
∫
C
ψ(z) g(z) dm(z) ∀ ψ ∈ C∞
0 (C) . (6)
As usual, here C∞
0 (C) denotes the class of all infinitely differentiable functions
ψ : C → R with compact support in C, △ = ∂2
∂x2
+ ∂2
∂y2
is the Laplace operator and
dm(z) corresponds to the Lebesgue measure in C.
2. Dirichlet problem with continuous data.
By Theorem 2 in [16] we come to the following result on the existence, regularity and
representation of solutions for the Dirichlet problem to the Poisson equation in arbitrary
Jordan domains D in C where we assume that the charge density g is extended by zero
outside of D.
Theorem 1. Let D be a Jordan domain in C, φ : ∂D → R be a continuous function
and g : D → R belong to the class Lp(D) for p > 1. Then the function
U := Ng − DN∗
g
+ Dφ , N∗
g := Ng|∂D , (7)
is continuous in D with U |∂D = φ, belongs to the class W 2,p
loc (D) and satisfies the
Poisson equation △U = g a.e. in D. Moreover, U ∈ W 1,q
loc (D) for some q > 2 and U
is locally Hölder continuous in D. Furthermore, U ∈ C1,α
loc (D) with α = (p − 2)/p if
g ∈ Lp(D) for p > 2.
Remark 1. Note also by the way that a generalized solution of the Dirichlet
problem to the Poisson equation in the class C(D) ∩W 1,2
loc (D) is unique at all, see e.g.
Theorem 8.30 in [13], and (7) gives the effective representation of this unique solution.
The case of quasilinear Poisson equations is reduced to the case of the linear Poisson
equations by the Leray–Schauder approach.
31
V. Gutlyanskĭı, V. Ryazanov, E. Yakubov
Theorem 2. Let D be a Jordan domain in C, φ : ∂D → R be a continuous function
and h : D → R be a function in the class Lp(D) for p > 1. Suppose that a continuous
function f : R → R has nondecreasing |f | of |t| and
lim
t→+∞
f(t)
t
= 0 . (8)
Then there is a continuous function U : D → R with U |∂D = φ, U |D ∈W 2,p
loc such that
△U(z) = h(z) · f(U(z)) for a.e. z ∈ D . (9)
Moreover, U ∈W 1,q
loc (D) for some q > 2 and U is locally Hölder continuous. Furthermore,
U ∈ C1,α
loc (D) with α = (p− 2)/p if p > 2.
In particular, the latter statement in Theorem 2 implies that U ∈ C1,α
loc (D) for all
α = (0, 1) if h is bounded.
Proof. If ∥h∥p = 0 or ∥f∥C = 0, then the Dirichlet operator Dφ gives the desired
solution of the Dirichlet problem for equation (9), see e.g. I.D.2 in [18]. Hence we may
assume further that ∥h∥p ̸= 0 and ∥f∥C ̸= 0.
By Theorem 1 and the maximum principle for harmonic functions, we obtain the
family of operators F (g; τ) : Lp(D) → Lp(D), τ ∈ [0, 1]:
F (g; τ) := τh · f(Ng −DN∗
g
+Dφ) , N
∗
g := Ng|∂D , ∀ τ ∈ [0, 1] (10)
which satisfies all groups of hypothesis H1-H3 of Theorem 1 in [22].
H1). First of all, F (g; τ) ∈ Lp(D) for all τ ∈ [0, 1] and g ∈ Lp(D) because by
Theorem 1 f(Ng −DN∗
g
+ Dφ) is a continuous function and, moreover, by Theorem 1
in [16]
∥F (g; τ)∥p ≤ ∥h∥p |f ( 2M ∥g∥p + ∥φ∥C) | < ∞ ∀ τ ∈ [0, 1] .
Thus, by Theorem 1 in combination with the Arzela–Ascoli theorem, see e.g. Theorem
IV.6.7 in [6], the operators F (g; τ) are completely continuous for each τ ∈ [0, 1] and
even uniformly continuous with respect to the parameter τ ∈ [0, 1].
H2). The index of the operator F (g; 0) is obviously equal to 1.
H3). By Theorem 1 in [16] and the maximum principle for harmonic functions, we
have the estimate for solutions g ∈ Lp of the equations g = F (g; τ):
∥g∥p ≤ ∥h∥p |f ( 2M ∥g∥p + ∥φ∥C) | ≤ ∥h∥p |f( 3M ∥g∥p)|
whenever ∥g∥p ≥ ∥φ∥C/M , i.e. then it should be
|f( 3M ∥g∥p)|
3M ∥g∥p
≥ 1
3M ∥h∥p
(11)
and hence ∥g∥p should be bounded in view of condition (8).
32
Dirichlet problem for Poisson equations in Jordan domains
Thus, by Theorem 1 in [22] there is a function g ∈ Lp(D) such that g = F (g; 1) and,
consequently, by our Theorem 1 the function U := Ng − DN∗
g
+ Dφ gives the desired
solution of the Dirichlet problem for the quasilinear Poisson equation (9). �
Remark 2. As it is clear from the proof, Theorem 2 is valid if f is an arbitrary
continuous bounded function. Moreover, condition (8) can be replaced by the weaker
lim sup
t→+∞
|f(t)|
t
<
1
3M∥h∥p
(12)
where M is the constant from the estimate (14) of Theorem 1 in [16].
Theorem 2 together with Remark 2 can be applied to some physical problems. The
first circle of such applications is relevant to reaction-diffusion problems. Problems of
this type are discussed in [5], p. 4, and, in detail, in [2]. A nonlinear system is obtained
for the density u and the temperature T of the reactant. Upon eliminating T the system
can be reduced to the equation
△u = λ · f(u) (13)
with h(z) ≡ λ > 0 and, for isothermal reactions, f(u) = uq where q > 0 is called the
order of the reaction. It turns out that the density of the reactant u may be zero in a
subdomain called a dead core. A particularization of results in Chapter 1 of [5] shows
that a dead core may exist just if and only if 0 < q < 1 and λ is large enough, see also
the corresponding examples in [15]. In this connection, the following statements may
be of independent interest.
Corollary 1. Let D be a Jordan domain in C, φ : ∂D → R be a continuous
function and let h : D → R be a function in the class Lp(D), p > 1. Then there exists
a continuous function u : D → R with u|∂D = φ such that u ∈W 2,p
loc (D) and
△u(z) = h(z) · uq(z) , 0 < q < 1 (14)
a.e. in D. Moreover, u ∈ W 1,β
loc (D) for some β > 2 and u is locally Hölder continuous
in D. Furthermore, u ∈ C1,α
loc (D) with α = (p− 2)/p if p > 2.
Corollary 2. Let D be a Jordan domain in C and φ : ∂D → R be a continuous
function. Then there is a continuous function u : D → R with u|∂D = φ such that
u ∈W 2,p
loc (D) for all p ≥ 1 and
△u(z) = uq(z) , 0 < q < 1 , (15)
a.e. in D. Moreover, u ∈ C1,α
loc (D) for all α ∈ (0, 1).
Note also that certain mathematical models of a thermal evolution of a heated
plasma lead to nonlinear equations of the type (13). Indeed, it is known that some of
them have the form △ψ(u) = f(u) with ψ′(0) = +∞ and ψ′(u) > 0 if u ̸= 0 as, for
33
V. Gutlyanskĭı, V. Ryazanov, E. Yakubov
instance, ψ(u) = |u|q−1u under 0 < q < 1, see e.g. [5]. With the replacement of the
function U = ψ(u) = |u|q · signu, we have that u = |U |Q · signU , Q = 1/q, and, with
the choice f(u) = |u|q2 · signu, we come to the equation △U = |U |q · signU = ψ(U).
Corollary 3. Let D be a Jordan domain in C and φ : ∂D → R be a continuous
function. Then there is a continuous function U : D → R with U |∂D = φ such that
u ∈W 2,p
loc (D) for all p ≥ 1 and
△U(z) = |U(z)|q−1U(z) , 0 < q < 1 , (16)
a.e. in D. Moreover, U ∈ C1,α
loc (D) for all α ∈ (0, 1).
Finally, we recall that in the combustion theory, see e.g. [3], [24] and the references
therein, the following model equation
∂u(z, t)
∂t
=
1
δ
· △u + eu , t ≥ 0, z ∈ D, (17)
takes a special place. Here u ≥ 0 is the temperature of the medium and δ is a certain
positive parameter.
We restrict ourselves here by the stationary case, although our approach makes
it possible to study the parabolic equation (17), see [15]. Namely, the equation (9) is
appeared here with h ≡ δ > 0 and the function f(u) = e−u that is bounded as in
Remark 2.
Corollary 4. Let D be a Jordan domain in C and φ : ∂D → R be a continuous
function. Then there is a continuous function U : D → R with U |∂D = φ such that
u ∈W 2,p
loc (D) for all p ≥ 1 and
△U(z) = δ · e−U(z) , δ > 0 , (18)
a.e. in D. Moreover, U ∈ C1,α
loc (D) for all α ∈ (0, 1).
Due to the factorization theorem in [15], we plan to extend these results to semi–
linear equations describing the corresponding physical phenomena in anisotropic and
inhomogeneous media in arbitrary Jordan domains.
3. The definition and preliminary remarks on the logarithmic capacity.
Given a bounded Borel set E in the plane C, a mass distribution on E is a
nonnegative completely additive function ν of a set defined on its Borel subsets with
ν(E) = 1. The function
Uν(z) :=
∫
E
log
∣∣∣∣ 1
z − ζ
∣∣∣∣ dν(ζ) (19)
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) . (20)
34
Dirichlet problem for Poisson equations in Jordan domains
It is also well-known the following geometric characterization of the logarithmic
capacity, see e.g. the point 110 in [23]:
C(E) = τ(E) := lim
n→∞
V
2
n(n−1)
n (21)
where Vn denotes the supremum of the product
V (z1, . . . , zn) =
l=1,...,n∏
k<l
|zk − zl| (22)
taken over all collections of points z1, . . . , zn in the set E. Following Fékete, see [9], the
quantity τ(E) is called the transfinite diameter of the set E.
Remark 3. Thus, we 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 quasiconformal mappings
on compact sets, see e.g. Theorem II.4.3 in [21].
In order to introduce sets that are measurable with respect to logarithmic capacity,
we define, following [7], inner C∗ and outer C∗ capacities :
C∗(E) : = sup
F⊆E
C(E), C∗(E) : = inf
E⊆O
C(O) (23)
where supremum is taken over all compact sets F ⊂ C and infimum is taken over
all open sets O ⊂ C. A set E ⊂ C is called measurable with respect to the
logarithmic capacity if C∗(E) = C∗(E), and the common value of C∗(E) and C∗(E)
is still denoted by C(E).
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} (24)
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 Nevanlinna, see Chapter V in [23].
Hence a set E is of (Hausdorff) length zero if C(E) = 0, see Theorem V.6.2 in [23].
However, there exist sets of length zero having a positive logarithmic capacity, see e.g.
Theorem IV.5 in [7].
Remark 4. 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 [7]. Moreover, as it follows from the definition, any set E ⊂ C of
finite logarithmic capacity can be represented as a union of a sigma-compactum (union
of countable collection of compact sets) and a set of logarithmic capacity zero. Thus,
35
V. Gutlyanskĭı, V. Ryazanov, E. Yakubov
the measurability of functions with respect to logarithmic capacity is invariant under
Hölder continuous change of variables.
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 [27]. 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 [7].
Dealing with measurable boundary functions φ(ζ) with respect to the logarithmic
capacity, we will use the abbreviation q.e. (quasi-everywhere) on a set E ⊂ C, if
a property holds for all ζ ∈ E except its subset of zero logarithmic capacity, see [19].
4. Dirichlet problem with measurable data in the unit disk.
In the paper [8], it was proved as Theorem 3.1 the following analog of the known
Luzin theorem in terms of logarithmic capacity, cf. e.g. Theorem VII(2.3) in [27].
Proposition 1. 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)
q.e. on (a, b). Furthermore, the function Φ can be chosen such that Φ(a) = Φ(b) = 0
and |Φ(x)| ≤ ε under arbitrary prescribed ε > 0 for all x ∈ [a, b].
Corollary 5. Let φ : ∂D → R be a measurable function with respect to logarithmic
capacity. Then there is a continuous function Φ : ∂D → R such that Φ′(eit) = φ(eit)
q.e. on R.
The Poisson–Stieltjes integral
ΛΦ(z) :=
1
2π
π∫
−π
Pr(ϑ− t) dΦ(eit) , z = reiϑ, r < 1 , ϑ ∈ R (25)
is well-defined for arbitrary continuous functions Φ : ∂D → R, see e.g. Section 2 in [26].
Directly by the definition of the Riemann–Stieltjes integral and the Weierstrass
type theorem for harmonic functions, see e.g. Theorem I.3.1 in [14], ΛΦ is a harmonic
function in the unit disk D := {z ∈ C : |z| < 1} because the function Pr(ϑ − t) is the
real part of the analytic function
Aζ(z) :=
ζ + z
ζ − z
, ζ = eit, z = reiϑ , r < 1 , ϑ and t ∈ R . (26)
Next, by Theorem 1 in [26] we have the following useful conclusion.
36
Dirichlet problem for Poisson equations in Jordan domains
Proposition 2. Let φ : ∂D → R be a measurable function with respect to logarithmic
capacity and Φ : ∂D → R be a continuous function with Φ′(eit) = φ(eit) q.e. on R. Then
ΛΦ has the angular limit
lim
z→ζ
ΛΦ(z) = φ(ζ) q.e. on ∂D . (27)
Finally, by Theorem 2 in [16], Proposition 2 and the known Poisson formula, see
e.g. I.D.2 in [18], we come to the following result on the existence, regularity and
representation of solutions for the Dirichlet problem to the Poisson equation in the
unit disk D. We assume that the charge density g is extended by zero outside of D in
the next theorem.
Theorem 3. Let a function φ : ∂D → R be measurable with respect to logarithmic
capacity and let a continuous function Φ correspond to φ by Corollary 5. Suppose that
a function g : D → R is in the class Lp(D) for p > 1. Then the following function in D
U := Ng − PN∗
g
+ ΛΦ , N∗
g := Ng|∂D , (28)
belongs to the class W 2,p
loc (D), satisfies the Poisson equation △U = g a.e. in D and has
the angular limit
lim
z→ζ
U(z) = φ(ζ) q.e. on ∂D . (29)
Moreover, U ∈ W 1,q
loc (D) for some q > 2 and U is locally Hölder continuous.
Furthermore, U ∈ C1,α
loc (D) with α = (p− 2)/p if g ∈ Lp(D) for p > 2.
Remark 5. Note that by the Luzin result, see also Theorem 3 in [26], the statement
of Theorem 3 is valid in terms of the length measure as well as the harmonic measure
on ∂D. However, by the well–known Ahlfors–Beurling example, see [1], the sets of
length zero as well as of harmonic measure zero are not invariant with respect to
quasiconformal changes of variables. The latter circumstance does not make it is possible
to apply the result in the future for the extension of the statement to generalizations
of the Laplace equation in anisotropic and inhomogeneous media. Hence we prefer to
use logarithmic capacity.
5. Dirichlet problem with measurable data in almost smooth domains.
We say that a Jordan curve Γ in C is almost smooth if Γ has a tangent q.e. Here
it is said that a straight line L in C is tangent to Γ at a point z0 ∈ Γ if
lim sup
z→z0,z∈Γ
dist (z, L)
|z − z0|
= 0 . (30)
In particular, Γ is almost smooth if Γ has a tangent at all its points except a countable
set. The nature of such Jordan curves Γ is complicated enough because the countable
set can be everywhere dense in Γ.
37
V. Gutlyanskĭı, V. Ryazanov, E. Yakubov
Now, given a domain D in C, kD(z, z0) denotes the quasihyperbolic distance,
kD(z, z0) := inf
γ
∫
γ
ds
d(ζ, ∂D)
, (31)
introduced in the paper [12]. Here d(ζ, ∂D) denotes the Euclidean distance from the
point ζ ∈ D to ∂D and the infimum is taken over all rectifiable curves γ joining the
points z and z0 in D.
Next, it is said that a domain D satisfies the quasihyperbolic boundary condi-
tion if
kD(z, z0) ≤ a ln
d(z0, ∂D)
d(z, ∂D)
+ b ∀ z ∈ D (32)
for constants a and b and a point z0 ∈ D. The latter notion was introduced in [10] but,
before it, was first applied in [4].
Remark 6. Given a Jordan domain D in C with the almost smooth boundary
satisfying the quasihyperbolic boundary condition. By the Riemann theorem, see e.g.
Theorem II.2.1 in [14], there is a conformal mapping f : D → D that is extended to
a homeomorphism f̃ : D → D by the Caratheodory theorem, see e.g. Theorem II.3.4
in [14]. Moreover, f∗ := f̃ |∂D, as well as f−1
∗ , is Hölder continuous by Corollary to
Theorem 1 in [4]. Thus, by Remark 4 a function φ : ∂D → R is measurable with
respect to logarithmic capacity if and only if the function ψ := φ ◦ f−1
∗ : ∂D → R is
so. Set Φ := Ψ ◦ f∗ where Ψ : ∂D → R is a continuous function corresponding to ψ by
Corollary 5.
Proposition 3. Let D be a Jordan domain in C with the almost smooth boundary
satisfying the quasihyperbolic boundary condition. Suppose that φ : ∂D → R is measu-
rable with respect to logarithmic capacity and Φ : ∂D → R is the continuous function
corresponding to φ by Remark 6. Then the harmonic function LΦ(z) := ΛΦ◦f−1
∗
(f(z))
has the angular limit φ q.e. on ∂D.
Proof. Indeed, by Remark 6 and Proposition 2 there is the angular limit
lim
w→ξ
ΛΨ(w) = ψ(ξ) q.e. on ∂D . (33)
By the Lindelöf theorem, see e.g. Theorem II.C.2 in [18], if ∂D has a tangent at a
point ζ, then
arg [f̃(ζ)− f̃(z)]− arg [ζ − z] → const as z → ζ .
After the change of variables ξ := f̃(ζ) and w := f̃(z), we have that
arg [ξ − w]− arg [f̃−1(ξ)− f̃−1(w)] → const as w → ξ .
In other words, the conformal images of sectors in D with a vertex at ξ is asymptotically
the same as sectors in D with a vertex at ζ. Thus, nontangential paths in D are
transformed under f̃−1 into nontangential paths in D.
38
Dirichlet problem for Poisson equations in Jordan domains
Recall that firstly the almost smooth Jordan curve ∂D has a tangent q.e., secondly
by Remark 6 the mappings f∗ and f−1
∗ are Hölder continuous, and thirdly by Remark 3
they transform sets of logarithmic capacity zero into sets of logarithmic capacity zero.
Consequently, (33) implies the desired conclusion. �
Finally, by Theorem 2 in [16], Proposition 3 and the Poisson formula, we come to
the following result on the existence, regularity and representation of solutions for the
Dirichlet problem to the Poisson equation in the Jordan domains. We assume here that
the charge density g is extended by zero outside of D in the next theorem.
Theorem 4. Let D be a Jordan domain in C with the almost smooth boundary
satisfying the quasihyperbolic boundary condition, a function φ : ∂D → R be measurable
with respect to logarithmic capacity and let a continuous function Φ correspond to φ by
Remark 6. Suppose that a function g : D → R is in the class Lp(D) for p > 1. Then
the following function in D
U := Ng − DN∗
g
+ LΦ , N∗
g := Ng|∂D , (34)
belongs to the class W 2,p
loc (D), satisfies the Poisson equation △U = g a.e. in D and has
the angular limit
lim
z→ζ
U(z) = φ(ζ) q.e. on ∂D . (35)
Moreover, U ∈ W 1,q
loc (D) for some q > 2 and U is locally Hölder continuous.
Furthermore, U ∈ C1,α
loc (D) with α = (p− 2)/p if g ∈ Lp(D) for p > 2.
Remark 7. Note that by the Luzin result, see also Theorem 3 in [26], the statement
of Theorem 4 is valid in terms of the length measure on rectifiable ∂D. Indeed, by the
Riesz theorem length f−1
∗ (E) = 0 whenever E ⊂ ∂D with |E| = 0, see e.g. Theorem
II.C.1 and Theorems II.D.2 in [18]. Conversely, by the Lavrentiev theorem |f∗(E)| = 0
whenever E ⊂ ∂D and length E = 0, see [20], see also the point III.1.5 in [25].
References
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V. Gutlyanskĭı, V. Ryazanov, E. Yakubov
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В. Гутлянский, В. Рязанов, Э. Якубов
Задача Дирихле для уравнений Пуассона в жордановых областях.
Прежде всего, мы изучаем задачу Дирихле для уравнений Пуассона △u(z) = g(z) с g ∈ Lp,
p > 1, и непрерывными граничными данными φ : ∂D → R в произвольных жордановых обла-
стях D ⊂ C и доказываем существование непрерывных решений u этой задачи в классе W 2,p
loc .
Кроме того, u ∈ W 1,q
loc для некоторого q > 2 и u локально непрерывны по Гельдеру. Более то-
го, u ∈ C1,α
loc с α = (p − 2)/p, если p > 2. Затем, на этой основе и применяя подход Лере–
Шаудера, мы получаем аналогичные результаты для задачи Дирихле с непрерывными гранич-
ными данными в произвольных жордановых областях для квазилинейных уравнений Пуассона
40
Dirichlet problem for Poisson equations in Jordan domains
вида △u(z) = h(z) · f(u(z)) с теми же предположениями о функции h как выше для g и непре-
рывных функций f : R → R, которые либо ограничены, либо с неубывающим |f | от |t|, таких,
что f(t)/t → 0 при t → ∞. Мы также приводим здесь приложения к математической физике,
которые относятся к задачам диффузии с абсорбцией, плазме и горению. В дополнение, мы рас-
сматриваем задачу Дирихле для уравнений Пуассона в единичном круге D ⊂ C с произвольными
граничными данными φ : ∂D → R, которые измеримы относительно логарифмической емкости.
Здесь мы устанавливаем существование неклассических решений этой проблемы в терминах уг-
ловых пределов в D п.в. на ∂D относительно логарифмической емкости с теми же локальными
свойствами как и выше. Наконец, мы распространяем эти результаты на почти гладкие жорда-
новы области D в C с квазигиперболическим граничным условием по Герингу–Мартио.
Ключевые слова: задача Дирихле, квазилинейные уравнения Пуассона, логарифмический по-
тенциал, логарифмическая емкость, угловые пределы.
В. Гутлянський, В. Рязанов, Е. Якубов
Задача Дiрихле для рiвнянь Пуасона у жорданових областях.
Перш за все ми вивчаємо задачу Дiрихле для рiвнянь Пуасона △u(z) = g(z) с g ∈ Lp, p > 1, та
неперервними граничними даними φ : ∂D → R в довiльних жорданових областях D ⊂ C та дово-
димо iснування неперервних рiшень u цiєї задачi в класi W 2,p
loc . Крiм цього, u ∈W 1,q
loc для деякого
q > 2 та u локально неперервнi за Гельдером. Бiльш того, u ∈ C1,α
loc з α = (p − 2)/p, якщо p > 2.
Потiм, на цiй основi, застосовуючи пiдхiд Лере–Шаудера, ми отримуємо аналогiчнi результати
для задачi Дiрихле з неперервними граничними даними в довiльних жорданових областях для
квазiлiнiйних рiвнянь Пуасона виду △u(z) = h(z) · f(u(z)) з тими же припущеннями про функцiї
h як вище для g та неперервних функцiй f : R → R, якi або обмеженi, або з неспадним |f | вiд
|t|, таких, що f(t)/t → 0 при t → ∞. Ми також наводимо тут додатки до математичної фiзики,
якi вiдносяться до задач дифузiї з абсорбцiєю, плазмi та горiнню. На додаток, ми розглядаємо
задачу Дiрихле для рiвнянь Пуасона в одиничному колi D ⊂ C з довiльними граничними даними
φ : ∂D → R, якi вимiрнi вiдносно логарифмiчної ємностi. Тут ми встановлюємо iснування некла-
сичних рiшень цiєї проблеми у термiнах кутових границь у D п.в. на ∂D вiдносно логарiфмiчної
ємностi з тими ж локальними властивостями як i вище. Нарештi, ми поширюємо цi результати
на майже гладкi жордановi областi D в C з квазiгiперболiчною граничною умовою за Герингом–
Мартiо.
Ключовi слова: задача Дiрихле, квазiлiнiйнi рiвняння Пуасона, логарифмiчна ємнiсть, куто-
вi межи.
Institute of Applied Mathematics and Mechanics
of the NAS of Ukraine, Slavyansk, Ukraine;
Holon Institute of Technology, Holon, Israel
vgutlyanskii@gmail.com, Ryazanov@nas.gov.ua,
yakubov@hit.ac.il, eduardyakubov@gmail.com
Received 17.12.2018
41
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