e-ISSN : 26203502
p-ISSN : 26153785
International Journal on Integrated Education
Carleman's Formula of a Solution of the Poisson Equation
Zuxro E. Ermamatova
Samarkand State University
Abstract
We suggest an explicit continuation formula for are a solution to the Cauchy problem for the
Poisson equation in a domain from its values and the values of its normal derivative on part of
the boundary. We construct an continuation formula of this problem based on the CarlemanYarmuhamedov function method.
Key words: Poisson equations, ill-posed problem, regular solution, Carleman -Yarmuhamedov
function, Green's formula, Carleman formula, Mittag-Le¬er entire function
Introduction
3
Introduce the following notations: R is the third dimensional real Euclidean space,
x  ( x1 , x2 , x3 ), y  ( y1 , y 2 , y 3 ) R 3 , x  ( x1 , x2 ), y   ( y1, y 2 )  R 2 ,
 2  y  x 2 ,  2 s, r 2  y  x 2   2  ( y3  x3 )2 ,   tg  ,   1,
2
G  y : y  y3 , y3  0, G  y : y  y3 , y3  0,
C   :     i ,      ,      .
and 𝜀2 sufficiently small positive constants,
𝐺𝜌𝜀 = {𝑦: |𝑦 ′ | < 𝜏(𝑦3 − 𝜀)}, 𝜕𝐺𝜌𝜀 = {𝑦: |𝑦 ′ | = 𝜏(𝑦3 − 𝜀)}, ̅𝐺̅̅𝜌̅𝜀 = 𝐺𝜌𝜀 ∪ 𝜕𝐺𝜌𝜀 ,
3
Ω𝜌 - is a bounded simply connected domain whose boundary in R whose boundary 𝜕Ω𝜌
consists of a part of the conic surface 𝑇 ≡ 𝜕𝐺𝜌 and a smooth surface S; lying inside the cone
𝐺𝜌̅ . The case 𝜌 = 1 is the limit case. In this case 𝐺1 is the half-space 𝑦3 > 0, 𝜕𝐺1 is the
hyperplane 𝑦3 = 0; and Ω1 is a bounded simply connected domain whose boundary consists of
a compact connected part of the hyperplane 𝑦3 = 0 and a smooth surface S in the half-space
̅ 𝜌 = Ω𝜌 ∪ 𝜕Ω𝜌 , 𝑆0 is the interior of 𝑆.
𝑦3 ≥ 0; Ω
In this article we propose an explicit formula for reconstruction of a solution of the Poisson
equation
𝜕2 𝑈
−∆𝑈(𝑥) = − ∑3𝑖=1 𝜕𝑥 2 = 𝑓(𝑥),
(0.1)
a domain from its values and the values of its normal derivative on part of the boundary
𝑈(𝑥) = 𝑓1 (𝑥),
{𝜕𝑈(𝑥)
= 𝑓2 (𝑥), 𝑥 ∈ 𝑆,
𝜕𝑛
(0.2)
i.e., we give an explicit continuation formula for a solution to the Cauchy problem for the
Poisson equation.
The Cauchy problem for the Poisson equation (0.1)-(0.2) is well-known to be ill-posed [1], [2].
It has applications in many different areas such as plasma physic, electrocardiography, and
corrosion non-destructive evaluation (e.g., [3], [13], [15], [7]). Traditionally, regularization
techniques, such as Tikhonov regularization [14] and the quasi-reversibility approach [8], were
used to provide robust numerical schemes [17].
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We construct a solution to the problem in the domain Ω𝜌 when the Cauchy data is given on a
part 𝑆 of the boundary 𝜕Ω𝜌 . The Cauchy problem for the Poisson equation relates to the class of
ill-posed problem [11].
We suppose that a solution to the problem exists (in this event it is unique) and continuously
differentiable in the closed domain and the Cauchy data are given exactly. In this case we
establish an explicit continuation formula. This formula enables us to state a simple and
convenient criterion for solvability of the Cauchy problem.
The result established here is a multidimensional analog of the theorems and Carleman type
formulas [12], by G.M.Goluzin, V.I.Krylov, V.A.Fok, and F.M.Kuni in the theory of
holomorphic functions of one variable [5], [16].
The method for obtaining these results bases on an explicit form of fundamental solution of the
Poisson equation which depends on a positive parameter vanishes together with its derivatives on
a xed cone and outside it as the parameter tends to innity, while the pole of the fundamental
solution lies inside the cone. Following to M.M.Lavrent'ev, a fundamental solution with these
properties is called a Carleman function for the cone [9]. Having constructed a Carleman
function explicitly, we write down a continuation formula. Existence of a Carleman function
follows from S.N.Mergelyan's approximation theorem [10]. However, this theorem shows no
way in writing the Carleman function explicitly.
1. The Mittag-Leffler entire function
The continuation formulas below are expressed explicitly in terms of the Mittag-Leffler entire
function: therefore, we now present its basic properties without proof. These properties as well
as detailed proofs can be found in [6, Chapter 3, § 2]. The Mittag-Leffler entire function is
defined by the series
∞
𝐸𝜌 (𝑤) = ∑
𝑛=0
𝑤𝑛
,
Г(1 + 𝑛⁄𝜌)
𝜌 > 0,
𝑤 ∈ 𝐶,
𝐸1 (𝑤) = exp(𝑤),
where Γ is the Euler gamma-function. Henceforth we suppose that 𝜌 > 1 . Denote 𝛾 =
𝜋
𝛾(1, 𝛽), 0 < 𝛽 < 𝜌 , 𝜌 > 1, the contour in the complex 𝑤 – plane which consists of the ray
𝑎𝑟𝑔𝑤 = −𝛽, |𝑤| ≥ 1, the arc −𝛽 ≤ 𝑎𝑟𝑔𝑤 ≤ 𝛽, of the circle |𝑤| = 1, and the ray, 𝑎𝑟𝑔𝑤 =
𝛽, , |𝑤| ≥ 1, and is traversed so that 𝑎𝑟𝑔𝑤 is non decreasing. The contour 𝛾 splits the
complex domain 𝐶 into the two simply connected infinite domains Ω− and Ω+ , lying
𝜋
𝜋
respectively to the left and to the right of 𝛾 . We suppose that 2𝜌 < 𝛽 < 𝜌 , 𝜌 > 1.
Under these conditions, the following integral representations are valid
 



 exp w    w, w    ,
E  w  


  w, w    ,


E ( w)    ( w) , w     ,
(1.1)
(1.2)
where
  w 
exp  

exp  


d

d .
,   w 
2i   (  )   w
2i  (  ) (  w) 2
(1.3)

Since 𝐸𝜌 (𝑤) takes real vales for real 𝑤, we obtain
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Re  w 
  w     w 
2


exp     Re w
d ,

2i   (  )   w  w 
(1.4)
  w    w  Im w
exp   
Im  w 

d ,

2i
2i   (  )   w  w 
(1.5)
 2 exp     Re w
Im

d .

Im w 2i    w2   w 2
  w
Henceforth we take

(1.6)
 2
 ,   1 , in the definition of the contour  1,   . It is clear
2 2
that if

  2  arg w   ,
2
(1.7)
then 𝑤 ∈ Ω𝜌− and E w    w .
Denote
𝑝
𝜌
𝜁𝑝 𝑒 𝜁
𝑇𝑘,𝑝 (𝑤) =
∫
𝑑𝜁, 𝑘 = 1,2, … , 𝑝 = 0,1, … .
2𝜋𝑖 (𝜁 − 𝑤)𝑘 (𝜁 − 𝑤
̅)𝑘
𝛾
The following inequalities are valid for
E  w  
M1
,
1 w

  2  arg w   :
2
E  w 
M2
1 w
2
,
(1.8)
𝑀
3
|𝑇𝑘,𝑝 (𝑤)| ≤ 1+|𝑤|
2𝑘 , 𝑘 = 1,2, …,
(1.9)
𝜋
𝜀
𝜋
where 𝑀1 , 𝑀2 , 𝑀3 are constants independent of 𝑤. Choose 𝛽 = 2𝜌 + 22 < 𝜌, 𝜌 > 1, in (1.1).
Then 𝐸𝜌 (𝑤) = 𝜓𝜌 (𝑤), where 𝜓𝜌 (𝑤) is defined by (1.3). Moreover, note that 𝑐𝑜𝑠𝜌𝛽 < 0 and
the integral converges:


P
P
  exp cos   d  ,

p  0,1,...
(1.10)
Given a sufficient large |𝑤| (𝑤 ∈ Ω+ 𝑎𝑛𝑑 𝑤 ∈ Ω− ), we now obtain
min   w  w sin

2
2
, min   w   w sin

2
2
.
(1.11)
From (1.2) and the compositions
1
1

 
,
 w
w w  w
1
1

 
 w
w w   w 
(1.12)
for w large we derive
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


11

1
const
P
,
E  w  Г 1 1   
 2   exp cos   d 
2


w
w


2 sin 2 w 

 

1 
Г 1 1   
exp  P d .

  2i 

Hence, the first inequality of (1.8) follows. Similarly, from (1.10), (1.3), and the decomposition
1
  w

2
1

2

2

w2
w 2   w w 2   w2
for w large we derive the inequality

1 1
const
.
E  w  Г 1 1   2 
3
w
w

The second inequality of (1.8) is proven. For 𝑘 = 1,2, … from (1.12) we obtain
1
1
(−1)𝑘
𝜁𝑘
= [ 𝑘 + ⋯+ 𝑘
]×
(𝜁 − 𝑤)𝑘 (𝜁 − 𝑤
̅)𝑘
𝑤
𝑤 (𝜁 − 𝑤)𝑘
(−1)𝑘
𝜁𝑘
1
−𝑘
× [ 𝑘 + ⋯+ 𝑘
]=
+
+ ⋯.
𝑘
2𝑘
2𝑘+1
|𝑤|
|𝑤|
𝑤
̅
𝑤 (𝜁 − 𝑤
̅)
(𝜁 − 𝑤)
The first term of the decomposition is principal for |𝑤| large, therefore, (1.10) and (1.11) yield
the inequality of (1.9):
1
1
𝑐𝑜𝑛𝑠𝑡
|𝑇𝑘,𝑝 − Г−1 (1 − 𝜌) |𝑤|2𝑘 | ≤ |𝑤|2𝑘+1 .
2. Construction of a Carleman-Yarmukhamedov function
According to [15], we define the Carleman-Yarmukhamedov function Ф𝜎 (𝑥, 𝑦) for 𝑠 ≥ 0, 𝑣 ≥ 1
by the equality:
∞
𝐾(𝑤)
𝑑𝑢
−2𝜋 2 𝐾(𝑥3 )Ф𝜎 (𝑥, 𝑦) = ∫0 𝐼𝑚 [𝑤−𝑥 ] √𝑢2
3
+𝛼2
, 𝑤 = 𝑖√𝑢2 + 𝛼 2 + 𝑦3 .
(2.1)
Here 𝐾(𝑤) is an entire function of complex variable which takes real values for real 𝑤, ( 𝑤 =
𝑢 + 𝑖𝑣, 𝑢, 𝑣 are reals numbers) such as 𝐾(𝑢) ≠ ∞, |𝑢| < ∞, K (0)  0 ,  R  0 CR  0
Sup
Re w  R , Im w  C R
( K ( w)  Im w K ( w)  Im w K ( w) )   .
2
(2.2)
For real 𝑤 from reality of K (w) we have K ( w)  K ( w) . Then (2.1) implies that for every
R  0


Sup K ( w)  (1  Im w ) K ( w)  (1  Im w ) K ( w)   .
Re w  R
2
(2.3)
Now we write (2.1) in the form
∞ (𝑦3 −𝑥3 )𝐼𝑚𝐾(𝑤)
−2𝜋 2 𝐾(𝑥3 )Ф𝜎 (𝑥, 𝑦) = ∫0 {
√𝑢2 +𝛼2
𝑑𝑢
− 𝑅𝑒𝐾(𝑤)} 𝑟 2 +𝑢2 ,
(2.4)
where
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 K ( w)  1  K ( w) K ( w)  w K ( w)  w K ( w)
Im


 

2 i (r 2  u 2 )
w 
 w  2i  w

( y3  x3 ) Im K ( w)  s  u 2 Re K ( w)
.
r2  u2
(2.5)
From (2.3) and (2.4) it follows that for 𝑦 ≠ 𝑥 the integral in (2.1) converges absolutely.
From 𝐾(𝑤) ≡ 1, then the function Ф𝜎 (𝑥, 𝑦) is the classical fundamental solution to the
Laplace equation, i.e.
Ф𝜎 (𝑥, 𝑦) ≡ Ф0 (𝑥, 𝑦) = 1⁄(4𝜋𝑟).
In the paper [15] proved
Theorem 2.1. The function Ф𝜎 (𝑥, 𝑦) defined for 𝛼 > 0, by (2.2) and (2.3) is representable as
Ф𝜎 (𝑥, 𝑦) = Ф0 (𝑥, 𝑦) + 𝐺𝜎 (𝑥, 𝑦)
(2.6)
where Ф0 (𝑥, 𝑦) = 1⁄(4𝜋𝑟) and the function 𝐺𝜎 (𝑥, 𝑦) – is harmonic in the variable 𝑦 in 𝑅 3 ,
including 𝑦 = 𝑥.
From theorem 2.1 followed that, the function Ф𝜎 (𝑥, 𝑦) for many 𝜎 > 0 the variable y emerge
̅𝜌)
fundamental solution of Poisson equation. Therefore for the function 𝑈(𝑦) ∈ 𝐶 2 (Ω𝜌 ) ∩ 𝐶 1 (Ω
and for every point 𝑥 ∈ Ω𝜌 is valid the Green’s formula [4] :
𝑈(𝑥) = ∫Ω Ф𝜎 (𝑥, 𝑦)𝑓(𝑦)𝑑𝑦 − ∫∂Ω [𝑈(𝑦)
𝜌
𝜕Ф𝜎 (𝑥,𝑦)
𝜕𝑛
𝜌
− Ф𝜎 (𝑥, 𝑦)
𝜕𝑈(𝑦)
𝜕𝑛
] 𝑑𝑆𝑦 , (2.7)
where 𝑓(𝑦) ∈ 𝐶 𝛼 (Ω𝜌 ), 0 < 𝛼 < 1.
3. Carleman formulas
Take the function 𝐾(𝑤) in the formula (2.1) to be the Mittag-Leffler enter function
𝐾(𝑤) = exp(𝑎𝑤 2 )𝐸𝜌 (𝜎𝑤),
where 𝜌 > 1, 𝑤 = 𝑖√𝑢2 + 𝑠 + 𝑦3 − 𝑥3 , 𝐾(0) = 𝐸𝜌 (0) = 1, 𝑎 > 0 and 𝜎 ≥ 0. Denote the
corresponding fundamental solution Ф𝜎 (𝑥, 𝑦) and its derivative with respect to the variable 𝜎
𝑑Ф (𝑦−𝑥)
by Ф𝜎 (𝑦 − 𝑥) and Ψ𝜎 (𝑦 − 𝑥) ≡ 𝜎𝑑𝜎 . It follows from the theorem 1.1 that Ψ𝜎 (𝑦 − 𝑥)
satisfy is Poisson equation in 𝑅 3 , then
∞
−2𝜋 2 Ф𝜎 (𝑥, 𝑦) = ∫0 𝐼𝑚 [
𝑤−𝑥3
𝑑𝑢
] √𝑢2
=.
+𝛼2
2
∞
2
exp(𝑎𝑤 2 ) 𝐸𝜌 (𝜎𝑤)
= 𝑒 𝜎(𝑦3 −𝑥3) ∫0 𝜑𝜎 (𝑦, 𝑥, 𝑢)
𝑒 −𝑎𝑠−𝑎𝑢
𝑢2 +𝑟 2
𝑑𝑢,
(3.1)
where
𝜑𝜎 (𝑦, 𝑥, 𝑢) = [
(𝑦3 − 𝑥3 )
√𝑢2 + 𝛼 2
(𝑦 −𝑥3 )
.+ [𝐼𝑚𝐸𝜌 (𝜎𝑤) + √𝑢32
Ψ𝜎 (𝑦 − 𝑥) ≡
+𝛼2
𝑅𝑒𝐸𝜌 (𝜎𝑤)] sin(𝜗√𝑢2 + 𝛼 2 ), 𝜗 = 2𝑎(𝑦3 − 𝑥3 ),
𝑑Ф𝜎 (𝑦−𝑥)
𝑑𝜎
𝐼𝑚𝐸𝜌 (𝜎𝑤) − 𝑅𝑒𝐸𝜌 (𝜎𝑤)] cos (𝜗√𝑢2 + 𝛼 2 ) +
∞
𝑑𝑢
= ∫0 𝐼𝑚[exp(𝑎𝑤 2 ) 𝐸′𝜌 (𝜎𝑤)] √𝑢2
+𝛼2
.
(3.2)
(3.3)
Theorem 3.1. Let Ω𝜌 be bounded domain and suppose that each point of ∂Ω𝜌 regular (with
respect to the Laplasian). If 𝑓 is a bounded, locally H𝑜̈ lder continuous function in Ω𝜌 , the
Cauchy problem (0.1)-(0.2) is uniquely solvable for any 𝑓1 (𝑦) and 𝑓2 (𝑦) are given functions of
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the class 𝐶(𝑆). Then the Carleman formulas
𝜕 𝑖 𝑈(𝑥)
𝜕𝑥𝑗𝑖
= lim
𝜎→∞
− ∫∂Ω {𝑓1 (𝑦)
𝜌
𝜕 𝑖 𝑈𝜎 (𝑥)
𝜕𝑥𝑗𝑖
𝜕𝑖 𝜕Ф𝜎 (𝑦−𝑥)
𝜕𝑥𝑗𝑖
𝜕𝑛
= lim [∫ 𝑓(𝑦)
𝜎→∞
− 𝑓2 (𝑦)
𝜕 𝑖 Ф𝜎 (𝑦 − 𝑥)
𝜕𝑥𝑗𝑖
Ω𝜌
𝜕𝑖 Ф𝜎 (𝑦−𝑥)
𝜕𝑥𝑗𝑖
} 𝑆𝑦 ]
𝑑𝑦 −
(3.5)
are valid for every 𝑥 ∈ Ω𝜌 , where 𝑖 = 0,1, 𝑗 = 1,2,3,
𝜕0 𝑈𝜎 (𝑥)
𝜕𝑥𝑗0
= 𝑈𝜎 (𝑥),
𝜕0 Ф𝜎 (𝑦−𝑥)
𝜕𝑥𝑗0
= Ф𝜎 (𝑦 − 𝑥),
and the convergence in (3.5) is uniform on compact sets in Ω𝜌 .
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