HP estimation for the Cauchy problem for nonlinear elliptic equation

TAẽP CH PHAT TRIEN KH&CN, TAP 20, SO T5- 2017

HP estimation for the Cauchy problem for
nonlinear elliptic equation
Le Duc Thang
University of Science, VNU- HCM
Ho Chi Minh City Industry and Trade College
(Received on 5th December 2016, accepted on 28 th November 2017)

ABSTRACT
In this paper, we investigate the Cauchy
Error estimates between the regularized solution
p
problem for a ND nonlinear elliptic equation in a
and the exact solution are established in H space
bounded domain. As we know, the problem is
under some priori assumptions on the exact
severely ill-posed. We apply the Fourier
solution.
truncation method to regularize the problem.
Key words: nonlinear elliptic equation, ill-posed problem, regularization, truncation method
INTRODUCTION
In this paper, we consider the Cauchy problem for a nonlinear elliptic equation in a bounded
domain. The problem has the form

u F (x
u (x , x N )
u (x ,T )
ux (x ,T )

, x N , u ( x , x N )),

0,
(x ),
0,
x

N

( x , xN )
( x , xN )
x
,

is a positive constant,
, N is a natural number and
N
2 , the function
L ( )
is known and F
is called the source function. It is well-known the
above problems is severely ill-posed
in the
sense of Hadamard. In fact, for a given final data,
we are not sure that a solution of the problem
exists. In the case a solution exists, it may not
depend continuously on the final data. The
problem has many various applications, for
example in electrocardiography [7], astrophysics
[6] and plasma physics [15, 16].
(0,

N

)

(1)

.

T

Where

( 0,T ),
( 0,T ),

1

2

In the past, there have been many studies on
the Cauchy problem for linear homogeneous
elliptic equations, [1, 5, 9, 10, 12]. However, the
literature on the nonlinear elliptic equation is quite
scarce. We mention here a nonlinear elliptic

problem of [13] with globally Lipschitz source
terms, where authors approximated the problem
by a truncation method. Using the method in
[13,14], we study the Cauchy problem for
nonlinear elliptic in multidimensional domain.

The paper is organized as follows. In Section
2, we present the solution of equation (1). In
Section 3, we present the main results on
regularization theory for local Lipschitz source
function. We finish the paper with a remark.
SOLUTION OF THE PROBLEM
Assume that problem (1) has a unique
solution u(x , x N ) . By using the method of
separation of variables, we can show that solution
of the problem has the form

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Science & Technology Development, Vol 5, No.T20- 2017
u(x , x N )
n1 1 n2 1

nN

T

x N ) n12
n12

xN

Indeed, let u(x , x N )

n1n2 ...nN

...

...

nN2 1 )

nN2

...

un n ...n

nN

1

1 2

1

2

(x )

1

n22

n22

...
n1 1 n2 1

orthonormal basis

n22

nN2 1 )

n1n2 ...nN

1

1

sinh((

+

x N ) n12

cosh((T

...

N 1

Fn n ...n
1 2

N 1

1

(u )( )d .

n1n2 ...nN

1

(x ).

(2)

1

(x N )

n1n2 ...nN

1

(x ) be the Fourier series in L2 ( ) with

N 1

sin(n1x 1 )sin(n2x 2 )...sin(nN 1x N 1 ) . From (1), we can obtain

the following ordinary differential equation

d2
2
N

dx
un n

1 2

un n

1 2

... nN

... nN

1

T
T

1

where Fn n ... n

N 1

1 2

n12
n1n2 ... nN

d
u
dx N n1n2 ... nN

un n ...n

1

n22

...

nN2

1

un n ... n
1 2

N

1

xN

Fn n

1 2

... nN

1

u xN ,

0,T ,

xN

,

(3)

0,

(u)(x N )

u(x , x N )

N 1

1 2

xN

1

F (x , x N , u(x , x N )
n1n2 ...nN

1

n1n2 ... nN

1

dx ,

n1n2 ...nN

(x )

1

n1n2 ...nN

1

(x )dx and

(x )dx .

The equation (3) is ordinary differential equations. It is easy to see that its solution is given by

un n ...n
1 2

N 1

(x N )

cosh((T
T

x N ) n12

n22
x N ) n12

sinh((

n12

xN

n22

nN2

...

n22

...
nN2

...

n1n2 ...nN

1

nN2

1

1

Fn n ...n
1 2

N 1

(u )( )d .

1

(4)
REGULARIZATION AND ERROR ESTIMATE FOR NONLINEAR PROBLEM WITH
LOCALLY LIPSCHITZ SOURCE
We know from (4) that, when n1, n2 ,..., nN
cosh((T

x N ) n12

n22

...

nN2

1

) and

1

become large , the terms

sinh((

x N ) n12
n12

n22

n22

...
nN2

...

nN2 1 )

increase rather

1

quickly. Thus, these terms are the cause for instability. In this paper, we use the Fourier truncated method.
The essence of the method is to eliminate all high frequencies from the solution, and consider the problem
only for n1, n2 ...nN 1 satisfying n12 n22 ... nN2 1 C . Here C is a constant which will be
selected appropriately as a regularization parameter which satisfies limC
0

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.


TAẽP CH PHAT TRIEN KH&CN, TAP 20, SO T5- 2017
Let the function F :

u, v

[0,T ]

0 and for any u, v satisfying

such that: for each M

M , there holds

F (x , x N , u)

F (x , x N , v)

where (x , x N )

KF (M ) u

v,

(5)

[0,T ] and
F (x , x N , u)

KF (M ) : sup

F (x , x N , v)

u

v

: u, v

M, u

We note that KF (M ) is increasing and lim KF (M )

v, (x ,x N )

.

0 , we approximate F by

. For all M

M

[0,T ]

FM defined by

F (x , x N , M ),
F (x , x N , u(x , x N )),
F (x , x N , M ),

FM (x , x N , u(x , x N ))

0 , we consider a parameter M

For each

u(x ,x N ) M ,
-M u(x ,x N ) M ,
u(x ,x N )
M.

0 . We shall use the following well-

as

posed problem

v

PC FM x , x N , v x , x N ,

v x , xN

x , xN
x , xN

0,

v x ,T

0,T ,

PC

x ,

vx

N

x ,T

0,T ,

x

0,

(6)

.

where

PC w

w,
n1 ,n2 ...,nN

1

1

n12 n22 ... nN2

We show that the solution u ,

u , (x , x N )

1

1

+

L2 ( ) .

for all w

of problem (6) satisfies the following integral equation

x N ) n12

n22

n22

nN2 1 )

...

nN2 1 )

n1n2 ...nN

1

C

1

sinh((

x N ) n12
n12

xN

1

1

n12 n22 ... nN2
T

n1n2 ...nN

1

C

cosh((T
n1 ,n2 ,...,nN

n1n2 ...nN

n22

...

...
nN2

1

(7)

FM

n1n2 ...nN

(u )( )d
1

n1n2 ...nN

1

(x ),

Lemma 1. For u1 (x , x N ), u2 (x , x N ) , we have

FM (x , x N , u2 (x , x N )
Proof. If u1 (x , x N )

FM (x , x N , u1(x , x N )

M and u2 (x , x N )

M then

FM (x , x N , u2 (x , x N )
If u1(x , x N )

M

u2 (x , x N )

KF (M ) u2 (x , x N ) u1(x , x N ) .

FM (x , x N , u1(x , x N )

0.

M then

FM (x , x N , u2 (x , x N ) FM (x , x N , u1(x , x N ))

FM (x , x N , u2 (x , x N ) FM (x , x N , M )
KF (M ) u2 (x , x N ) u1(x , x N ) .

If u1(x ', x N )

M

M

u2 (x ', x N ) then

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Science & Technology Development, Vol 5, No.T20- 2017
FM (x , x N , u2 (x , x N ) FM (x , x N , u1(x , x N ))

FM (x , x N , M ) FM (x , x N , M )
KF (M ) u2 (x , x N ) u1(x , x N ) .

If M

u1(x , x N ), u2 (x , x N )

M then

FM (x , x N , u2 (x , x N ) FM (x , x N , u1(x , x N ))

F (x , x N , u2 (x , x N ) F (x , x N , u1(x , x N ))
KF (M ) u2 (x , x N ) u1(x , x N ) .

This completes the proof.
Lemma 2. Let u be the exact solution to problem (1). Then we have the following estimate

u , (x N )

PC u(x N )

2 exp(2(T

2

L( )

2

x N )C )

L2 ( )

T

+2K F2 (M )(T

x N ) exp(2(

x N )C ) u , ( )

2

u( )

L2 ( )

xN

and u , we have

Proof. From the definition of u ,
u , (x N )

PC u(x N )

2
2

L( )

+2
n1 ,n2 ,...,nN

1

1

n12 n22 ... nN2

n1 ,n2 ,...,nN

1

n12

n22

n22

...

nN2 1 (

1

x N ) n12

sinh((

xN
1

x N ) n12

cosh((T

2
2
n12 n22 ... nN

T

d .

1

2
n1n2 ...nN

n1n2 ...nN

1

1

C

n22

nN2

...

nN2 1 )

...

2

((FM )n n ...n
1 2

N 1

(u , )( )

Fn n ...n
1 2

N 1

(u )( ))d

1

C

T

2

2exp(2(T -x N )C )

2

L( )

+2(T -x N ) exp(2(

x N )C FM ( , u , ( ))

F ( , u( ))

xN

2

d .

L2 ( )

(8)
Since lim M

0 , there exists M such that M

, for a sufficiently small

0

For M we have FM (x , x N , u(x , x N ))

u

L ([0,T ];L2 ( ))

.

F (x , x N , u(x , x N )) . Using the Lipschitz property of FM as in

Lemma 1, we get
FM ( , u , ( ))

F ( , u( ))

2

K F2 (M ) u , ( )

L2 ( )

u( )

2
L2 ( )

.

(9)

Combining (8) and (9), we complete the proof of Lemma 2.
Theorem 1. Let

0 and let F be the function defined in (5). Then the problem (6) has a unique
solution u ,
C ([0,T ]; L2 ( )) .
C ([0,T ]; L2 ( )) . Put

Proof. We prove the equation (7) has a unique solution u ,
(u , )(x , x N )

(x , x N )

G(x , x )

where

(x , x N )

cosh((T
n1 ,n2 ,...,nN

1

1

n12 n22 ... nN2

Trang 196

1

C

)

x N ) n12

n22

...

nN2 1 )

n1n2 ...nN

1

n1n2 ...nN

1

(x )


TAẽP CH PHAT TRIEN KH&CN, TAP 20, SO T5- 2017
and
T

n1 ,n2 ,...,nN

1

1

n12

xN

n12 n22 ... nN2

x N ) n12

sinh((

G(x , x N )

n22

n22

nN2 1 )

...
nN2

...

FM

1

n1n2 ...nN

1

(u , )( )d

n1n2 ...nN

1

(x )

C

1

.
We claim that
p

p

(v , )(x N )

(w

)(x N )

,

K F2 (M )T exp(2TC )

p

v

p!

L2 ( )

w

,

(10)

,

for p 1 , where
is the sup norm in C ([0,T ]; L2 ( )) . We shall prove the above inequality by
induction.
For p 1 , using the inequality
T

x N ) n12

sinh((

2
1

n

xN

n

2
2

n22
n

...

nN2 1 )
d

...
2
N 1

exp(2 n12

n22

nN2 1T )T

...

and using Lemma 1, we have

(v , )(x N )

(w
T

n1 ,n2 ,...,nN

1

1

n12 n22 ... nN2

,

)(x N )

L2 ( )

x N ) n12

sinh((

n12

xN
1

2

n22

n22
...

...
nN2

2

nN2 1 )

FM

n1n2 ...nN

1

1

(v , )( )

FM

T

n1 1 n2 1

nN

1

1

n1n2 ...nN

1

(v , )( )

T

exp(2TC )T

FM ( , v , ( ))

FM ( , w , ( ))

xN

Thus (10) holds for
p k 1 .
We have
,

)(x N )

k 1

p

(w

2

FM

d

2

)(x N )

1

,

)( ) d

K F2 (M )exp(2TC )T 2 v

L( )

p

1 . Suppose that (10) holds for

,

(w

n1n2 ...nN

2

w

,

k . We prove that (10) holds for

2

L ( )

exp(2TC )T

FM ( ,
xN

k

(v

( )))

,

k

FM ( ,

(w

T
k

(v

,

)( )

k

(w

,

(v , )(x N )

(w

,

)(x N )

2
2

L( )

2
F

K (M )K

2k
F

exp(2TC )T exp(2TC k )

T

)( )

xN
k

( )))

,

xN

k 1

.

,

T

2

K F2 (M ) exp(2TC )T

k 1

,

2

FM

...
xN

(v

1

)( ) d

C

exp(2TC )T

k 1

n1n2 ...nN

(w

2
L2 ( )

2

d

L2 ( )

d

k 1

1!

2

v,

w

,

Therefore, we get

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Science & Technology Development, Vol 5, No.T20- 2017

p

p

(v , )(x N )

(w

)(x N )

,

K F2 (M )T exp(2TC )

C ([0,T ]; L2 ( )) .

Let us consider

: C ([0,T ]; L2 ( ))

lim

w

,

p0

(u)

In fact, one has

0.

(u , )

u has a unique solution u ,
p0

(

(u

(u

))

,

(11)

C ([0,T ]; L2 ( )) . It is easy to see that

p0

As a consequence, there exists a positive integer number p0 such that
the equation

,

,

p

p!

p

v

p!

L2 ( )

for all v , , w ,

K F2 (M )T exp(2TC )

p

,

C ([0,T ]; L2 ( )) . We claim that

p0

) . By the uniqueness of the fixed point of

(u)

u , , i.e., the equation

is a contraction. It follows that
(u , )

u, .

, one has

C ([0,T ]; L2 ( )) .

u has a unique solution u ,

To show error estimates between the exact solution and the regularized solution, we need the exact
solution belonging to the Gevrey space.
Definition 1. (Gevrey-type space). (see [2, 3]) The Gevrey class of functions of order s 0 and index
0 is denoted by G s /2 and is defined as

G s /2

L2 ( ) :

f

(n12

...
n1 1 n2 1

nN

1

n22

nN2 1 )s /2 exp(2

...

n12

n22

nN2 1 ) | f ,

...

1

n1n2 ...nN

1

It is a Hilbert space with the following norm

f

(n12

...

G s /2

n1 1 n2 1

nN

1

n22

...

nN2 1 )s /2 exp(2

n12

n22

...

nN2 1 ) | f ,

1

For a Hilbert space H , we denote L (0,T ; H )

f : [0,T ]

n1n2 ...nN

1

|2 .

H | ess sup|f(t)|H
0 t T

and
f

ess sup f (t ) .

L (0,T ;H )

H

0 t T

We consider some assumptions on the exact solution as the following:

ess sup
0 xN T

(n12

...
n1 1 n2 1

nN

ess sup
0 xN

(13)
for all x N

1

exp(2(x N

nN

n22

... nN2

) n12

n22

...nN2 1 )un2 n ...n
1 2

N 1

, I 1, I 2 , are positive constants.

k

G , we have the following inequality

w

Trang 198

nN2 1 ) exp 2x N n12

1

1

[0,T ] , where ,

Lemma 3. For any w

...

1

un2 n ...n
1 2

1

...
n1 1 n2 1

n22

PC w

L2 ( )

C ke

C

w

Gk

.

(x N )

I2 ,

N 1

(x N )

I 1 , (12)

|2

.


TAẽP CH PHAT TRIEN KH&CN, TAP 20, SO T5- 2017
G k , we get

Proof. For w

w

PC w

2
2

L( )

w,

=
n1 ,n2 ,...,nN
n12

C

2k

e

n22

...

e

2
nN
1

1

2 C

w

Gk

1

C

n12

n22

nN2

...

k
1

2

n12

exp 2

n22

nN2

...

1

w,

1

n12 n22 ... nN2

2k

1

2 C
n1 ,n2 ,...,nN

C

1

2
n1n2 ...nN

1

n1n2 ...nN

1

C

.

This completes the proof.
The following theorem provides some error estimates in the L2 norm when the exact solution
belongs to the Gevrey space.
Theorem 2. Assume that the problem (1) has a unique solution u which satisfies (12). If C and M are
chosen such that lim eTC
0

TC

0 and lim exp(2KF2 (M )T 2 )C

lim exp(2KF2 (M )T 2 ) e

0

0 ,

0

then we have

u , (x N ) u(x N )
Proof. Since u

2

2C

L2 ( )

I 22

4 2e

2TC

exp(2K F2 (M )T 2 )e

xNC

.

(14)

Gx then using Lemma 3, we get
N

u(x N )

PC u(x N )

2

C

2

L ( )

2

2x N C

e

w

2
Gx

.
N

Lemma 2 and the triangle inequality lead to
u , (x N )
2C

2

e

u(x N )
2x N C

2

2 u , (x N )

2

L( )

u(x N )

2
Gx

4 exp(2(T

PC u(x N )

2

2 u(x N )

2

L( )

PC u(x N )

2
L2 ( )

2

x N )C

L2 ( )

N

T

+4K F2 (M )(T

xN )

x N )C ) u

exp(2(

,

( )

u( )

xN

Multiplying (15) by e

e

2x N C

2x N C

2
L2 ( )

d .

(15)

and applying Gronwalls inequality, we get

u , (x N ) u(x N )

2
2

L( )

2C

2

sup u(x N )

0 xN T

2
Gx

4e

2TC

2

exp 4K F2 (M )T 2 ,

N

which leads to the desired result

u , (x N )

u(x N )

2

L2 ( )

2C

2

I 22

4e

2TC

2

exp 2K F2 (M )T 2 e

xNC

.

This completes the proof.

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Science & Technology Development, Vol 5, No.T20- 2017
The next theorem provides an error estimate in the Hilbert scales {H p ( )}p

which is equipped

with a norm defined by
2

f

(n12

...

p

H ( )

n1 1 n2 1

nN

n22

2

nN2 1 )p f ,

...

n1n2 ...nN

1

1

.

1

Theorem 3. Assume that the problem (1) has a unique u which satisfies (13). Let us choose C  and M
2

such that lim C peTC

)T 2

0 and lim eKF (M

0

0

u , (x N ) u(x N )

1)e

( 2

p

H ( )

e

C

KF2 (M )T 2

KF2 (M )T 2

Cp

0

C

e

TC

C pe

lim e

I3

2e

...

nN2

KF2 (M )T 2

TC

0 , then we have
xN C

C pe

e

xN

,

[0,T ].

Proof. First, we have
u , (x N )

2

PC u(x N )

n12

p

H ( )

n1 ,n2 ,...,nN
n12

u , (x N )

PC u(x N )

2

n22

n22

1

1

... nN2

u , (x , x N ) u(x , x N ),

n1n2 ...nN

1

(x )

2

C

1

C 2p

Hp( )

p
1

u
n1 ,n2 ,...,nN

,

1

1

2
n12 n22 ... nN

1

(x , x N )

u(x , x N ),

n1n2 ...nN

1

(x )

2

C 2p u

,

(x N )

u(x N )

C

It follows from theorem 2 that
u

,

(x N )

PC u(x N )

Hp( )

On the other hand, we consider the function
G( )
p 1

Since G ( )

D

e

i.e., 2(x N

n12

n22

p

e

2e

2 C

sup u(x N )

0 xN T

D

2
Gx0

nN2

...

p

n22

1

exp

...

nN2

...

2(x N

nN2

4 2e

2TC

(16)

N

D>0.

,

D ) , it follows that G is decreasing when D

(p

p , then for n12

)C

xN C

exp(2K F2 (M )T 2 )C pe

p . Thus if

e

p (T
2

)

C 2 , we get

1

) n12

n22

nN2

...

1

C 2 pe

2(x N

)C

,

and

u(x N )

PC u(x N )

2
Hp( )

n12
n1 ,n2 ,...,nN
n12

n22

...

C 2 p exp

1

n22

p
1

nN2 1

C e

1

(x )

2

C

2(x N

)C

)C

) n12

exp 2(x N
1

1

n12 n22 ... nN2
(x N

n1n2 ...nN

1

n1 ,n2 ,...,nN

2p

u(x , x N ),

sup u(x N )

0 xN T

Gx0

1

n22

...

nN2

1

u(x , x N ),

n1n2 ...nN

1

(x )

C

.

N

Therefore

u(x N )

PC u(x N )

Hp( )

C pe

(x N

Combining (16) and (17), we get

Trang 200

)C

sup u(x N )

0 xN T

Gx0

N

(17)

2

2
L2 ( )

.


TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017
u , (x N )

u(x N )

u , (x N )

Hp( )

exp 2K F2 (M )T 2
a2

The inequality

b2

2e

2 C

PC u(x N )

sup u(x N )

0 xN T

a

u , (x N ) u(x N )

b for a, b
p

H ( )

0,

4 2e

2TC

sup u(x N )

0 xN T

KF2 (M )T 2

1)e

1

ln

C

e

r ln

2T

2e

T

CONCLUSION

C

xNC

.

C pe

2I

2
2

In this paper, we investigate the Cauchy
problem for a ND nonlinear elliptic equation in a
bounded domain. We apply the Fourier truncation
method for regularizing the problem. Error
estimates between the regularized solution and
exact solution are established in HP space under
some priori assumptions on the exact solution. In
future, we will

4

TC

e

C pe

2 2

T

1

xNC

.

,
TC

lim exp(2KF2 (M )T 2 ) e
0

Then (14) becomes
L2 ( )

e

N

KF2 (M )T 2

ln

0

u(x N )

Gx0

(0,1) and M such that

, for

1

I3

. It is easy to check that lim exp(2KF2 (M )T 2 )C
u , (x N )

Hp( )

0 leads to

T

KF M
for r

2
Gx0

PC u(x N )

N

( 2

Remark 1. In theorem 2, let us choose C

u(x N )

Hp( )

ln

1

2

T

ln

1

r

0.

xN
T

.

consider the Cauchy problem for a coupled
system for nonlinear elliptic equations in three
dimensions.
Acknowledgment: The author thanks the
anonymous referees for their valuable suggestions
and comments leading to the improvement of the
paper..

Đánh giá HP cho bài tốn Cauchy cho
phương trình elliptic phi tuyến
 Lê Đức Thắng
Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM
Trường Cao Đẳng Cơng Thương TPHCM

TĨM TẮT
Trong bài báo này, chúng tơi nghiên cứu bài
tốn Cauchy cho phương trình elliptic phi tuyến
trên miền bị chặn trong khơng gian nhiều chiều.
Như đã biết, bài tốn này là khơng chỉnh. Chúng
tơi sử dụng phương pháp chặt cụt Fourier để
chỉnh hóa nghiệm của bài tốn. Đánh giá sai số

giữa nghiệm chỉnh hóa và nghiệm chính xác được
thiết lập trong khơng gian HP với các giả thiết cho
trước về tính trơn của nghiệm chính xác.

Trang 201

Science & Technology Development, Vol 5, No.T20- 2017
Từ khóa: phương trình elliptic phi tuyến, bài toán không chỉnh, chỉnh hóa, phương pháp chặt cụt

TÀI LIỆU THAM KHẢO
[1]. L. Bourgeois, J. Dard, About stability and
regularization of ill-posed elliptic Cauchy problems:
the case of Lipschitz domains, Appl. Anal., 89,
1745–1768 (2010).
[2]. L. Elden and F. Berntsson, A stability estimate for a
Cauchy problem for an elliptic partial differential
equation, Inverse Problems, 21, 1643–1653 (2005).
[3]. X.L. Feng, L. Elden, C.L. Fu, A quasi-boundaryvalue method for the Cauchy problem for elliptic
equations with nonhomogeneous Neumann data, J.
Inverse Ill-Posed Probl., 18, 617–645 (2010).
[4]. J. Hadamard, Lectures on the Cauchy Problem in
Linear Differential Equations, Yale University
Press, New Haven, CT (1923).
[5]. D.N. Hao, N.V. Duc, D. Sahli, A non-local
boundary value problem method for the Cauchy
problem for elliptic equations, Inverse Problems,
25:055002 (2009).
[6]. Dan Henry, Geometric Theory of Semilinear
Parabolic Equations, Springer-Verlag, Berlin
Heildellberg, Berlin (1981).
[7]. V. Isakov, Inverse Problems for Partial Differential
Equations, volume 127 of Applied Mathematical
Sciences. Springer, New York, second edition,
(2006).
[8]. Z. Qian, C.L. Fu, Z.P. Li, Two regularization
methods for a Cauchy problem for the Laplace

equation, J. Math. Anal. Appl., 338, 479–489
(2008).
[9]. T. Reginska, R. Kazimierz, Approximate solution of
a Cauchy problem for the Helmholtz equation,
Inverse Problems, 22, 975–989 (2006).

Trang 202

[10]. D.D. Trong, N.H. Tuan, Regularization and error
estimate for the nonlinear backward heat problem
using a method of integral equation, Nonlinear
Anal., 71, 4167–4176 (2009).
[11]. N.H. Tuan, T.Q. Viet, N.V. Thinh, Some remarks
on a modified Helmholtz equation with
inhomogeneous source. Appl. Math. Model, 37,
793–814 (2013).
[12]. N.H. Tuan, D. D. Trong and P. H. Quan, A note on
a Cauchy problem for the Laplace equation:
regularization and error estimates, Appl. Math.
Comput ., 217, 2913–2922 (2010).
[13]. H. Zhang, T. Wei, A Fourier truncated
regularization method for a Cauchy problem of a
semi-linear elliptic equation, J. Inverse Ill-Posed
Probl., 22, 143–168 (2014).
[14]. N.H. Tuan, L.D. Thang, V.A. Khoa, T. Tran, On
an inverse boundary value problem of a nonlinear
elliptic equation in three dimensions. J. Math. Anal.
Appl. 426 , 1232–1261 (2015).
[15]. V.B. Glasko, E. A. Mudretsova, V. N. Strakhov,
Inverse problems in the gravimetry and

magnetometry, Ill-Posed Prob-lems in the Natural
Science ed A N Tikhonov and A V Goncharskii
(Moscow: Moscow State University Press), C. 89–
102 (1987).
[16]. L. Bourgeois, A stability estimate for ill-posed
elliptic Cauchy problems in a domain with corners,
C. R. Math. Acad. Sci. Paris, 345, 385–390 (2007).