#### Abstract

We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

#### 1. Introduction

Throughout this paper denotes a complex Hilbert space endowed with the inner product , and the norm , stands for the Banach algebra of bounded linear operators on .

Let be a linear unbounded operator with dense domain . Assume that is self-adjoint, positive definite in , which has a continuous spectrum , .

We consider the elliptic Cauchy problem (ECP) of finding such that where is some prescribed data in the Hilbert space .

This problem is an abstract version of Cauchy problem, which generalizes Cauchy problem for second-order elliptic partial differential equations in a cylindrical domain; for example, we mention the following problem.

*Example 1. *An example of (1) is the Cauchy problem for the modified Helmholtz equation in the infinite strip [1]:
where the operator is given by
It is well known that this operator is self-adjoint with continuous spectrum
We note here that the discrete eigenfunctions expansion method cannot be used, but we can use the Fourier diagonalization method to deal with this kind of problems.

Such problem arises in many practical situations, nondestructive testing techniques [2], geophysics [3], cardiology [4], and other applications. There are many various monographs about the historical development of this topic, for more details, we refer the reader to [5, 6]. Recently there has been an excellent topic review [7] of this problem.

Because problem (1) is severely ill-posed; that is, a small perturbation in the given Cauchy data may result in a very large error on the solution. In order to overcome this instability character, the regularization methods are required.

Some regularization methods for the Cauchy problem for elliptic equations have been proposed by many authors. For instance: Tikhonov regularization method [8], the quasi-reversibility method [9], the quasi-boundary-value method [10–13], Kozlov-Maz’ya iteration method [14], and the mollification method [15].

This work is mainly devoted to theoretical aspects of the spectral regularization methods to problem (1) in the abstract setting, by considering more general self-adjoint operators when is positive and induces the elliptic case, that is, has the following properties: for any , the resolvent exists and satisfies the estimates

In the case when is a linear positive self-adjoint operator with compact inverse, problem (1) has been treated by a different method and there is a large literature in this direction. However, in the case where has a continuous spectrum the literatures are quite scarce.

In the present paper we shall use two spectral regularization methods to construct a stable solution to our original ill-posed problem.

#### 2. Preliminaries and Basic Results

In this section we present the notation and the functional setting which will be used in this paper and prepare some material which will be used in our analysis.

##### 2.1. Spectral Theorem and Properties

By the spectral theorem, for each positive self-adjoint operator , there is a unique right continuous family of orthogonal projection operators such that with

In our case, we have because , .

Theorem 2 (see [16, Theorem 6, XII.2.5, pages 1196–1198]). *Let be the spectral resolution of the identity associate to and let be a complex Borel function defined -almost everywhere on the real axis. Then is a closed operator with dense domain. Moreover*(i)*,*(ii)*, , ,*(iii)*, ,*(iv)*. In particular, if is real Borel function, then is self-adjoint, *(v)*the operator is bounded if and only if is bounded on . In this case, .*

We denote by , , the -semigroup generated by . Some basic properties of are listed in the following theorem.

Theorem 3 (see [17, chapter 2, Theorem 6.13, page 74]). *For this family of operators one has:*(1)*, ; *(2)*the function , , is analytic; *(3)*for every real and , the operator ; *(4)*for every integer and , ; *(5)*for every , , one has .*

Theorem 4. *For , is self-adjoint and one to one operator with dense range (, ).*

*Proof. *Let , . Then by virtue of (iv) of Theorem 2, we can write .

Let , , then , which implies that , . Using analyticity, we obtain that , . Strong continuity at now gives . This shows that .

Thanks to
we conclude that is dense in .

*Remark 5. *For , , this Theorem ensures us that is self-adjoint and one to one operator with dense range . Then we can define its inverse , which is an unbounded self-adjoint strictly positive definite operator in with dense domain

*Definition 6 (Hilbert scales). *Let be an unbounded self-adjoint strictly positive definite operator in . Following the definition in [18], one introduces the Hilbert scale according to definition
Here is a Hilbert space with inner product and norm .

In our setting we take . In this case, one has

Following [15], one defines the following.

*Definition 7 (mollification operator). *For and , one introduces the Yosida approximation of identity

*Remark 8. *The idea of the mollification method is very simple and natural: if the data are given inexactly, then we try to find a sequence of mollification operators which map the improper data into well-posed classes of the problem (mollify the improper data). Within these mollified data our problem becomes well-posed.

Theorem 9. *One has*(1)* and , *(2)*for all , ,*(3)*for all , .*

*Proof. *(1) .

(2) .

(3) Assume that . Then
and thus, .

Now for , there exists such that , (since is dense in (see Remark 5)). We have
Thus, , , and so .

Using the change of variables

Cauchy’s problem (1) reduces to the two Cauchy problems Thus, the solution of the original problem (1) can be written in the form It is well known that the operator generates a strongly continuous analytic semigroup . In addition, the spectral radius of the semigroup for any . Hence, it follows that the Cauchy problem (15) is well-posed and its solution may be written in the form As opposed to problem (15), Cauchy problem (15) (backward parabolic equation) is not correctly posed, and its (unique) formal solution is given by

*Remark 10 (see [19, page 375]). *The uniqueness solvability of problem (16) results from the logarithmic convexity of the function :

A useful characterization of the admissible set for which problem (16) has a solution is as follows.

Lemma 11. *Problem (16) has a solution if and only if , and its unique solution is represented by
*

*Proof. *Suppose that problem (16) has a solution . Then
The function if and only if
Observing that the function is increasing, then
This last inequality is exactly equivalent to .

As a consequence, we have the following corollary.

Corollary 12. *Problem (1) has a solution if and only if , and its unique solution is represented by
*

#### 3. Regularization and Error Estimates

##### 3.1. The Truncation Method

From (25) we can see that the term is the cause of unstability. In order to overcome the ill-posedness of problem (1), we modify the solution by filtering the high frequencies using a suitable method and instead consider (25) only for , where is some constant which satisfies .

According to spectral theory of self-adjoint operators [20], for any bounded Borel set , we can define the orthogonal projection

To solve (1) in a stable way we approximate by its projection , and instead of considering (1) with we take its projected version where is the characteristic function of the interval for . The quantity is referred to as a cut-off frequency.

Let (resp., ) be the exact (resp., the measured data) at , such that .

The approximated solution corresponding to the measured data is denoted by For simplicity, we denote the solution of problem (1) by , and the regularized solution associated to the data by .

Our first main theorem is the following theorem.

Theorem 13. *The solution defined in (27) depends continuously in on the data ; that is, if and are two regularized solutions corresponding to and , respectively, then one has
*

This inequality implies that the solution of the regularized problem (27) depends continuously on the data .

Now we compute the difference between the original solution and the approximate solution .

Theorem 14. *Let be a solution of problem (1) with the exact data ; then the following estimate holds:
*

*Proof. *From relations (25) and (27) we have
then

Using the inequality
we derive

Using (29), (30) and the triangle inequality, we obtain

This completes the proof.

*Remark 15. *If we choose , where , then we have the error bound

From (36) we see that (28) is an approximation of the exact solution . The approximation error depends continuously on the measurement error for fixed . However, as , the accuracy of the regularized solution becomes progressively lower. Consequently, we have not any information about the continuous dependence of the solution if is close to .

In the theory of ill-posed Cauchy problems, we can often obtain continuous dependence on the data for the closed interval by assuming additional smoothness and using a stronger norm.

Now we show two error estimates under the following conditions:(H1),(H2), .

*Remark 16. *In practice, we know that it is very difficult to verify the conditions (H1) and (H2), so we give different assumptions on the given data as follows:

Theorem 17. *If (resp., ), , , then one has the following estimates:
*

*Proof. *From the expansions
we have

Then
Using Theorem 17 and the triangle inequality, we can write
By choosing , we obtain the desired inequality.

Using the same techniques we have
hence
Using (29) and the triangle inequality, we obtain
By choosing , we obtain

##### 3.2. The Mollification Method

Now, we approximate the original problem (1) by the sequence of problems

Theorem 18. *If the approximate Cauchy problem (47) admits a unique solution , which depends continuously upon the data with respect to uniform topology of .*

*Proof. *From the representation
we have
(i)If , we obtain
(ii)If , the function with achieves its maximum at , , from which we deduce

From this bound, we derive

From the linear property of our problem, stability estimate of problem (47) may be written precisely in the following corollary.

Corollary 19. *If (resp., ) is the approximate solution corresponding to (resp., ), then
*

*Remark 20. *We have
It is easy to show that

This remark shows that for all .

*Proof. *The inclusion is equivalent to . We have
where .

Theorem 21. *If , then
*

*Proof. *We compute
where and .

This implies that and by virtue of of Theorem 9, we conclude the desired convergence.

The following technical lemmas play the key role in our analysis and calculations.

Lemma 22. *Let
**
where , , , , , and . Then one has
**
where , .*

*Proof. *Differentiating the expression and setting the derivative equal to zero, we find
The function admits a unique solution
Therefore
We have
By using the inequality (, ), then for , we obtain and we can write
which implies that . Hence, we obtain
where , .

Lemma 23. *Let
**
where , , , , and . Then one has the following.**If , then
**If , , , then
*

*Proof. *By a simple differential calculus, we show that the function achieves its maximum at . Consequently

Now we assume the following a priori bounds hold:

Theorem 24. *Let (resp., ) be the solution of problem (1) (resp., (47)) with the exact data . If (71) (resp., (72)) is satisfied, then one has the following error estimates:
*

*Proof. *Putting
Using the change of variables , we obtain the new expressions

By virtue of Lemma 22 (inequality (60) and Lemma 23 (inequalities (68) and (69)), we can write
where , . Consider

We have
Using (78) and (79), we derive

Combining (53), (73), and (74) with the help of triangle inequality we deduce the following corollary.

Corollary 25. *Let (resp., ) be the solution of problem (1) (resp., (47)) with the exact data (resp., the inexact data ) such that . If (71) (resp., (72)) is satisfied, then one has the following error estimates:
**
where
*

If we choose with , then we have

##### 3.3. Example: Cauchy Problem for the Modified Helmholtz Equation

In this paragraph, we give a concrete example to see how to apply the theoretical results developed in this study.

Let us consider the Cauchy problem (modified Helmholtz equation) in the infinite strip : where is a real positive constant.

Let be the Fourier transform of : With the help of the Fourier transformation, problem (1) can be transformed to an equivalent problem in the frequency domain: It is easy to check that the formal solution of problem (92) has the form or equivalently, the formal solution of problem (90) is given by

Putting . Then as . From this remark, it is easy to see that a small perturbation in the data may cause a dramatically large error in the solution . In addition, the magnifying factor is , hence, the problem is severely ill-posed.

Since the data are based on (physical) observations and are not known with complete accuracy, we assume that and satisfy where and belong to , denotes the measured data, and denotes the noise level.

For this problem, we define the regularized solutions with noisy data : where is the characteristic function of the interval . Consider where . The quantities and are the parameters which were defined in Sections 3.1 and 3.2.

#### 4. The Nonlocal Boundary Value Problem Method and Some Extensions

In this section we give the connection between the mollification method and the nonlocal boundary value problem method; also we give some extensions to our investigation.

##### 4.1. The Nonlocal Boundary Value Problem Method

Consider the following problem: (i)The case coincides with the method treated by Hào et al. in [10]. (ii)In the case , the solution of is coincides with the solution resulting from the mollification method. (iii)The error estimates obtained in our analysis by using the mollification method are similar to those obtained in [8, 10].

This shows that our study framework is more general and includes many results obtained in this direction.

##### 4.2. Generalization

Let us consider where is a self-adjoint, linear unbounded operator in and changes the sign with ( exists and ).

We assume , . The spectral theory of self-adjoint operators enables us to write

that is, the Hilbert space decomposes into the direct sum , and

This decomposition gives us two problems: one is elliptic (ill-posed) and the other is hyperbolic (well-posed). Consider

The formal solution of problem (99) is

We define We follow the same methodology developed in the previous Sections 3.1 and 3.2, we show that and are two stable approximations to problem (99), and we establish the same results of error estimates.

*Remark 26. *We define the mollification operator , where satisfies
Under certain conditions on and with the help of a technical calculation, we can extend the results obtained in Section 3.2.

#### Acknowledgments

The authors are grateful to the editor and the anonymous referees for their valuable comments and helpful suggestions which have much improved the presentation of the paper. This work is supported by the MESRS of Algeria (CNEPRU Project B01120090003).