Cauchy problem for Laplace equation in a strip is considered. The optimal error bounds between the exact solution and its regularized approximation are given, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. We also provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds.

The Cauchy problem for the Laplace equation in particular, and for other elliptic equations in general, occurs in the study of many practical problems in areas such as plasma physics [

In this paper, we will concretely consider the following Cauchy problem for Laplace equation in a strip:

In [

The motivation of this paper is inspired by Tautenhahn, in [

Let

In order to obtain explicit stability estimate for problem (

This paper is organized as follows: In Section

We consider arbitrary ill-posed inverse problem [

Assume we want to solve (

Let us assume that

In order to derive explicitly the optimal error bounds for the worst case error

The function

Under Assumption

Let

In the following we consider two special methods: the method of generalized Tikhonov regularization and the method of generalized singular value decomposition.

For the method of generalized Tikhonov regularization, a regularized approximation

Let

The regularized approximation

For this method the following result holds [

Let

Let us formulate the problem (

Due to Parseval formula,

Due to the equivalence of conditions (

For operator equation (

From (

The function

The function

(i)

(ii)

(iii)

(iv)

(v) For the inverse function

(vi) The function

The continuity of function

(i) From (

(ii) From (

(iii) From (ii) we know that

(iv) According to (iii),

(v) From (

(vi) We know that the function

The proof is finished.

Now we formulate our main result of this section concerning the best possible worst case error

Applying Theorem

Let

(i) In case

(ii) In case

(iii) In case

From (

(i) For

(ii) In case

(iii) In case

In this section we consider two special regularization methods, apply them to problem (

The method of generalized Tikhonov regularization (

Applying Theorem

Consider the operator equation (

(i) In the case

(ii) In the case

From Theorem

(i) In the case

(ii) In the case

Now we consider the method of generalized singular value decomposition. Due to (

Applying Theorems

Consider the operator equation (

(i) In the case

(ii) In the case

From Theorem

(i) In the case

(ii) In the case

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partly supported by the Fundamental Research Funds for the Central Universities No. 27R1410016A.