H estimation for the Cauchy problem for nonlinear elliptic equation

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

Where T is a positive constant, 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].
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 ( , ) N u x x . By using the method of separation of variables, we can show that solution of the problem has the form  (3) is ordinary differential equations. It is easy to see that its solution is given by

REGULARIZATION AND ERROR ESTIMATE FOR NONLINEAR PROBLEM WITH LOCALLY LIPSCHITZ SOURCE
We know from (4) that, when 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 .  We show that the solution , u of problem (6) satisfies the following integral equation This completes the proof.
Lemma 2. Let u be the exact solution to problem (1). Then we have the following estimate 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)  Proof. We prove the equation (7) has a unique solution for all We consider some assumptions on the exact solution as the following: exp(2 ( ) ) .
Proof. Since N x uG then using Lemma 3, we get Lemma 2 and the triangle inequality lead to

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The next theorem provides an error estimate in the Hilbert scales { ( )} It follows from theorem 2 that On the other hand, we consider the function 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 H P space under some priori assumptions on the exact solution. In future, we will consider the Cauchy problem for a coupled system for nonlinear elliptic equations in three dimensions.