Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation

A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven; a convergence estimate of Hölder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution; some numerical results show that this method works well.


Introduction
Let Ω be a bounded, connected domain in ( ) with a smooth boundary ∂Ω and assume that H is a real Hilbert space.We consider the following Cauchy problem of a semi-linear elliptic partial differential equation Further, we suppose ( ) be the eigenvalues of the operator x L , i.e., for the boundary value problem in , 0, on , there exists a nontrivial solution n X H ∈ .And ( ) 0 and lim .
Our problem is to determine ( ) , u y ⋅ from problem (1.1).Problem (1.1) is severely ill-posed, i.e., a small perturbation in the given Cauchy data may result in a dramatic error on the solution [1].Thus regularization techniques are required to stabilize numerical computations, (see [1] [2]).We know that, as the right term 0 f = , it is the Cauchy problem of the homogeneous elliptic equations.For the homogeneous problem, there have many regularization methods to deal with it, (see [3]- [8]).We note that, these references mainly consider the Cauchy problem of linear homogeneous elliptic operator equation, but the literature which involves the semi-linear cases is quite scarce.In 2014, [9] considered the problem (1.1), where the authors used Fourier truncated method to solve it and derived the convergence estimate of logarithmic type.Recently, there are some similar works about the Cauchy problem for nonlinear elliptic equation, and they have been published, such as [10] [11].
In the present paper, we adopt a filtering function method to deal with this problem.The idea of this method is similar to the ones in [4] [5] [12] [13], etc.However, note that our method here is new and different from them in the above references (see Section 2).Meanwhile we will derive the convergence estimate of Hölder type for this method, which is an improvement for the result in [9].This paper is organized as follows.In Section 2, we use the filtering function method to treat problem (1.1) and prove some well-posed results (the existence, uniqueness and stability for the regularization solution).In Section 3, a Hölder type convergence estimate for the regularized method is derived under an a-priori bound assumption for the exact solution.Numerical results are shown in Section 4. Some conclusions are given in Section 5.

Filtering Function Method
We assume there exists a solution to problem (1.1), then it satisfies the following nonlinear integral equation (see [9]) here, n X are the orthonormal eigenfunctions for the operator x L , and From (2.1), we can see that the functions ( ) so in order to guarantee the convergence of solution ( ) , u y x , the high frequencies( n → ∞ ) of two functions need to be eliminated.Therefore, a natural way is to use a filter function ( ) , n q α λ to filter out the high frequencies of ( ) where δ is the error level, ⋅ is the H-norm.According to the above description, for 0 r > , we choose the filter function ( ) , and define the following regularization solution ( ) where, , , , , , In fact, it can be verified that (2.4) satisfies the following mixed boundary value problem formally .
Our idea is to approximate the exact solution (2.1) by the regularization solution (2.4), i.e., using the solution of (2.5) to approximate the one of (1.1).

Some Well-Posed Results
Let 0 1 α < < , 0 x > , for the fixed 0 y T r τ ≤ ≤ ≤ + , we define the function x , and from ( ) note that, when 0 τ = , it can be obtained that ( ) Now, we prove that the problem (2.4) is well-posed (existence, uniqueness and stability for the regularization solution), the proof mentality of Theorem 2.1 mainly comes from the references [14], which describes the existence and uniqueness for the solution of (2.4).

Convergence Estimate
In this section, under an a-priori bound assumption for the exact solution a convergence estimate of Hölder type for the regularization method is derived.The corresponding result is shown in Theorem 3.
Proof.Denote u α be the solution of problem (2.4) with exact data ϕ denotes a linear densely defined self-adjoint and positive-definite operator with respect to x.The function ϕ is known, and 1 :

τ
attain unique maximum at the point 0

2 . 2
By the uniqueness of the fixed point of , we give and prove the stability of the regularization solution.Theorem Suppose f satisfies (1of problem (2.4) corresponding to the measured datum 1 δ