Regularization method for an ill-posed Cauchy problem for elliptic equations

Abstract

Abstract The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by u ′′ ⁢ ( t ) - A ⁢ u ⁢ ( t ) = 0 {u^{\prime\prime}(t)-Au(t)=0} , 0 < t < T {0<t<T} , u ⁢ ( 0 ) = φ {u(0)=\varphi} , u ′ ⁢ ( 0 ) = 0 {u^{\prime}(0)=0} , where A is a positive self-adjoint and unbounded linear operator. The problem is severely ill-posed in the sense of Hadamard [23]. We shall give a new regularization method for this problem when the operator A is replaced by A α = A ⁢ ( I + α ⁢ A ) - 1 {A_{\alpha}=A(I+\alpha A)^{-1}} and u ⁢ ( 0 ) = φ {u(0)=\varphi} is replaced by a nonlocal condition. We show the convergence of this method and we construct a family of regularizing operators for the considered problem. Convergence estimates are established under a priori regularity assumptions on the problem data. Some numerical results are given to show the effectiveness of the proposed method.