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

The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by u" (t) - Au( t) = 0, 0 < t < T, u(0) = phi, u' (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 alpha = A( I + alpha A)(-1) and u(0) = phi 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.