A non-local boundary value problem method for the Cauchy problem for elliptic equations

Let H be a Hilbert space with norm || ⋅ ||, A:D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and given positive numbers. The ill-posed Cauchy problem for elliptic equations is regularized by the well-posed non-local boundary value problem with a ⩾ 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.