A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

Abstract

We investigate the Cauchy problem for linear elliptic operators with C ^∞–coefficients at a regular set Ω ⊂ R ², which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ⊂ ∂Ω and our goal is to reconstruct the trace of the H ¹(Ω) solution of an elliptic equation at ∂Ω/Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

^†Supported by CNPq under grant 200049-94.1 and FWF project F–1308 within Spezial-forschungsbereich 13.

Acknowledgments

Notes

^†Supported by CNPq under grant 200049-94.1 and FWF project F–1308 within Spezial-forschungsbereich 13.

¹For a formal definition of well posed problems, see e.g. [Ba] or [EHN].

²The Cauchy data (f, g) is called consistent if the corresponding problem CP has a solution. Otherwise (f, g) is called inconsistent Cauchy data.

³For details on the definition of the Sobolev spaces see [Ad] or [DaLi].

⁴The corresponding definitions can be found in [BrPe].

⁵See [Je] for a complete proof.

⁶It is obvious that this Picard iteration may fail to converge in this general framework.

⁷Due to the geometrical interpretation, we adopt the notation used in [Gr] and [EnSc]. In [Do], matrices satisfying iv) are called normal matrices.

⁸For simplicity we denote by ∥·∥ the norm of the Hilbert space H, but meant is the norm ∥·∥_* introduced in Definition 4.

⁹See URL http://www.netlib.org for details.