Recovery of harmonic functions in planar domains from partial boundary data respecting internal values

Juliette Leblond (1) , Dmitry Ponomarev (1)

1 APICS - Analysis and Problems of Inverse type in Control and Signal processing

We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary ∂Ω. Assuming Dirichlet and Neumann data available on Γ ⊂ ∂Ω to be real-valued functions in W 1/2,2 (Γ) and L 2 (Γ) classes, respectively, we develop a non-iterative method for solving this ill-posed Cauchy problem choosing L 2 bound of the solution on ∂Ω \ Γ as a regularizing parame- ter. The present complex-analytic approach also naturally allows imposing additional pointwise constraints on the solution which, on practical side, can help incorporating outlying boundary measurements without changing the boundary into a less regular one.

Mots clés: Partially overdetermined boundary value problems; Laplace PDE; best constrained approxima- tion; interpolation in Hardy spaces; ill-posed inverse problems; regularization techniques; representation of analytic functions from partial boundary data20

Domaines: Mathématiques [math]

Juliette Leblond, Dmitry Ponomarev. Recovery of harmonic functions in planar domains from partial boundary data respecting internal values. Journal of Inverse and Ill-posed Problems, 2016, ⟨10.1515/jiip-2015-0089⟩. ⟨hal-01242160⟩