Recovery of harmonic functions from partial boundary data respecting internal pointwise values
Juliette Leblond, Dmitry Ponomarev · Journal of Inverse and Ill-Posed Problems · 2016
Abstract We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary ∂ Ω ${\partial\Omega}$ . Assuming Dirichlet and Neumann data available on Γ ⊂ ∂ Ω ${\Gamma\subset\partial\Omega}$ to be real-valued functions in W 1 / 2 , 2 ( Γ ) ${W^{1/2,2}(\Gamma)}$ and L 2 ( Γ ) ${L^{2}(\Gamma)}$ classes, respectively, we develop a non-iterative method for solving this ill-posed Cauchy problem choosing L 2 ${L^{2}}$ bound of the solution on ∂ Ω ∖ Γ ${\partial\Omega\setminus\Gamma}$ as a regularizing parameter. 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.