A regularization method for a Cauchy problem of Laplace's equation in an annular domain
The Cauchy problem \[ \Delta u=0\quad \text{in} \quad S, \] \[ u= f \quad on \quad \Gamma,\quad \frac{du}{d\nu}= g \quad \text{on} \quad\Gamma, \] is defined in an annular domain \( S\subset \mathbb R^{2}\) with the boundary \( dS=\Gamma\;U \gamma \), where \( \Gamma\) and \(\gamma\) are two circles of radius \( r_{1}>r_{2}\). The functions \(f\) and \(g\) cannot be known exactly and it is assumed that \[ \|f^{\delta} - f\|\leq \delta, \quad \|g^{\delta} -g \|\leq \delta. \] To obtain a stable approximate solution of this problem the authors use the Tikhonov regularization parameter \(\alpha\) in the minimization of the functional \[ I(v)= \|v - f^{\delta}\|^{2} + \omega\|\frac{dv}{d\nu}-g^{\delta}\|^{2}+\alpha\|v\|^{2} \] in the space of fundamental solutions corresponding to uniformly located points \(Q_{1i}\) and \(Q_{2i}\) on the circles \(R_{1}\) and \(R_{2}\).
scientific article; zbMATH DE number 6107081
scientific article