Analytic solutions of extremal problems for the Laplace's equation (Q2795502)

The authors discuss two extremal problems for the Laplace equation in the unit disk \(U\). Given \(f\in L_2(U)\), the first problem is to find a harmonic function \(u\in L_2(U)\) such that \(\int_U |u-f|^2 dx \to \min\). The second problem has the form \(\lambda \int_U |u\;- \;f|^2 dx + \int_{\partial U} |u-f|^2 ds \to \min\) with a given \(\lambda>0\). For collection, see the Adamjan-Arov-Krein theorem [\textit{V. M. Adamjan} et al., Math. USSR, Sb. 15, 31--73 (1972; Zbl 0248.47019)].