Interpolation of scattered data in ${\mathbb{R}}^3$ using minimum $L_p$-norm networks, $1<p<\mathrm{\infty }$

We consider the extremal problem of interpolation of scattered data in ${\mathbb{R}}^3$ by smooth curve networks with minimal $L_p$-norm of the second derivative for $1<p<\mathrm{\infty }$. The problem for $p=2$ was set and solved by Nielson (1983). Andersson et al. (1995) gave a new proof of Nielson’s result by using a different approach. Partial results for the problem for $1<p<\mathrm{\infty }$ were announced without proof in (Vlachkova (1992)). Here we present a complete characterization of the solution for $1<p<\mathrm{\infty }$. Numerical experiments are visualized and presented to illustrate and support our results.