Interpolation by functions from a Sobolev space with minimum $L_p$-norm of the Laplace operator

We consider an interpolation problem with minimum value of the $L_p$-norm ($1\leq p<\infty$) of the Laplace operator of interpolants for a class of interpolated sequences that are bounded in the $l_p$-norm. The data are interpolated at nodes of the grid formed by points from $\mathbb{R}^n$ with integer coordinates. It is proved that, if $1\leq p$<$n/2$, then the $L_p$-norm of the Laplace operator of the interpolant can be arbitrarily small for any sequence that is interpolated. Two-sided estimates for the $L_2$-norm of the Laplace operator of the best interpolant are found for the case $n=2$.