Interpolation in a ball with a minimum value of the $L_p$-norm of the Laplace operator

We consider the problem of interpolating finite sets of numerical data bounded in $l_p$-norms ($1\leq p<\infty$) by smooth functions that are defined in an $n$-dimensional Euclidean ball of radius $R$ and vanish on the boundary of the ball. Under some constraints on the location of interpolation nodes, we obtain two-sided estimates with a correct dependence on $R$ for the $L_p$-norms of the Laplace operators of the best interpolants.