On an interpolation problem with a minimum value of the Laplace operator

We consider the problem of interpolation of finite sets of numerical data by smooth functions that are defined on a plane square and vanish on its boundary. Under some constraints on the location of interpolation points inside the square, close upper and lower estimates with the same dependence on the number of interpolation points are obtained for the $L_\infty$-norms of the Laplace operator of the best interpolants on the class of bounded interpolation data. Exact solutions are found for the cases of interpolation at one point and at two points.