Interpolation on a square with a minimum value of the uniform norm 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, we obtain two-sided estimates with a correct dependence on the number of interpolation points for the $L_\infty$-norms of the Laplace operator of the best interpolants. For the case of interpolation at one point, which is the center of the square, we find an exact solution.