The best approximation of Laplace operator by linear bounded operators in the space $L_p$

We obtain close two-sided estimates for the best approximation of the Laplace operator by linear bounded operators on the class of functions, for which the second degree of the Laplace operator belongs to the $L_p$-space. We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class defined with an error. In a particular case ($p=2$) we solve all three problems exactly.