A Posteriori Error Estimates for Approximate Solutions to the Obstacle Problem for the $p$ -Laplacian

The paper is concerned with a functional identity and estimates that are fulfilled for the measures of deviations from exact solutions of the obstacle problem for the $p $-Laplacian. They hold true for any functions from the corresponding (energy) functional class, which contains the generalized solution to the problem as well. We do not use any special properties of approximations or numerical methods nor information on the exact configuration of the coincidence set. The right-hand side of the identities and estimates contains only known functions and can be explicitly calculated, and the left-hand side represents a certain measure of the deviation of the approximate solution from the exact one. The right-hand side of the identity and estimates contains only known functions and can be explicitly calculated, while and the left-hand side represents a certain measure of the deviation of the approximate solution from the exact one. The obtained functional relations allow one to estimate the error of any approximate solutions to the problem regardless of the method of how they are obtained. In addition, they enable one to compare the exact solutions to problems with different data. The latter provides the possibility to estimate the errors of mathematical models.