Local estimates for functionals depending on the gradient with a perturbation

This paper concerns minimization problems from Calculus of Variations depending on the gradient and with a linear perturbation. Inspired in qualitative properties that are valid for elliptic partial differential equations, it presents some local estimates near non extremum points as well as extremum points. These estimates are inspired on a class of functions given by A. Cellina in [2]. Also, a comparison result with respect to these functions is presented. Finally, some local estimates are obtained for the difference between the supremum and the infimum of any solution to the problems considered.