A characterization for solutions to autonomous obstacle problems with general growth

Abstract

Obstacle problems are by now well studied in the context of regularity theory, but a still pending issue is the relation between minima and extremals. The regularity of the solutions is often proved thanks to the fact that the minimizers are extremals too, i.e. they solve a variational inequality related to the problem. While in the context of standard growth conditions the literature is well established, under more general growth hypotheses there are still issues in this regard. Considering an autonomous obstacle problem, whose Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity, I will present a characterization for the unique solution in terms of extremality and a primal-dual formulation of the problem.