A Priori Gradient Bounds and Local $C^{1, \alpha}$-Estimates for (Double) Obstacle Problems under Non-Standard Growth Conditions

Abstract

We prove local gradient bounds and interior Hölder estimates for the first derivatives of functions u \in W^1_{1, loc} (\Omega) which locally minimize the variational integral I(u) = \int _{\Omega} f (\nabla u) dx subject to the side condition \psi _1 ≤ u ≤ \psi_2 . We establish these results for various classes of integrands f with non-standard growth. For example, in the case of smooth f the (s, \mu, q) -condition is sufficient. A second class consists of all convex functions f with (p, q) -growth.