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

We prove local gradient bounds and interior Hölder estimates for the first derivatives of functions u ∈ W 1 1,loc(Ω) which locally minimize the variational integral I(u) = R Ω f(∇u) dx subject to the side condition Ψ1 ≤ u ≤ Ψ2. We establish these results for various classes of integrands f with non-standard growth. For example, in the case of smooth f the (s, μ, q)condition is sufficient. A second class consists of all convex functions f with (p, q)-growth.