A priori gradient bounds and local C1,α-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 ∈ W1,loc1(Ω) which locally minimize the variational integral I(u) = ∫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. © Heldermann Verlag.