## M. Bildhauer - M. Fuchs - G. Mingione

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

created on 17 Dec 2001

modified on 06 Jul 2002

[

BibTeX]

*Published Paper*

**Inserted:** 17 dec 2001

**Last Updated:** 6 jul 2002

**Journal:** Z. Anal. Anwendungen

**Volume:** 20

**Number:** 4

**Pages:** 959-985

**Year:** 2001

**Abstract:**

We prove local gradient bounds and interior HÃ¶lder
estimates for the first derivatives of functions $u \in W^{1,1}_{loc} (\Omega)$
which locally minimizes the variational integral $$I(u) = \int f (Du) \ dx$$
subject to the double side condition $\phi_1 \leq u \leq \phi_2$.
We establish these results for various classes of integrands $f$
with non-standard growth. For example, in the case of smooth $f$ the
$(s,m,q)$-condition is sufficient, for a suitable choice of the paramenters.
A second class consists of all convex functions $f$ with
$(p,q)$-growth, where $f$ is not supposed to be differentiable everywhere; the condition on
$(p,q)$ is $q/p<(n+1)/n$.