Calculus of Variations and Geometric Measure Theory
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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


Published Paper

Inserted: 17 dec 2001
Last Updated: 6 jul 2002

Journal: Z. Anal. Anwendungen
Volume: 20
Number: 4
Pages: 959-985
Year: 2001


We prove local gradient bounds and interior Hölder 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 $q/p<(n+1)/n$.

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