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Zeitschrift für Analysis und ihre Anwendungen


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Volume 20, Issue 4, 2001, pp. 959–985
DOI: 10.4171/ZAA/1054

Published online: 2001-12-31

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

M. Bildhauer, Martin Fuchs and Giuseppe Mingione[1]

(1) Università di Parma, Italy

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 $I(u) = \int _{\Omega} f (\bigtriangledown u) dx$ 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.

Keywords: Non-standard growth, (double) obstacle problems, a priori estimates, regularity of minimizers

Bildhauer M., Fuchs Martin, Mingione Giuseppe: A Priori Gradient Bounds and Local $C^{1, \alpha}$-Estimates for (Double) Obstacle Problems under Non-Standard Growth Conditions. Z. Anal. Anwend. 20 (2001), 959-985. doi: 10.4171/ZAA/1054