JournalszaaVol. 20, No. 4pp. 959–985

A Priori Gradient Bounds and Local C1,αC^{1, \alpha}-Estimates for (Double) Obstacle Problems under Non-Standard Growth Conditions

  • M. Bildhauer

    Universität des Saarlandes, Saarbrücken, Germany
  • Martin Fuchs

    Universität des Saarlandes, Saarbrücken, Germany
  • Giuseppe Mingione

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

We prove local gradient bounds and interior Hölder estimates for the first derivatives of functions uW1,loc1(Ω)u \in W^1_{1, loc} (\Omega) which locally minimize the variational integral I(u)=Ωf(u)dxI(u) = \int _{\Omega} f (\bigtriangledown u) dx subject to the side condition ψ1uψ2\psi _1 ≤ u ≤ \psi_2. We establish these results for various classes of integrands ff with non-standard growth. For example, in the case of smooth ff the (s,μ,q)(s, \mu, q)-condition is sufficient. A second class consists of all convex functions ff with (p,q)(p, q)-growth.

Cite this article

M. Bildhauer, Martin Fuchs, Giuseppe Mingione, A Priori Gradient Bounds and Local C1,αC^{1, \alpha}-Estimates for (Double) Obstacle Problems under Non-Standard Growth Conditions. Z. Anal. Anwend. 20 (2001), no. 4, pp. 959–985

DOI 10.4171/ZAA/1054