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

Abstract

We prove local gradient bounds and interior Hölder estimates for the first derivatives of functions $u\in W_{1,loc}^1(\Omega )$ which locally minimize the variational integral $I(u)={\int }_{\Omega }f(\nabla u)dx$ subject to the side condition ${\psi }_1\le u\le {\psi }_2$. 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.