A priori gradient bounds and local \(C^{1,\alpha}\)-estimates for (double) obstacle problems under non-standard growth conditions
The paper deals with local gradient bounds and interior Hölder regularity for the first derivatives of functions \(u\in W^{1,1}_{\text{ loc}}(\Omega)\) which minimize locally the variational integral \[ \int_\Omega f(\nabla u)dx \] and are subject to the side condition \(\Psi_1\leq u\leq \Psi_2\) almost everywhere in a bounded domain \(\Omega\subset {\mathbb R}^n,\) \(n\geq 2.\) The function \(f\colon {\mathbb R}^n\to [0,\infty)\) is a strictly convex one and satisfies suitable non-standard growth conditions as \(|Z|\to\infty.\) In particular, these include the case of \(f(Z)=|Z|^p\log (1+|Z|),\) its iterated version and also integrands of \((p,q)\)-growth.
scientific article; zbMATH DE number 1747399
scientific article