The regularity of minima of variational problems with graph obstacles (Q1102522)

\textit{F. Tomi} [Math. Z. 128, 43-74 (1972; Zbl 0243.49023)] showed that continuous minima \(u: {\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N\), n,N\(\geq 2\), of Dirichlet's integral \[ E(u):=\int _{\Omega}| Du| ^ 2 dx, \] under the side condition \(u^ N(x)\geq f(x,u^ 1(x),...,u^{N-1}(x))\) are of class \(C^{1,\alpha}(\Omega)\) provided \(f: \Omega\) \(\times {\mathbb{R}}^{N-1}\to {\mathbb{R}}\) satisfies some mild regularity assumptions. In our paper we prove that the initial regularity hypothesis of Tomi holds for weak minimizers of Sobolev class \(H^{1,2}\), hence weak solutions of the graph obstacle problem are regular on the whole domain of definition.