On minima of variational problems with some nonconvex constraints (Q1084329)

Suppose \(u: \Omega\subset {\mathbb{R}}^ n\to {\mathbb{R}}^ N\) is a minimizer of the Dirichlet-integral in a set K, defined by a graph-obstacle: \[ K=\{u| u-u_ 0\in \overset\circ H^ 1_ 2(\Omega)^ N,\quad u^ N(x)\geq g(x,u^ 1(x),...,u^{N-1}(x))\quad a.e.\}. \] \textit{F. Tomi} [Math. Z. 128, 43-74 (1972; Zbl 0243.49023)] proved \(C_{1,\alpha}\)-regularity of u in the case of \(\Omega \subset {\mathbb{R}}^ 2\), while the higher-dimensional case and generalized problems were studied and partial regularity proved in several papers by \textit{F. Duzaar} and \textit{M. Fuchs} [see e.g. Manuscr. Math. 56, 209-234 (1986; Zbl 0587.49012) and Math. Z. 191, 585-591 (1986; Zbl 0568.49009)]. In our paper \(C_{\alpha}\) (and hence also \(C_{1,\alpha})\) a-priori-estimates for minimizers are given under certain conditions on g, as e.g. \(g=g(x,| \tilde u|^ 2)\) with \(\partial g(x,t)/\partial t\geq 0\) or \(N=2\) and \(\partial g(x,u^ 1)/\partial u_ 1<0\). With a different technique the regularity of minimizers with respect to arbitrary graph-obstacles -- but without a-priori-estimates -- can be proved, which will be presented in a forthcoming paper by M. Fuchs and the author.