Variational problem with an obstacle in $\mathbb R^N$ for a class of quadratic functionals

A variational problem with an obstacle for a certain class of quadratic functionals is considered. It is assumed that admissible vector-valued functions satisfy the Dirichlet boundary condition and the obstacle is a given smooth $(N-1)$-dimensional surface $S$ in $\mathbb R^N$. It is not supposed that the surface $S$ is bounded.
It is proved that any minimizer $u$ of such an obstacle problem is a partially smooth function up to the boundary of prescribed domain. It is shown that $(n-2)$-Hausdorff measure of the set of singular points is zero. Moreover, $u$ is a weak solution of quasilinear system with two kinds of quadratic nonlinearities in the gradient. This is proved by a local penalty method. Bibl. – 25 titles.