A problem with an obstacle that goes out to the boundary of the domain for a class of quadratic functionals on $\mathbb R^n$

A variational problem with obstacle is studied for a quadratic functional defined on vector-valued functions $u\colon\Omega\to\mathbb R^N$, $N>1$. It is assumed that the nondiagonal matrix that determines the quadratic form of the integrand depends on the solution and is “split”. The role of the obstacle is played by a closed (possibly, noncompact) set $\mathcal K$ in $\mathbb R^N$ or a smooth hypersurface $S$. It is assumed that $u(x)\in\mathcal K$ or $u(x)\in S$ a.e. on $\Omega$. This is a generalization of a scalar problem with an obstacle that goes out to the boundary of the domain. It is proved that the solutions of the variational problems in question are partially smooth in $\overline\Omega$ and that the singular set $\Sigma$ of the solution satisfies $H_{n-2}(\Sigma)=0$.