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

Abstract

Abstract. A variational problem with obstacle is studied for a quadratic functional defined on vector-valued functions u : Ω → 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 K in R N or a smooth hypersurface S. It is assumed that u(x) ∈ K or u(x) ∈ S a.e. on Ω. 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 s Ω and that the singular set Σ of the solution satisfies H n−2 (Σ) = 0.