Estimate of the singular set of the solution to variational problem with a nonconvex obstacle that goes out to the boundary

Abstract

We consider a variational problem for a quadratic functional on the class of vectorvalued functions $ u:\bar{\varOmega}\to {{\mathbb{R}}^N} $ , N > 1, such that $ u\left( {\bar{\varOmega}} \right)\subset $ , where Ω is a bounded domain in $ {{\mathbb{R}}^n} $ , n ≥ 2, and is the closure of a nonconvex bounded domain in $ {{\mathbb{R}}^N} $ with smooth boundary of class C 2. Under the assumption that is a bounded set diffeomorthic to a closed ball in $ {{\mathbb{R}}^N} $ , we show that it is possible to improve the known estimate for the singular set of the solution to the variational problem. In particular, in the case n = 3, the solution may have only isolated singularities. Bibliography: 16 titles.