ESTIMATE OF THE SINGULAR SET OF THE SOLUTION TO VARIATIONAL PROBLEM WITH A NONCONVEX OBSTACLE THAT GOES OUT TO THE BOUNDARY

摘要

We consider a variational problem for a quadratic functional on the class of vector-valued functions u : Ω- →R~n, N > 1, such that u(Ω) ⊂ X, where Ω is contained in R~n , where Ω is a bounded domain in R~n, n≥2, and X is the closure of a nonconvex bounded domain in R~n with smooth boundary of class C~2. Under the assumption that X is a bounded set diffeomorthic to a closed ball in 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.