VARIATIONAL PROBLEM WITH AN OBSTACLE IN R~N FOR A CLASS OF QUADRATIC FUNCTIONALS

摘要

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