OPTIMALITY CONDITIONS FOR ELLIPTIC VARIATIONAL-INEQUALITIES

By the penalty method, Ekeland's Variational Principle and lower-semicontinuity of some set-valued mappings, necessary conditions of optimal control for some abstract elliptic variational inequalities are obtained in the cases where the convex sets satisfy some smoothness conditions. The idea is to find optimality conditions first for some penalized problems by Ekeland's Variational Principle then to pass limits to obtain the optimality conditions. It is shown that these conditions lead to some known optimality conditions in many cases. They also yield new necessary conditions for some problems. Our results give uniform forms for several known optimality conditions for elliptic variational inequalities.