Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints

We present a method for deriving first order necessary conditions of optimal controls for semilinear second order elliptic partial differential equations and variational inequalities with various kinds of state constraints.The main tools used are the Ekeland variational principle and the spike variation technique for elliptic equations which is a combination of the ideas from vector valued measure theory and the representation of solutions for second order partial differential equations via Green's functions.