Lipschitz continuity of solutions and corresponding multipliers to distributed and boundary semilinear elliptic optimal control problems with mixed pointwise control-state constraints

This paper is concerned with the existence and regularity of mininizers as well as of corresponding multipliers to an optimal control problem governed by semilinear elliptic equations, in which mixed pointwise control-state constraints are considered in a quite general form and the controls act simultaneously in the domain and on the boundary. Under standing assumptions, the minimizers and the corresponding multipliers do exist. Furthermore, by applying the bootstrapping technique and establishing some calculation tools for functions in Sobolev spaces of fractional order, the optimal solutions and the associated Lagrange multipliers are shown to be Lipschitz continuous.