Continuity regularity of optimal control solutions 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 minimizers 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. The L2- and Lp-type regularization is considered for both distributed and boundary controls. Under standing assumptions, the minimizers and the corresponding multipliers do exist. Furthermore, by applying the bootstrapping technique and using some calculation tools for functions in Sobolev spaces of fractional order, the optimal solutions are shown to be Lipschitz continuous when the L2-type regularization is applied and they are proven to be Hölder continuous with the exponent θ=1p−1 if only Lp-type regularization is used.