Boundary control of semilinear elliptic equations - existence of optimal solutions

A class of semilinear elliptic boundary control problems is considered. Owing to measurable and bounded controls we use the weak solution approach for the state equation. For the derivation of solvability results we present two methods. The first method works with the complete continuity of the state mapping, whereas in the second one theorems about the separation of convex sets and measurable selection are applied to overcome the complete continuity of the state mapping.