Necessary conditions for elliptic control problems with control and state constraints

Let $\Omega \subset \rm I\! R^2\,$ be a bounded domain with piecewise smooth boundary $\, \Gamma = \partial\,\bar{\Omega}\,$. The derivative in the direction of the outward unit normal $\,\nu\,$ of $\,\Gamma\,$ will be denoted by $\,\partial_{\nu}\,$. Suppose that the boundary is partioned as $\, \Gamma = \Gamma_1 \cup \Gamma_2\,$ with disjoints sets $\,\Gamma_1, \Gamma_2 \subset \Gamma\,$ that consist of finitely many connected components.