Optimal boundary control problem for ill-posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions

The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet-Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first-order optimality system assuming the optimal solution is slightly more regular.