  In this paper we study an optimal control problem associated to a linear
degenerate elliptic equation with mixed boundary conditions. The equations of
this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak
solutions. We adopt the weight function as a control in $L^1(\Omega)$. Using
the direct method in the Calculus of variations, we discuss the solvability of
this optimal control problem in the class of weak admissible solutions.
