  We consider optimal control problems, where the control appears in the main
part of the operator. We derive the Pontryagin maximum principle as a necessary
optimality condition. The proof uses the concept of topological derivatives. In
contrast to earlier works, we do not need continuity assumptions for the
coefficient or gradients of solutions of partial differential equations.
Following classical proofs, we consider perturbations of optimal controls by
multiples of characteristic functions of sets, whose scaling factor is send to
zero. For $2d$ problems, we can perform an optimization over the elliptic
shapes of such sets leading to stronger optimality conditions involving a
variational inequality of a new type.
