  In this paper we perform the rigorous derivation of the topological
derivative for optimization problems constrained by a class of quasi-linear
elliptic transmission problems. In the case of quasi-linear constraints,
techniques using fundamental solutions of the differential operators cannot be
applied to show convergence of the variation of the states. Some authors
succeeded showing this convergence with the help of technical computations
under additional requirements on the problem. Our main objective is to simplify
and extend these previous results by using a Lagrangian framework and a
projection trick. Besides these generalisations the purpose of this manuscript
is to present a systematic derivation approach for topological derivatives.
