This paper is concerned with the study of a class of nonsmooth cost functions subject to a quasi-linear PDE in Lipschitz domains in dimension two. We derive the Eulerian semiderivative of the cost function by employing the averaged adjoint approach and maximal elliptic regularity. Furthermore we characterize stationary points and show how to compute steepest descent directions theoretically and practically. Finally, we present some numerical results for a simple toy problem and compare them with the smooth case. We observe faster convergence rates in the nonsmooth case in our tests.