  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 semi-derivative of the cost function by employing the averaged
adjoint approach and maximal elliptic regularity. Furthermore we characterise
stationary points and show how to compute steepest descent direc- tions
theoretically and practically. Finally, we present some numerical results for a
simple toy problem and compare them with the smooth case. We also compare the
convergence rates and obtain higher rates in the nonsmooth case.
