Differentiability of the Dirichlet to Neumann map under movements of   polygonal inclusions with an application to shape optimization
  In this paper we derive rigorously the derivative of the Dirichlet to Neumann map and of the Neumann to Dirichlet map of the conductivity equation with respect to movements of vertices of triangular conductivity inclusions. We apply this result to formulate an optimization problem based on a shape derivative approach. 
E. Beretta, E. Francini, S. Vessella
math.AP