Shape differentiability of the Neumann problem of the Laplace equation in the half - space

Shape optimization for the Neumann problem of the Laplace equation is important for application and also from the numerical point of view. Mathematical analysis of such problem in the half space is not available. In this paper we prove the shape differentiability of solutions in appropriate weighted Sobolev spaces which describe the behavior of solutions at infinity. We will consider two different perturbations of domain to get the existence of weak Gateaux material derivative and in the second case the existence of Fréchet material derivatives.