On Computation of the Shape Hessian of the Cost Functional Without Shape Sensitivity of the State Variable

A framework for calculating the shape Hessian for the domain optimization problem, with a partial differential equation as the constraint, is presented. First and second order approximations of the cost with respect to geometry perturbations are arranged in an efficient manner that allows the computation of the shape derivative and Hessian of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state and adjoint variables are only required to be Hölder continuous with respect to geometry perturbations.