Hadamard Variational Formula for the Green Function for the Velocity and Pressure of the Stokes Equations

We consider the Hadamard variational formula for the Green function of the Stokes equations {G, R} which describes the motion of the incompressible fluids moving slowly in the bounded domain Omega with the smooth boundary partial derivative Omega an. Under the perturbation of domains keeping those volumes and topological types invariant, we not only refine the proof of its formula for the velocity G but we also develop a new formula for the pressure R. Our result may be regarded as the Hadamard variational formula for the Green functions as an example of the elliptic system of equations with the Dirichlet boundary condition.