Shape Derivative in the Wave Equation with Dirichlet Boundary Conditions

The aim of this paper is to give a full analysis of the the shape differentiability for the solution to the second order hyperbolic equation with Dirichlet boundary conditions. The implicit function theorem does not work to solve the problem of weak regularity of the data; nevertheless by a more technical approach we prove an analogous result. We will first prove the theorem under strong regularity of the right hand side, then using the hidden regularity we will prove the shape derivative continues to exist under weak condition of regularity. We end up with a second order shape derivative for this problem.