Second-Order Shape Derivative for Hyperbolic PDEs
In this paper we study the second-order shape derivative for the solution to the wave equation with Dirichlet boundary conditions. We first prove the shape derivative exists for this problem. Specific complications arise when differentiating with respect to the domain hyperbolic PDEs; this requires an exclusive approach using the hidden regularity described in . This study has been started in  and announced in . We present here a new result where the data are no longer defined on the whole D and then restricted to the subsets of D but rather defined differently on each domain of D. Moreover, the differentiability is improved under stronger regularity of those functions. The Dirichlet boundary condition is now nonhomogeneous which was also necessary to consider the second-order shape derivative.