Shape derivative in the wave equation with Dirichlet boundary conditions (Q1818843)

The authors study the second-order Dirichlet hyperbolic problem \[ \begin{aligned} & \partial^2_ty-\text{div}(K\nabla y)=f\quad \text{on } (0,T)\times \Omega,\\ & y=g\quad \text{on }(0,T)\times \partial\Omega,\\ & y(0)=\varphi,\quad \partial_ty(0)=\psi\quad\text{on }\Omega,\end{aligned} \] where \(K\) be a coercive and symmetric \(N\times N\)-matrix of functions, and \(\Omega\subset\mathbb{R}^n\) is a a domain with the boundary \(\partial\Omega\in C^k\) for \(k \geq 2\). First of all the authors prove that the solution of the given problem is shape and material differentiable under a strong regularity of the data. Then using the hidden regularity they prove the existence of a shape derivative under weak regularity conditions of the data. The implicit function theorem used in the smooth cases does not work to solve the problem of a weak regularity of the data. Nevertheless by a more technical approach the authors prove analogical results as for the regular data.