Hyperbolic Equations with Mixed Boundary Conditions: Shape Differentiability Analysis

We consider the wave equation with Dirichlet–Neumann boundary conditions on a family of perturbed domains Ω s . We discuss the shape differentiability analysis associated with the above mentioned problem, namely the existence of strong material and shape derivatives of the solution, and the rendering of the new wave problem whose solution is given by the shape derivative. The study shows that the Neumann boundary conditions completely change the focus and strategy involved in the shape differentiability analysis, in comparison to the case of the wave equation with purely Dirichlet boundary conditions. In this paper we show that for the existence of weak material derivative, the classical sensitivity analysis of the state can be bypassed by using parameter differentiability of a functional expressed in the form of Min–Max of a convex–concave Lagrangian with saddle point. Then we analyze the strong material derivative via a brute force estimate on the differential quotient, using known regularity results on the solution of the wave problem.