Mathematics

In this talk I will consider the hyperbolic system $\ddot u-{\rm div}\,(\mathbb A\nabla u)=f$ in the time varying cracked domain $\Omega\setminus\Gamma_t$. Here $u$ is an $\mathbb R^d$--valued vector function, $\mathbb A$ satisfies the usual assumptions in linear elasticity, and the cracks $\Gamma_t$ are increasing closed subsets of $\overline{\Omega}$ contained in a prescribed $C^2$ manifold. I will prove existence and uniqueness of weak solutions assuming that there is a regular change of variables which reduces $\Omega\setminus\Gamma_t$ to the fixed domain $\Omega\setminus\Gamma_0$. Moreover, I will show an energy equality and, as a consequence, a continuous dependence result on the cracks for the solutions of this system.