01514nas a2200109   4500008004100000245013600041210006900177520106400246100002101310700002201331856005101353       2015                        en  d00aThe wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data0 awave equation on domains with cracks growing on a prescribed pat3 aGiven a bounded open set $\Omega \subset \mathbb R^d$ with Lipschitz boundary and an increasing family $\Gamma_t$, $t\in [0,T]$, of closed subsets of $\Omega$, we analyze the scalar wave equation $\ddot{u} - div (A \nabla u) = f$ in the time varying cracked domains $\Omega\setminus\Gamma_t$. Here we assume that the sets $\Gamma_t$ are contained into a prescribed $(d-1)$-manifold of class $C^2$.
Our approach relies on a change of variables: recasting the problem on
the reference configuration $\Omega\setminus \Gamma_0$, we are led to
consider a hyperbolic problem of the form $\ddot{v} - div (B\nabla v) + a \cdot \nabla v - 2 b \cdot \nabla \dot{v} = g$ in $\Omega \setminus \Gamma_0$. Under suitable assumptions on the regularity of the change of variables that transforms $\Omega\setminus \Gamma_t$ into $\Omega\setminus \Gamma_0$, we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks.1 aDal Maso, Gianni1 aLucardesi, Ilaria  uhttp://urania.sissa.it/xmlui/handle/1963/3462900419nas a2200121   4500008004100000245007200041210006900113260001800182100002100200700002100221700002000242856003500262       1985                        en  d00aWeak convergence of measures on spaces of semicontinuous functions.0 aWeak convergence of measures on spaces of semicontinuous functio  bSISSA Library1 aDal Maso, Gianni1 aDe Giorgi, Ennio1 aModica, Luciano  uhttp://hdl.handle.net/1963/463