Calculus of Variations and Geometric Measure Theory

G. Dal Maso - I. Lucardesi

The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data

created by lucardesi on 13 Oct 2015
modified by dalmaso on 16 Mar 2018


Published Paper

Inserted: 13 oct 2015
Last Updated: 16 mar 2018

Journal: Appl. Math. Res. Express. AMRX
Number: 1
Pages: 184-241
Year: 2017
Doi: 10.1093/amrx/abw006

Preprint SISSA


Given 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.