Well-posedness and stability of the Lagrange representation of the n-D wave equation via boundary triples

We study the Lagrange representation of the wave equation with generalized Laplacian $\mathrm{div}T\mathrm{\nabla }$. We allow the coefficients – the Young modulus $T$ and the density $\rho$ – to be ${\mathrm{L}}^{\mathrm{\infty }}$ or even nonlocal operators. Moreover, the Lipschitz boundary of the domain $\mathrm{\Omega }$ can be split into several parts admitting Dirichlet, Neumann and/or Robin-boundary conditions of displacement, velocity and stress. We show well-posedness of this classical model of the wave equation utilizing boundary triple theory for skew-adjoint operators. In addition we show semi-uniform stability of solutions under slightly stronger assumptions by means of a spectral result.