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

Abstract

We study the Lagrange representation of the wave equation with generalized Laplacian $\mathrm{div}T\nabla$. We allow the coefficients -- the Young modulus $T$ and the density $\rho$ -- to be ${\mathrm{L}}^{\infty }$ or even nonlocal operators. Moreover, the Lipschitz boundary of the domain $\Omega$ 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.

22 pages

References in corpus (4)

• Semi-uniform stability of operator semigroups and energy decay of damped waves
• Well-posedness of linear first order Port-Hamiltonian Systems on multidimensional spatial domains
• M-dissipative boundary conditions and boundary tuples for Maxwell operators
• Semi-uniform stabilization of anisotropic Maxwell's equations via boundary feedback on split boundary