The Wave Equation

Abstract

In this chapter we study the wave equation

$${u_{tt}}\, = \,\Delta u$$

on an open subset Ω of ℝⁿ. We will use the theory of cosine functions and work on L ² (Ω). We first consider the Laplacian with Dirichlet boundary conditions. This is a selfadjoint operator and well-posedness is a consequence of the spectral theorem. A further aim is to replace the Laplace operator by a general elliptic operator. This will be done by a perturbation theorem for selfadjoint operators which we prove in Section 7.1.