Variable Coefficient Wave Equations and Systems

Consider, in ${\rm R}_x^n \times {\rm R}_t$, a second order partial differential operator of the form where all coefficients are real and C ^∞, and L ₁ is a first order operator. We would like L to be an operator similar to the wave operator, and to enjoy the same properties: Finite speed of propagation, energy inequalities, etc. We saw in Chapter 2 that, for an operator in the plane, it is natural to require that its principal part should be the principal part of a product a real vector fields. Here, suppose first that L is homogeneous (that is, L ₁ ≡ 0) with constant coefficients.