I was reading about the finite propagation speed of the wave equation on a Riemannian manifold. I was wondering if instead of the LaplaceBeltrami operator $\Delta$ we consider the equation $$\partial_t^2 u = Lu$$ where $L$ is a negativedefinite selfadjoint elliptic operator of order 2, would we still have finite speed of propagation property? Thanks a lot for your help.

1$\begingroup$ Searching for "finite speed" and "elliptic" on MathSciNet turns up ams.org/mathscinetgetitem?mr=1205177 Section 1.3 of which is "1.3. Finite speed propagation and essential selfadjointness". $\endgroup$ – Willie Wong Feb 3 '14 at 9:22

$\begingroup$ I have the impression that any textbook on linear hyperbolic PDE's discusses this. $\endgroup$ – Deane Yang Feb 3 '14 at 20:57
Your example is a special case of an operator with a wavelike principal symbol on a Lorentzian manifold. Namely, if the coefficients of the second order derivatives in $L$ are $L = h^{ij} \partial_i \partial_j + \cdots$, then $h^{ij}$ is an inverse Riemannian metric and it is always possible to write $L = \Delta_h + B$, with $\Delta_h$ the $h$Laplacian and $B$ a first order differential operator. Your equation then becomes $\Delta_g \psi = B\psi$, where $\Delta_g = \partial_t^2  \Delta_h$ is the $g$Laplacian and $g=\mathrm{diag}(1,h)$ is a Lorentzian metric. In other words, $\Delta_g$ is a wave operator. Thus, you get the usual finite speed of propagation results for your equation, because the lower order operator $B$ does not affect them.
An excellent treatment of equations with wavelike principal symbols can be found in Wave Equations on Lorentzian Manifolds and Quantization by Christian Baer, Nicolas Ginoux and Frank Pfaeffle.