A sub-elliptic estimate for a class of invariantly defined elliptic systems
Lesley M. Sibner, Robert J. Sibner · Pacific Journal of Mathematics · 1981
We consider a certain invariantly defined nonlinear system of partial differential equations on a Riemannian manifold.Since a special case describes a steady, irrotional, compressible flow on the manifold, it is natural to refer to the (square of) the pointwise norm of the solution as the speed of the flow and to the density of the flow.Under appropriate restrictions on the density, the system is elliptic and we obtain a sub-elliptic estimate and a maximum principle for the speed of the flow in terms of the curvature of the manifold.