A sub-elliptic estimate for a class of invariantly defined elliptic systems

Lesley Millman Sibner and Robert Jules Sibner

Vol. 94 (1981), No. 2, 417–421

DOI: 10.2140/pjm.1981.94.417

Abstract

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.

Mathematical Subject Classification 2000

Primary: 58G99

Secondary: 35J99, 76N15

Milestones

Received: 8 May 1980

Published: 1 June 1981