Vol. 94, No. 2, 1981

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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

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
Received: 8 May 1980
Published: 1 June 1981
Lesley Millman Sibner
Robert Jules Sibner