Journal of Inequalities and Applications
Volume 5 (2000), Issue 4, Pages 343-349
A generalized 2-D Poincaré inequality
1Osservatorio Geofisico Sperimentale, P.O. Box 2011, Trieste 34016, Italy
2lstituto Talassografico del CNR, Trieste 34123, Italy
Received 29 April 1999; Revised 10 July 1999
Copyright © 2000 Fabio Cavallini and Fulvio Crisciani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Two 1-D Poincaré-like inequalities are proved under the mild assumption that the integrand function is zero at just one point. These results are used to derive a 2-D generalized Poincare inequality in which the integrand function is zero on a suitable arc contained in the domain (instead of the whole boundary). As an application, it is shown that a set of boundary conditions for the quasi geostrophic equation of order four are compatible with general physical constraints dictated by the dissipation of kinetic energy.