Journal of Inequalities and Applications 
Volume 5 (2000), Issue 4, Pages 343-349
doi:10.1155/S1025583400000175

A generalized 2-D Poincaré inequality

Fabio Cavallini1 and Fulvio Crisciani2

 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

Abstract

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.