A generalized 2-D Poincare inequality

Two 1-D Poincare-like inequalities are proved under the mild assumption tha t 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 func tion is zero on a suitable are contained in the domain (instead of the whol e boundary). As an application, it is shown that a set of boundary conditio ns for the quasi geostrophic equation of order four are compatible with gen eral physical constraints dictated by the dissipation of kinetic energy.