THE POINCARÉ INEQUALITY FOR A VECTOR FIELD WITH ZERO TANGENTIAL OR NORMAL COMPONENT ON THE BOUNDARY

Abstract

Poincaré's inequality is well known: given a bounded domain G, ∥u∥_p ⋚ c∥∇u∥_p provided u(x) vanishes on the boundary ∂G. The case where u(x) is a vector field u(x) that does not vanish on the boundary ∂G is considered. It is shown that when either the tangential component or the normal component vanishes on the boundary ∂G, then the Poincaré inequality is satisfied.