Sur une inégalité de coercivité

AbstractWe give here the proof of an inequality originally due to K. O. Friedrichs [2]. If Ω is a regular bounded open set of Rn, each vector-valued function u ϵ L2(Ω), such that div u ϵ L2(Ω) and rot u ϵ L2(Ω), belongs to H1(Ω), if n · uΓ = 0 or n ∧ uΓ = 0, Γ being the boundary of Ω. Moreover, for such u, we have ∥ u ∥H1(Ω) ⩽ C[∥ u ∥L2(Ω) + ∥ div u ∥L2(Ω) + ∥ rot u ∥L2(Ω)], when C does not depend of u