ON THE POINCARE LEMMA ON DOMAINS

We prove the Poincare lemma on a domain Omega with a Dirichlet boundary condition under a natural assumption on the regularity of Omega: a closed form f in the Holder space C-r,C-a is the differential of a C-r +1,C-alpha form, provided that the domain Omega itself is C-r +1,C-alpha The proof is based on a construction by approximation, together with a duality argument, in the spirit of the strategy introduced by Bourgain and Brezis [3] to solve the divergence equation in the Sobolev space W-0(1,p) (Omega). We also establish the corresponding statement in the setting of higher order Sobolev spaces.