Estimating ∇u by div u and curl u

In this paper we obtain necessary and sufficient conditions for the validity of the estimate The constant c does not depend on u. u is a vector field with values in ?³. It is defined on a bounded set G of ?³ or an unbounded one, denoted by ?. The boundary conditions are as follows: either the normal component of u vanishes or the tangential one does. Our conditions are expressed in terms of the Betti numbers of G or ?.