Estimating ∇u by div u and curl u

In this paper we obtain necessary and sufficient conditions for the validity of the estimate magnified image The constant c does not depend on u . u is a vector field with values in ℝ 3 . It is defined on a bounded set G of ℝ 3 or an unbounded one, denoted by Ĝ. 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 Ĝ.