Estimating ∇u in terms of div u,curl u,either (ν, u) or ν×u and the topology

In the present paper we prove C^α-estimates for ∇ u using components of boundary values of u , div u , curl u and quantities given by components of boundary values of u as well as boundary values of elements belonging to de Rhams cohomology modules. The vector field u is defined on a bounded set G¯⊂ℝ³, meanwhile the cohomology group will be defined with regard to ℝ³−G. Our inequalities turn out to be a priori estimates concerning well-known boundary value problems for vector fields. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.