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If  G  ⊂ ℝ 2  is either a bounded or an exterior domain weak  L   q  -solutions of Δ  w   = rot ∗  p  in  G ,   w   | ∂ G   = 0 ( p  ∈  L   q  ( G ) given), are regarded. Because of the simple relation between rot and div in case  n  = 2 we deduce immediately from the corresponding results for Δ  v   = ∇ p  in  G ,   v   | ∂ G   = 0, all results for the problem considered above. In addition we derive a complete characterization of all solenoidal vector fields. This depends on topological properties of  G .

analysis

The Cosserat problem related to the curl and a complete characterization of all solenoidal vector fields vanishing at the boundary in case of space dimension  n  = 2