On distributions independent of image in certain non-cylindrical domains and a de Rham lemma with a non-local constraint Original Research Article

We study distributions not depending on the xNxN-variable in an open set Ω⊂RNΩ⊂RN. It is assumed that ΩΩ may be described through a very general function D:ω↦RD:ω↦R, where ω⊂RN-1ω⊂RN-1 is any open set. We give a representation theorem for this kind of distributions and show how they are related to distributions defined in ωω. A direct application of this theorem is the derivation of a de Rham-like lemma with a non-local constraint. These results can be applied to the analysis of hydrostatic approximation of Navier–Stokes equations.