Regularity for Vector Fields, Hodge Decomposition and Green Formula in Lipschitz Domains

For a bounded Lipschitz domain Ω ⊂ R³, neither necessarily simply connected nor with connected boundary Γ, we prove a new Hodge-type decomposition: every u ∈ H(curl, Ω) splits uniquely into a gradient and a field in H¹²(Ω) ∩ H(curl, Ω) with tangential L²-trace. As applications we remove the additional topological hypotheses from Costabel’s H¹²-regularity theorem, and extend the classical Green formula to two H(curl)-fields. We define tangential differential operators on the non-smooth boundary Γ and revisit the space H⁻¹²(div_Γ, Γ).