33.5. HELMHOLTZ DECOMPOSITIONS 1153
Theorem 33.5.7 There exists a continuous projection
π : Lp (U ;Rn)→ Lp (U ;Rn)
such that πF is a gradient and∇ · (F−πF) = 0
in the sense of weak derivatives.
Note this theorem shows that any Lp vector field is the sum of a gradient and a partwhich is divergence free. F = F−πF+πF.