Approximation of exact and closed differential forms by finite ones

Sobolev established in 1963 [i] that smooth finite functions form a dense set in a space with a seminorm Lp~Rn). For the case E = i this result asserts the possibility of approximating in Lp(Rn) potential vector fields by means of finite potential vector fields. Since the classes of solenoidal and potential vector fields coincide in R n, Sobolev~s result indicates also the possibility of approximating solenoidal vector fields by means of finite solenoidal vector fields. However, even for regions in R n, the problems of approximating potential and solenoidal vector fields are generally different.