L^r-variational inequality for vector fields and the helmholtz-weyl decomposition in bounded domains

We show that every L^r-vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in ℝ³ with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u-v _∂Ω = 0 and u × _∂Ω = 0, where v denotes the unit outward normal to ∂Ω. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C∞-forms on compact Riemannian manifolds into L^r-vector fields on Ω. As an application, the generalized Biot-Savart law for the incompressible fluids in Ω is obtained. Furthermore, various bounds of u in L^r for higher derivatives are given by means of rot u and div u.