$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains

Abstract

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 R 3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u . v |∂Ω = 0 and u x ν|∂Ω = 0, where ν 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.