Decomposition of Vector-Valued Divergence Free Sobolev Functions and Shape Optimization for Stationary Navier–Stokes Equations

Abstract

We establish a divergence free partition for vector-valued Sobolev functions with free divergence in R n , n ≥ 1. We prove that for any domain Ω of class 𝒞 in R n ,n = 2,3, the space and the space , which is the completion of in the H 1(Ω) n -norm, are identical. We will also prove that a.e. in Ω}, where D is a bounded Lipschitz domain such that Ω ⊂ ⊂ D. These results, together with properties for domains of class 𝒞, are used to solve an existence problem in the shape optimization theory of the stationary Navier–Stokes equations.