Decomposition of vector-valued divergence free Sobolev functions and shape optimization for stationary Navier-Stokes equations

We establish a divergence free partition for vector-valued Sobolev functions with free divergence in ${\bf R}^n, n\geq 1$. We prove that for any domain $\om$ of class $\cal C$ in ${\bf R}^n,n=2,3$, the space $D_0^1(\om)\equiv\{{\mathbf{v}} \in H^1_0(\Omega)^n ; {div}{\mathbf{v}}=0\}$ and the space $H_{0,\sigma}^1(\om)\equiv \bar{\{{\mathbf{v}}\in C^{\infty}_0(\Omega)^n;{div}{\mathbf v}=0\}}^{\|\cdot\|_{H^1(\om)^n}}$, which is the completion of $\{{\mathbf{v}} \in C^{\infty}_0(\Omega)^n; {div}{\mathbf v}=0\}$ in the $H^1(\Omega)^n$-norm, are identical. We will also prove that $H_{0,\sigma}^1(D\setminus\bar\om)=\{{\mathbf v}\in H_{0,\sigma}^1(D); {\mathbf v}=0 {a.e. in} \om\}$, where $D$ is a bounded Lipschitz domain such that $\om\subset\subset D$. These results, together with properties for domains of class $\mathcal C$, are used to solve an existence problem in the shape optimization theory of the stationary Navier-Stokes equations.