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 ${\mathbf{R}}^n,n\ge 1$. We prove that for any domain $\text{om}$ of class $\mathcal{C}$ in ${\mathbf{R}}^n,n=2,3$, the space $D_0^1(\text{om})\equiv \{\mathbf{v}\in H_0^1(\mathrm{\Omega }{)}^n;div\mathbf{v}=0\}$ 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}$} \in C^{\infty}0(\Omega)^n; {div}{\mathbf v}=0}$inthe$H^1(\Omega)^n$-norm,areidentical\mathrm{.}Wewillalsoprovethat$H{0,\sigma}^1(D\setminus\bar\om)={{\mathbf v}\in H_{0,\sigma}^1(D); {\mathbf v}=0 {a.e. in} \om}$,where$D$isaboundedLipschitzdomainsuchthat$\om\subset\subset D$. 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.