Control region and nested real cutoffs

PDF p. 7: (3.1)–(3.3); PDF p. 3: Figure 1(a)

Let $n\ge 2$, and let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty compact set with smooth boundary and connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$. Choose $R>0$ such that $\mathrm{\Theta }\subset B_R(0)$, where $B_r(0)=\{x:\mathrm{\mid }x\mathrm{\mid }<r\}$. A smooth function is one with continuous derivatives of every order; its support is the closure of the set on which it is nonzero.

Choose a real-valued $\phi \in C^{\mathrm{\infty }}({\mathbb{R}}^n;[0,1])$ such that $\phi (x)=0$ for $\mathrm{\mid }x\mathrm{\mid }\le R$, $\phi (x)>0$ for $\mathrm{\mid }x\mathrm{\mid }>R$, and $\phi (x)=1$ for $\mathrm{\mid }x\mathrm{\mid }\ge R+1$. Define $\chi =1-\phi$ and the control region $\omega =\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }>R\}=\{x\in \mathrm{\Omega }:\phi (x)>0\}$. Thus $\chi$ is smooth and compactly supported, with $\mathrm{supp}\chi \subseteq \overline{B_{R+1}(0)}$.

Choose $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;[0,1])$, meaning a smooth real-valued function with compact support, equal to one on $B_{R+2}(0)$ and supported in $B_{R+3}(0)$. Then $0\le \chi \le \psi$ and $\psi =1$ on a neighborhood of $\mathrm{supp}\mathrm{\nabla }\phi$. Set ${\mathrm{\Omega }}^{\mathrm{\prime }}=\mathrm{\Omega }\cap B_{R+3}(0)$; it is a bounded open set with smooth boundary $\mathrm{\partial }\mathrm{\Theta }\cup \mathrm{\partial }{B}_{R+3}(0)$.

Exterior and truncated obstacle domains

The standard smooth step function

Test function cutoffs and euclidean localization

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

The exterior geometry, boundary convention and existence of an enclosing ball are as in Exterior and truncated obstacle domains. For the first cutoff, let $\sigma$ be the standard smooth step of The standard smooth step function and put $\phi (x)=\sigma (\mathrm{\mid }x\mathrm{\mid }-R)$. The step is smooth, takes values in $[0,1]$, vanishes on $(-\mathrm{\infty },0]$, is positive on $(0,\mathrm{\infty })$ and equals one on $[1,\mathrm{\infty })$. The composition is smooth away from the origin and is identically zero near the origin because $R>0$; hence it is smooth everywhere and has the required values.

Apply math:lem-test-function-cutoffs-and-euclidean-localization to the compact set $\overline{B_{R+2}(0)}$ inside the open ball $B_{R+3}(0)$. It gives a smooth $[0,1]$-valued function supported in that open ball and equal to one on a neighborhood of the compact set; take this function as $\psi$.

On $B_{R+1}(0)$, $\psi =1$ and $0\le \chi \le 1$; outside that ball, $\chi =0\le \psi$. Thus $0\le \chi \le \psi$ everywhere.

The gradient is the vector of coordinate partial derivatives, as in The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. Since $\phi$ is constant on each of the open sets $\mathrm{\mid }x\mathrm{\mid }<R$ and $\mathrm{\mid }x\mathrm{\mid }>R+1$, all these partial derivatives vanish there. Thus $\mathrm{supp}\mathrm{\nabla }\phi$ lies in the closed annulus $R\le \mathrm{\mid }x\mathrm{\mid }\le R+1$, which is contained in $B_{R+2}(0)$. Therefore $\psi =1$ on an open neighborhood of that support.

Finally, applying the boundary identity in Exterior and truncated obstacle domains with enclosing radius $R+3$ gives $\mathrm{\partial }{\mathrm{\Omega }}^{\mathrm{\prime }}=\mathrm{\partial }\mathrm{\Theta }\cup \mathrm{\partial }{B}_{R+3}(0)$. The two pieces are disjoint and smooth, so their union is a smooth boundary; boundedness and openness follow from the same exterior construction.

The article supplies the control-region and cutoff notation in (3.1)–(3.3). Its reference here is attribution; the existence of the functions is justified by the independent suppliers above.