How to show that $\exists$ $\phi\in C_c^{\infty}(\Omega)$ s.t. $\int_{\Omega_1}D\phi(x)dx$ has unit norm?

Asked Nov 01 '18 at 15:27

Active Apr 17 '19 at 15:32

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Let $\Omega\subset \mathbb R^n$ be open, connected and $\Omega=\Omega_1 \cup \Omega_2$ where $\Omega_1\cap \Omega_2=\emptyset$, $\mu(\Omega_1)>0,\mu(\Omega_2)>0.$ Then show that there exists $\phi\in C_c^{\infty}(\Omega)$ such that $\displaystyle \int_{\Omega_1}D\phi(x)dx\in S^{n-1}.$

There is one theorem in Sobolev Space called :

Let $\Omega\subset \mathbb R^n$ be an open set and $K\subset \Omega$ be a compact set. Then there exists $\phi\in C_c^{\infty}(\Omega)$ such that

$0\leq \phi(x)\leq 1$ for all $x\in \Omega$,

$\phi(x)=1$ on $K$.

Here $C_c^{\infty}(\Omega)$ is the set of all $C^{\infty}$ functions in $\Omega$ which are compactly supported.

$S^{n-1}=\{x\in \mathbb R^n:||x||_2=1\}$

Any help is appreciated . Thank you

edited Jun 12 '20 at 10:38

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asked Nov 01 '18 at 15:27

Mini_me

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What is the purpose of the sets $\Omega_1$ and $\Omega_2$? – Umberto P. Nov 01 '18 at 20:02
Does $D\phi$ denote ordinary gradient, i.e. $\nabla \phi$? – md2perpe Nov 01 '18 at 21:38
@UmbertoP. Actually this is a part of a theorem where the original domain $\Omega$ is taken like this way. I think we don't have to use $\Omega_1$ or $\Omega_2.$ – Mini_me Nov 02 '18 at 12:31
@md2perpe Yeah... – Mini_me Nov 02 '18 at 12:32
Duplicate of https://math.stackexchange.com/a/3007295/168433 – md2perpe Apr 18 '19 at 10:01

How to show that $\exists$ $\phi\in C_c^{\infty}(\Omega)$ s.t. $\int_{\Omega_1}D\phi(x)dx$ has unit norm?

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