# Decompostition of a Lipschitz domain

We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if:

$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \|})\}$$ and $$\partial \Omega = \{x\in \mathbb R ^n : \left \| x \right \| = g(\frac{x}{\left \| x \right \|})\}$$ with $g$ is a continuous, positive function on the unit sphere.

In this paper, Bramble uses the fact that : Any Lipschitz domain can be written as the union of strongly star shaped Lipschitz domains: $\Omega=\cup_{i=1}^{M}\Omega_i$

Can you help me to find why do we have this result? Do you have any references in which I can find this proposition?

PS: Sorry to ask this question again, but this did not get answered (as I had desired) on M.SE.

• You can always use a partition of unity such that locally each $\Omega_i$ is star-shaped. May 15, 2017 at 6:47
• @ T. Amdeberhan : I do not know, if I understood what you want to say, but if you saw the paper of Bramble he used this proposition (and via the partion of unity) to show an inequality.. May 15, 2017 at 6:54
• Here is link to the Math.SE question. May 15, 2017 at 23:44
• The property is just a trivial consequence of the definition of Lipschitz domain. May 16, 2017 at 6:01

A Lipschitz domain $\Omega$ is an open set and any open set is a union of balls, which are strongly star-shaped. So I assume you meant $\overline\Omega$. By definition, any point of $\partial\Omega$ has a nbd in $\overline\Omega$ which is isometric to a sub-graph of a positive $k$-Lipschitz function $f:B(0,r)\to(0,+\infty)$, $$\{(x,t)\, :\, |x|<r, \;0<t<f(x)\}.$$ We can take $r<{f(0)\over 2k+1}$, which makes the latter set strongly star-shaped w.r.to the point $(0,r)$ as it is easy to check.

• @ Pietro Majer:I totally agree with what you have written Sir, But I can not see how to construct the function $g$ of the definition above^^. Can you explain a little? May 16, 2017 at 12:08
• If $U\subset\mathbb{R}^n$ is the above set, and $p:=(0,r)$, define, for any $x\in\mathbb{R}^n$ of unit norm, $g(x)$ to be the unique $t>0$ such that $p+tx\in\partial U$. You need to prove that this $t$ is actually unique, and that depends continuously from $x$. May 16, 2017 at 19:37
• @ Pieotro Majer: Thanks for your reply, I didn't get the idea, if $\forall x\in \partial \Omega$, there is a function $f$ and a nbd (Let's say $B(0,r)$) such that: $\partial \Omega\cap B(0,r) =\{X=(x,t):\left \| X \right \|< r, 0<t<f(x)\}$. So what is $p=(0,r)$ for you (r?)? can you explain more I will try to complete the proof, Thanks ! May 17, 2017 at 9:50
• $p:=(0,r)\in B(0,r) \times \mathbb{R}\subset \mathbb{R}^{n+1}$ refers to the above answer, where $B(0,r)\subset \mathbb{R}^n$ is the ball of radius $r>0$ centered at the origin $0\in\mathbb{R}^n$ (so also $\Omega$ is an open set of $\mathbb{R}^{n+1}$, in these notations ) Note that the sub-graph of $f$ is an open set in $B(0,r)\times \mathbb{R}\subset \mathbb{R}^{n+1}$, and $(0,r) \in \mathbb{R}^{n+1}$ is a point in this sub-graph. For more details I think you may post a suitable question in StackExchange. May 17, 2017 at 11:39
• k-Lipschitz ensures that the line {p+tx, t>0} has unique intersection with the boundary of the sub-graph set defined in the answer. The continuity of g follows by standard compactness argument. It's an easy computation. Try asking Stack Exchange. May 17, 2017 at 14:52

You may like to check such results, in particular, Proposition 2.5.4 of the below monograph. Hope you have access to it.

Proposition 2.5.4. Let $\Omega \in \mathcal A_0$ have Lipschitz boundary. Then there exists a finite open covering $\{\Omega_j\}_{j\in\{1,\dots,m\}}$ of $\overline\Omega$ such that, for every $j\in\{1,\dots,m\}$, $\Omega_j \cap \Omega$ is strongly star shaped with Lipschitz boundary.

Carbone L. and De Arcangelis R., Unbounded functionals in the calculus of variations: Representation, relaxation, and homogenization, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 125, Chapman & Hall/CRC, Boca Raton, FL, 2002