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Defining functions for unbounded $$C^m$$ domains. (English) Zbl 1288.26008
For an open set $$\Omega \subset \mathbb{R}^n$$ a real-valued $$C^m$$ function $$\rho$$ defined on a neighborhood $$U$$ of $$\partial \Omega$$ such that $$\{x\in U\mid\rho(x)<0\}=\Omega \bigcap U$$ and $$\nabla \rho \neq 0$$ on $$\partial \Omega$$ is called $$C^m$$ defining function ($$m\geq 1$$), $$\Omega$$ is a $$C^m$$ domain if it has a $$C^m$$ defining function. If $$\rho$$ is a $$C^m$$ defining function for $$\Omega \subset \mathbb{R}^n$$ defined on a neighborhood $$U$$ of $$\partial \Omega$$ and such that 1) dist$$(\partial \Omega, \partial U)>0$$, 2) $$\|\rho\|_{C^m(U)}<\infty$$, 3) $$\inf _U|\nabla \rho|>0$$, then $$\rho$$ is $$C^m$$ uniformly. A $$C^m$$ uniformly function for all $$m\in \mathbb{N}$$ is a $$C^{\infty}$$ uniformly function. The following notions are introduced:
For $$\Omega \subset \mathbb{R}^n$$ with $$C^m$$ boundary
$\widetilde{\delta}(x)=\begin{cases} d(x,\partial \Omega), \;\;x \overline{\in} \Omega\,,\\ -d(x,\partial \Omega), \;\;x \in \overline{\Omega}\,, \end{cases}$ $$\mathrm{Unp}(\partial \Omega)=\{x\in \mathbb{R}^n\mid \text{there is a unique point}\; y\in \partial \Omega, \text{such that}\; \delta(x)=|y-x| \}$$.
For $$y\in \partial \Omega$$, $$\mathrm{Reach}(\partial \Omega,y)=\sup\{r\geq 0\mid B(y,r)\subset \mathrm{Unp}(\partial \Omega)\}$$.
Reach$$(\partial \Omega)=\inf\{\text{Reach}(\partial \Omega,y)\mid y\in \partial \Omega\}$$.
The main result of the article is presented in the following
{Theorem. } Let $$\Omega \subset \mathbb{R}^n$$ be a $$C^m$$ domain, $$m>2$$. Then the following assertions are equivalent:
{$$1^0$$.}
$$\Omega$$ has a uniformly $$C^m$$ defining function.
{$$2^0$$.}
$$\partial \Omega$$ has positive reach, and for any $$0<\varepsilon<\text{Reach}(\partial \Omega)$$ the signed distance function satisfies $$\|\widetilde{\delta}\|_{C^m(U_{\varepsilon})}<\infty$$ on $$U_{\varepsilon}=\{x\in \mathbb{R}^n\mid\delta(x)<\varepsilon\}$$.
{$$3^0$$.}
There exists a $$C^m$$ defining function $$\rho$$ for $$\Omega$$ and a constant $$C>0$$ such that for every point $$p \in \partial \Omega$$ with local coordinates $$\{y_1,\ldots,y_n\}$$ satisfying $$\frac{\partial \rho}{\partial y_j(p)}=0$$ for $$1\leq j\leq n-1$$ $|\nabla \rho(p)|^{-1}|\frac{\partial^k \rho(p)}{\partial y^I}|<C$ where $$I$$ is a multi-index of length $$k$$, $$2\leq k\leq m$$, and $$0\leq I_n\leq \min\{m-k,k\}$$.

##### MSC:
 26B25 Convexity of real functions of several variables, generalizations 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 32T15 Strongly pseudoconvex domains
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