Uniform C^m-Regularity Condition

Definition¹

If there exists a locally finite open cover $\left\{ U_{j} \right\}$ of $\mathrm{bdry}\Omega$, and a sequence $\left\{ \Phi_{j} \right\}$ of $m$-smooth transformations taking $U_{j}$ onto the ball $B=\left\{ y\in \mathbb{R}^n : |y| \lt 1 \right\}$, with an inverse transformation $\Psi _{j}=\Phi_{j}^{-1}$ existing and satisfying $\text{(i)}$ ~ $\text{(iv)}$, then the open set $\Omega \subset \mathbb{R}^n$ satisfies the uniform $C^{m}$-regularity condition.

$\text{(i)}$ For any $\delta >0$, $\Omega_{<\delta}$$\subset \bigcup \nolimits_{j=1}^\infty \Psi \Big( \left\{y\in \mathbb{R}^n : |y| \lt \frac{1}{2} \right\} \Big)$ is true.

$\text{(ii)}$ For each $j$, $\Phi_{j}(U_{j} \cap \Omega )=\left\{ y \in B : y_{n} \gt 0 \right\}$

$\text{(iii)}$ If $(\phi_{j,1}, \dots, \phi_{j,n})$ and $(\psi_{j,1}, \dots, \psi_{j,n})$ are components of $\Phi_{j}$ and $\Psi_{j}$, respectively, then for all $\alpha$, $1\le i \le n$, and each $j$, there exists a positive constant $M$ that satisfies the following condition:

$$ | D^{\alpha} \phi_{j,i}(x) | \le M,\quad x\in U_{j} \\ | D^{\alpha} \psi_{j,i}(y) | \le M,\quad y\in B $$

$\text{(iv)}$ There exists some positive constant $R$ such that every collection of $R+1$ sets of $U_{j}$ has an empty intersection.