Strong Local Lipschitz Condition

Definition¹

If there exists a locally finite open cover $\left\{ U_{j} \right\}$ of $\delta \gt 0$, $M \gt 0$, and $\mathrm{bdry}\Omega$, such that for each $j$, there is a real-valued function $f_{j}$ with $n-1$ variables satisfying $\text{(i)}$ ~ $\text{(iv)}$, then the open set $\Omega \subset \mathbb{R}^n$ satisfies the strong local Lipschitz condition.

For all pairs $x,y\in$ $\Omega_{\lt \delta}$ that satisfy $|x-y| \lt \delta$, there exists $j$ that satisfies the condition below.

$$ x,y\in V_{j}=U_{j\gt\delta}=\left\{ z\in U_{j} : \mathrm{dist}(z,\ \mathrm{bdry}U_{j}) \gt \delta\right\} $$

$\text{(ii)}$ Each function $f_{j}$ satisfies the Lipschitz condition with Lipschitz constant $M$. That is, if

$$ |f(\xi)-f(\rho)|\le M|\xi-\rho| $$

$\text{(iii)}$ For some orthogonal coordinate system $(\zeta_{j,1},\ \cdots,\ \zeta_{j,n})\in U_{j}$, $\Omega \cap U_{j}$ is represented by the following inequality:

$$ \zeta_{j,n} \gt f_{j}(\zeta_{j,1},\ \cdots,\ \zeta_{j,n-1}) $$

$\text{(iv)}$ There exists some positive number $R$, such that the intersection of all collections of $R+1$ of the sets $U_{j}$ is empty.