I know Whitney Covering Lemma. But I do not know how it can be applied to PDE. More specifically, I do not know what is the intuition behind Whitney Covering Lemma.
First, let me state the lemma:
Let $\Omega$ be an open non-empty proper subset of $\mathbb{R}^n.$ Then there exists a family of closed cubes $\{Q_j\}_j$ such that
$(1)$ the union of the cube is the whole space, and the cubes have disjoint interiors.
$(2)$ $\sqrt{n}l(Q_j) \leq\operatorname{dist}(Q_j,\Omega) \leq 4 \sqrt{n} l (Q_j).$
$(3)$ If the boundaries of two cubes $Q_j$ and $Q_k$ touch then $\frac{1}{4} \leq \frac{l(Q_j)}{l(Q_k)} \leq 4$.
$(4)$For a given $Q_j,$ there exist at most $12^n Q_k$'s that touch it
Here, we define $l(Q)$ as the length of a cube.
My question is why the decomposition pick the constant like $4$. I want to know whether it will work if $4$ change to $3$ or other numbers. Why there are $12^n$ not let us say $2^n$?
Another question will be what the intuition of the lemma. Why we will use it in PDE?