Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of this paper.
Let $G$ be an bounded open connected subset of $\mathbb{R}^n$ with $n\geqslant 2$. Let $\mathcal{F}=\{S_i\}_{i=1}^m$ be open, pairwise disjoint subsets of $G$. Let $E,F$ be compact subsets of $G$ such that $\mathcal{F}\cup\{E,F\}$ are pairwise disjoint. Fix $x\in E$ and $y\in F$. Define \begin{equation*} u_i(z)=\inf_{\lambda\in \Gamma_G(z,x)}\mathrm{len}(\lambda^\ast\cap S_i),\quad z\in G,\quad 1\leqslant i\leqslant m, \end{equation*} and, assuming $u_i(y)>0$ for all $1\leqslant i\leqslant m$, where ''$\Gamma_G(z,x)$'' stands for the set of all rectifiable curves $\lambda\colon [a,b]\to G$ with the endpoints $z$ and $x$, $\lambda^\ast$ stands for the image of $\gamma$, i.e. $\lambda^\ast=\gamma([a,b])$, and ''$\mathrm{len}$'' stands for the curve length. Precisely, \begin{equation*} \mathrm{len}(\lambda^\ast):=\sup\sum_{i=1}^k |\lambda(t_i)-\lambda(t_{i-1})|,\quad \text{where\ } a=t_0<t_1<\cdots<t_k=b, \end{equation*} with the supremum taken over all partitions of $\{t_i\}$, ''rectifiable'' means $\mathrm{len}(\lambda^\ast)<\infty$, and ''$\mathrm{len}(\lambda^\ast\cap S_i)$'' means the length of $\lambda$ in $S_i$. Define \begin{equation*} u(z)=\dfrac{1}{m}\sum_{i=1}^m \dfrac{u_i(z)}{u_i(y)},\quad z\in G. \end{equation*} Then the paper said $u$ is Lipschitz (this paper does not give the exact definition of ''Lipschitz''), is constantly zero on $E$, and is constantly $1$ on $F$. In particular, if $\mathcal{F}=\{S_i\}_{i=1}^m$ satisfies \begin{equation*} \mathrm{len}(\lambda^\ast\cap S_i)\geqslant \mathrm{diam}(S_i)/C,\quad \forall \lambda\in \Gamma_G(z,x),\quad 1\leqslant i\leqslant m, \end{equation*} where ''$\mathrm{diam}$'' stands for the diameter of $S_i$, that is, \begin{equation*} \mathrm{diam}(S_i)=\sup\{|x-y|: x,y\in S_i\}, \end{equation*} and ``$C$'' is a constant bigger than $1$ and irrelevant with any points of $G$ and any $S_i$, then \begin{equation*} \|\nabla u_i(\cdot)/u_i(y)\|_{L^\infty(G)}\leqslant C/\mathrm{diam}(S_i) \end{equation*} and thus \begin{equation}\tag{*} \sum_{i=1}^m \int_{\overline{S_i}}|\nabla u|^n \leqslant \sum_{i=1}^m \dfrac{C^n |\overline{S_i}|}{m^n\mathrm{diam}(S_i)^n} \leqslant m^{1-n}C^n, \end{equation} where $\overline{S_i}$ stands for the closure of $S_i$ and $|\overline{S_i}|$ stands for the Lebesgue measure of $\overline{S_i}$.
My questions are:
(1) Why $u$ is Lipschitz?
(2) Why $u_i$ is differentiable?
(3) (*) seems implies that $|\nabla u_i|$ is support on $S_i$. So, why this is true? Here ``support'' means $|\nabla u_i|$ is nonzero on $\overline{S_i}$, but is zero on $G\setminus \overline{S_i}$. And $|\nabla u_i|$ \begin{equation*} |\nabla u_i|=\Bigl(\sum_{k=1}^n (\partial_k u_i)^2\Bigr)^\frac{1}{2}. \end{equation*}
About the first and second questions, I have some considerations. First I give the definition of ``Lipschitz''. A function $f\colon X\to Y$ from a metric space $X=(X,d_X)$ to a metric space $Y=(Y,d_Y)$ is said to be $L$-Lipschitz if there exists a constant $L\geqslant 0$ such that \begin{equation*} d_Y(f(a),f(b))\leqslant Ld_X(a,b) \end{equation*} for each pair of points $a,b\in X$. We also say that a function is Lipschitz if it is $L$-Lipschitz for some $L$. As I know, \begin{equation*} d_G(x,y):=\inf_{\gamma\in \Gamma_G(x,y)}\mathrm{len} (\gamma),\quad x,y\in G \end{equation*} induces a metric, which is called the intrinsic metric. Then following the proof to show that $d_G(x,y)$ is a metric, I think \begin{equation*} d_{S_i}(x,y):=\inf_{\gamma\in \Gamma_G(x,y)}\mathrm{len} (\gamma\cap S_i),\quad x,y\in G \end{equation*} is also a metric. Then by the triangle inequality, we know that for a fix point $x_0\in G$, \begin{equation}\tag{**} |d_{S_i}(x,x_0)-d_{S_i}(y,x_0)|\leqslant d_{S_i}(x,y). \end{equation} Thus we know that $u_i$ is a $1$-Lipschitz function with respect to the intrinsic metric. But the former process cannot explain why $u_i$ is differentiable. As we know, Rademacher's theorem yields that a Lipschitz function is differentiable almost everywhere. But Rademacher's theorem requires that the metric is the usual Euclidean metric, that is, replace $d_{S_i}(x,y)$ with $|x-y|$ in (**). So it seems that we cannot use Rademacher's theorem directly. Maybe there is an extension that Rademacher's theorem is also true with respect to the intrinsic metric?
