Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem
Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} \subset U$ with $\overline{V}$ compact). My problem is to show that $$\|f\|_{W^{m,p}(U)}\leq\delta\tag{2}.$$
Evans book (p. 252) says that $(2)$ is obtained by taking the supremum in $(1)$ over sets $V\subset\subset U$ and Leoni's book (p. 285), as well as the original paper, by applying the Lebesgue monotone convergence theorem. However, none of them gives the details. So, I'd like to know if the proof below is correct.
Take a sequence of sets $\{V_{k}\}_{k=1}^\infty$ such that
- $V_k\subset\subset U$ for all $k\in\mathbb{N}$;
- $V_k\subset V_{k+1}$ for all $k\in\mathbb{N}$;
- $U=\bigcup_{i=1}^{\infty} V_k$.
Fix $\alpha$ such that $|\alpha|\leq m$. For each $k\in\mathbb{N}$, define $f_k:U\to\mathbb{R}$ by setting $f_k(x)=|D^\alpha f(x)|^p$ if $x\in V_k$, and $f_k(x)=0$ if $x\in U\setminus V_k$. We have $$0\leq f_k(x)\leq f_{k+1}(x)$$ for all $k\in\mathbb{N}$ and all $x\in U$. Furthermore, $$\lim_{k\to\infty} f_k(x)=|D^\alpha f(x)|^p$$ for all $x\in U$. Thus, by the Lebesgue monotone convergence theorem, $$\|D^\alpha f\|_{L^p(U)}^p=\int_U|D^\alpha f(x)|^p\; dx=\int_U\lim_{k\to\infty}f_k(x)\; dx=\lim_{k\to\infty}\int_{U}f_k(x)\; dx$$ $$=\lim_{k\to\infty}\int_{V_k}f_k(x)\; dx=\lim_{k\to\infty}\int_{V_k}|D^\alpha f(x)|^p\; dx=\lim_{k\to\infty}\|D^\alpha f\|_{L^p(V_k)}^p.$$ Summing with respect to $\alpha$, we obtain $$\|f\|_{W^{m,p}(U)}^p=\sum_{|\alpha|\leq m}\|D^{\alpha}f\|_{L^p(U)}^p= \sum_{|\alpha|\leq m} \lim_{k\to\infty}\|D^\alpha f\|_{L^p(V_k)}^p=\lim_{k\to\infty}\sum_{|\alpha|\leq m} \|D^\alpha f\|_{L^p(V_k)}^p$$ $$=\lim_{k\to\infty}\|f\|_{W^{m,p}(V_k)}^p\leq \delta^p.$$
The proof looks correct to me, well done!
A couple of nitpicks: in the definition of $V \subset \subset U$ you forgot to include that $\overline{V}$ needs to be compact; the very last $\delta$ should be $\delta^p$.
The only thing that I would like to add is an explicit definition for $V_k$: $$V_k := \Big\{x \in U : \text{dist}(x, \partial U) > \frac{1}{k}\Big\} \cap B(0,k)$$