Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and let $ p\geq 1$. Let $(u_n)_n$ be a bounded sequence of $W_0^{1, p}(\Omega)$ and let $\Omega_n\subset\Omega$ be a subset of $\Omega$ which depends only 0n $n$ and such that $$meas(\Omega_{n})\longrightarrow 0 \quad \mbox{ as } n\to +\infty.$$ Could I conclude that $$\int_{\Omega_{n}} \vert\nabla u_n\vert^{p} dx\longrightarrow 0 \quad \mbox{ as } n\to +\infty?$$ If not, what additional assumptions I need?
Could anyone please help? Thank you in advance!
Consider $n=1$, $\Omega=(0,1)$ and a sequence $u_{n}$ of piecewise linear bumps of height $1$ on $(0,\frac{1}{n})$: $$ u_{n}(x) = \begin{cases} 2nx, & \text{for } 0<x\le \frac{1}{2n}\\ 1-2n\cdot(x-\frac{1}{2n}), & \text{for } \frac{1}{2n}\leq x\leq \frac{1}{n}\\ 0, & \text{for } \frac{1}{n} \le x < 1 \end{cases} $$ The sequence is bounded in $W^{1,1}(\Omega)$, as $||u_{n}||_{L^{1}} \le \frac{1}{n}$ and $||u_{n}'||_{L^{1}} = 2$ for all $n$.
(Note that $|u_{n}'(x)|=2n$ on $(0,\frac{1}{n})$ and $0$ everywhere else).
Now choose $\Omega_{n}=(0,\frac{1}{n})$.
Then $vol(\Omega_{n}) \rightarrow 0$, but $||u_{n}'||_{L^{1}(\Omega_{n})} = 2$ for all $n \in \mathbb{N}$.