Boundedness in $H^1(\Omega) $ and tightness

Question

Let $\Omega$ be a bounded subset of $\Bbb R^d$. And let $ (u_n)_n$ be a bounded sequence of the Sobolev space $H^1(\Omega)$.

Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: For very $\varepsilon>0$ there exists a compact set $K_\varepsilon\subset \Omega$ such that $$ \sup_{n}\int_{\Omega\setminus K_\varepsilon }|u_n(x)|^2dx<\varepsilon$$

I failed to find a counter example. Any help or reference in which I can find related topic is welcome.

Answer 1

The Sobolev embedding gives that

$$ \| u_n\|_{L^p} \le C \| u_n\|_{H^1},$$

where $p = 2^*>2$ when $d \ge 3$ and is any $p >2$ when $d=1, 2$. Thus we have

$$ \int_{\Omega\setminus K} |u_n|^2 \le \left( \int_{\Omega\setminus K} |u_n|^p\right)^{2/p} \big(\operatorname{Vol} (\Omega \setminus K)\big)^{1/q}\le C_1 \big(\operatorname{Vol} (\Omega \setminus K)\big)^{1/q}$$

for any set $K$. Now for any $\epsilon >0$, choose $K_\epsilon$ so that

$$C_1 \big(\operatorname{Vol} (\Omega \setminus K_\epsilon)\big)^{1/q}<\epsilon$$

and you are done.