How does Sobolev norm change with respect to domain size?

Question

Suppose $u\in H^s(0,r)$, let's say $s\ge 0$. It then follows that $u|_{(0,r')}\in H^s(0,r')$ for any $r'<r$. But is it true that $|u|_{H^s(0,r)}\rightarrow 0$ as $r\rightarrow 0$? How about negative s, too?

Answer 1

I believe it is true using Lebesgue dominated convergence for $s \geq 0$. For exemple for $s=1$ you write, for $r' < r$ :

$$\int_{B_{r'}} |u|^2 \ \mathrm{d}x + \int_{B_{r'}} |\nabla u|^2 \ \mathrm{d}x = \int_{B_{r}} \mathcal{1}_{B_{r'}}|u|^2 \ \mathrm{d}x + \int_{B_{r}} \mathcal{1}_{B_{r'}}|\nabla u|^2 \ \mathrm{d}x $$ and you use Lebesgue dominated convergence when $r' \rightarrow 0$ on each integrals.