1

let $D(\Omega)$ is the space of $C^{\infty}$ functions with compact support, with the usual notion of convergence we give in the context of distributions.

In my notes, before showing that $H^{-1} \subset D'(\Omega)$, my professor wrote that

$D(\Omega) \hookrightarrow H_0^1(\Omega)$ is a continuous embedding

I can't understand why this is true. If I consider the immersion $\mathcal{i}: D(\Omega) \rightarrow H_0^1(\Omega)$ with $i(v)=v$, I should be able to prove $$||i(v)||_{H_0^1}=||v||_{H_0^1} \leq ||v||_{D(\Omega)}$$ but I do not know what is the norm on $D(\Omega)$! I am missing some fundamental thing here, and indeed in my book I can't find what is the norm given on $D(\Omega)$. What am I missing?

bobinthebox
  • 185
  • 1
  • 12
  • The usual topology is not normable iirc. You’re gonna have to check a family of seminorms – Calvin Khor Nov 14 '21 at 09:23
  • I'm sorry, I don't understand the last part. What do you mean ? Which seminorms do I have to consider, and why ? @CalvinKhor – bobinthebox Nov 14 '21 at 09:24
  • 1
    As mentioned by @CalvinKhor the space $\mathscr{D}(\Omega)$ is not a normed space. To understand its topology and for a simple criterion of continuity you can apply to your map see https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Nov 14 '21 at 09:50
  • So, I should check $||v||{H_0^1} \leq C \max{x \in K} |D^{\alpha} v(x)|$ ? I'm pretty new to the topic @AbdelmalekAbdesselam – bobinthebox Nov 14 '21 at 10:19
  • Posted a partial answer to clarify this. – Abdelmalek Abdesselam Nov 15 '21 at 11:24

1 Answers1

1

As per Exercise 3 in the answer I linked to

Doubt in understanding Space $D(\Omega)$

what one must show is that for every continuous seminorm $\eta$ on $H_0^1(\Omega)$, there exists a continuous seminorm $\rho$ on $\mathscr{D}(\Omega)$ such that, for all $v\in\mathscr{D}(\Omega)$, $$ \eta(i(v))\le \rho(v)\ . $$ Now it is enough to do this $\eta$'s which define the locally convex topology of $H_0^1(\Omega)$, namely just $\eta=\|\cdot\|_{H_0^1}$.

To construct $\rho$, just take $\rho(v):=\|i(v)\|_{H_0^1}$. To show it is a continuous seminorm, it is enough to show it is an admissible seminorm on $\mathscr{D}(\Omega)$, i.e., belongs to the set $B$ given in Example 2 of the post I mentioned.

More concretely, what you need to show is that for all compact $K\subset\Omega$, there exists $C\ge 0$ and $N\ge 0$, such that for all $v\in\mathscr{D}(\Omega)$ with support contained in $K$, we have $$ \|v\|_{H_0^1}\le C\sup_{x\in K}\max_{|\alpha|\le N} |D^{\alpha}v(x)| $$ which a multivariate calculus exercise.

Abdelmalek Abdesselam
  • 5,278