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As the title say, I'm trying to figure out why is the following true: $$v \in H_2(\Omega) \Rightarrow v \in C(\Omega)$$ with $H_2(\Omega) = \{f: \Omega \to \mathbb{R}: \|f^{(k)}\|_{L_2} < \infty, \forall 0 \leq k \leq 2\}$ (say $\Omega = [0,1]$). I may have missed something in the statement but I think the idea is here. Google wasn't very helpful :'(

We use that to show that the solution of elliptic PDE's (which are in $H_2$ if the right-hand side is in $L_2$) are continuous.

Thanks a lot !

Shawn
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1 Answers1

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This is indeed the Sobolev embedding theorem. Perhaps view the bottom of page 160 here: https://www.math.ucdavis.edu/~hunter/book/ch7.pdf

user288742
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  • Thanks a lot, I think that's it. Only thing is that it seems to depend on the dimension of $\Omega$ and in $\mathbb{R}$, $H^1$ seems enough. We need $H^2$ in 2D though. At least I know the name of the property now :) – Shawn Jun 04 '16 at 03:24