Let $U\subseteq \mathbb{R}^n$ be an open set and let $f\in\mathcal{C}^k(U)$ for some positive integer $k$.
Are the following definitions of $\mathcal{C}^k$ regularity "up to boundary" equivalent?
(1) There exists an open set $V$ containing $U$ and a $\mathcal{C}^k$ extension of $f$ to $V$.
(2) $D^{\alpha}f$ is uniformly continuous on every bounded subset of $U$ for each $|\alpha|\leq k$.
I have always interpreted the class $\mathcal{C}^k\left(\overline{U}\right)$ using definition (1) but Evans-PDE uses definition (2) in its appendix (at least in the edition I am working from).
I can easily show that (1)$\Rightarrow$(2) but it would be nice to know if Evans definition is more general than mine.
Best Answer
They are equivalent using a differentiable version of the Tietze extension theorem: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077492502