2

Let $\Omega\subset\mathbb{R}^d$ be an open set. $C^k(\Omega)$ is defined as the space of functions $f:\Omega\to\mathbb{R}$ such that $\partial^nf$ is continuous for $0\leq|n|\leq k$.

There are usually two equivalent definitions for the space $C^k(\overline\Omega)$ one can see in references: $$ C^k(\overline\Omega)=\{f\in C^k(\Omega)\mid \partial^\alpha f \textrm{ has a continuous extension on } \overline\Omega\textrm{ for }|\alpha|\leq k\} $$

$$ C^k(\overline\Omega)=\{f\in C^k(\Omega)\mid \partial^\alpha f \textrm{ is uniformly continuous on } \Omega\textrm{ for }|\alpha|\leq k\} $$ See e.g. Folland's Introduction to Partial Differential Equations or Evans's Partial Differential Equations. I read somewhere that $C^{k}(\overline\Omega)$ is defined explicitly as the following $$ C^k(\overline\Omega)=\{g|_\Omega\mid g\in C^k(\mathbb{R}^d)\}. $$ But I really don't remember where. Could anybody come up with a reference with this definition?

  • $\Omega$ is bounded (so that the uniform continuity on $\overline{\Omega}$ follows from the continuity), and all your definition are equivalent – reuns Sep 22 '16 at 01:51

1 Answers1

0

In Hermann Sohr's The Navier-Stokes Equations — An Elementary Functional Analytic Approach (page.23), $C^k(\overline\Omega)$ means the space of all restrictions $u|_{\overline\Omega}$ to $\overline\Omega$ of functions $u\in C^k(\mathbb{R}^n)$ such that $$ \sup_{|\alpha|\leq k,x\in\mathbb{R}^n}|D^\alpha u(x)|<\infty. $$ Here $|\alpha|\leq k$ is replaced by $|\alpha|<\infty$ if $k=\infty$.