Sobolev space,notation

Question

There are two notions of Sobolev space: $H_0^{k,p}(\Omega)$ and $H^{k,p}(\Omega)$ as a closure of $C_0^\infty(\Omega)$ and $C^\infty(\Omega)$, respectively,w.r.t. $\|\cdot\|_{W^{k,p}(\Omega)}$. In my source the definitions of $C_0^\infty(\Omega)$ and $C^\infty(\Omega)$ have been omitted. What do they mean by this?

Answer 1

$C^\infty(\Omega)$ denotes the set of all infinitely differentiable real-valued functions defined on $\Omega$.

$C_0^\infty(\Omega)$ denotes the set of all functions in $C^\infty(\Omega)$ that vanish off a compact subset of $\Omega$.

Answer 2

• $C^{\infty}(\Omega)$ is the space of all functions that are smooth (infinitely continuously differentiabl) on $\Omega$, for example $$ C^{\infty}(\mathbb{R}) = \left\{ f:\mathbb{R}\to\mathbb{C} \;\middle|\; \text{$f^{(n)}$ exists for all $n$} \right\}. $$

• $C_0^{\infty}(\Omega)$ is a little ambiguous, and could mean a couple of things, depending on the author. To some authors, it means the space of smooth which tend to zero on the boundary. For example, $$ C_0^{\infty}(\mathbb{R}) = \left\{ f:\mathbb{R}\to\mathbb{C} \;\middle|\; \text{$f^{(n)}$ exists for all $n$ and $\lim_{|x|\to\infty} f(x) = 0$} \right\}. $$ In my experience, this is the more common meaning of the notation.

  That being said, other authors will take $C_0^{\infty}(\Omega)$ to denote the set of all smooth functions which have compact support in $\Omega$. That is, $f\in C_0^{\infty}(\Omega)$ if $$ \operatorname{supp}(f) = \overline{ \{x\in \Omega : f(x) \ne 0 \}} $$ is a compact subset of $\Omega$. You will sometimes see the notation $$ \operatorname{supp}(f) \Subset \Omega \qquad\text{or}\qquad \operatorname{supp}(f) \subset\subset \Omega, $$ which can be read as "the support of $f$ is compactly contained in $\Omega$." Essentially, this ensures that functions which are compactly supported in $\Omega$ vanish at the boundary, so that the space of compactly supported functions is contained in the space of functions which vanish at the boundary.

  You will also see this space denoted by $C_c^{\infty}(\Omega)$, and the potential ambiguity suggests that you should check whatever book or paper you are reading for some clarification here. If you are lucky, there is an index of notation used somewhere.