What does this ensemble of symbols mean: $C^{2}(\bar{\Omega})$?

Question

For Context, I'll provid the sentence in which the symbols lie; it sets up the conditions for a proof of Green's Identity.

Let $\Omega \subset \Re^n$ be domain with a smooth boundary $\partial \Omega$. Let u,v $\in $ $C^{2}(\bar{\Omega})$, where $\bar{\Omega}$ denotes the closure of $\Omega$.

What does this ensemble of symbols mean: $C^{2}(\bar{\Omega})$ ?

Continuously twice-differentiable on some neighborhood of the closure of the domain $\Omega$. — Ted Shifrin, May 10 '18 at 05:01
Yes, an open set to make sense of partial derivatives along the boundary. — Ted Shifrin, May 10 '18 at 05:32
Eyy thanks a lot. Green's Identity makes use of the Laplacian, so having twice-differentiability as a given makes sense. — Ten-Thousand-Hands, May 10 '18 at 05:48

score 0 · Accepted Answer · answered May 10 '18 at 05:52

0

From [Adams-Fournier, p. 80]:

$C^j(\bar{\Omega})$ denotes the closed subspace of $C^j_b(\Omega)$ consisting of functions having bounded, uniformly continuous derivatives up to order $j$ on $\Omega$ normed by $$ \|\phi\|_{C^j(\bar{\Omega})} = \max_{0 \leq \alpha \leq j} \sup_{x\in \Omega} |D^{\alpha}\phi(x)|. $$

Here $C^j_b(\Omega)$ is the space of functions having bounded, continuous derivatives up to order $j$ on $\Omega$.

answered May 10 '18 at 05:52

Julienne Franz

265

Thank you kindly; I'll be looking at that book as well. – Ten-Thousand-Hands May 10 '18 at 05:54
You're welcome. – Julienne Franz May 10 '18 at 05:59

Osama Ghani · Answer 2 · 2018-05-10T05:04:21.887

-1

$C^2 (\bar{\Omega})$ represents the $C^2$ functions on this domain i.e. functions whose second derivatives (both pure and mixed) exist and are continuous.

Edit: As Shifrin points out below, this is incorrect.

edited May 10 '18 at 05:04

answered May 10 '18 at 05:02

Osama Ghani

2,862

1

No, that would be $C^2(\Omega)$. – Ted Shifrin May 10 '18 at 05:03
Ah you’re right! – Osama Ghani May 10 '18 at 05:03

What does this ensemble of symbols mean: $C^{2}(\bar{\Omega})$?

2 Answers2