Does anybody know the definition of $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$, where $0<\alpha<1$?

Question

I hope someone can give me the definition of the following:

$C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$ for some $0<\alpha<1$ and some domain $\Omega$.

In this context they also talk about $\Omega$ as a bounded and $C^{2+\alpha}$ domain. What does this mean? . (bounded is clear ;) )

I found this in a paper about non-local diffusion. (It is used in many theorems, but there is no definition...)

Does this rooms have an specific name? Sorry if this something I should know, but if you don't know the name if this it is really hard to find with Google ore something else ;).

Thanks for your help.

Answer 1

$C^{2,1}(\Omega)$ usually refer to the set of function that are $C^2$ on $\Omega$ with respect to the first variable, and $C^1$ with respect to the second one.

In case you have "fractional" numbers, it usually refers to the Holder condition as mentioned in the other answer.

We say that a certain domain (open and convex) $\Omega$ is $C^2$ if some specific conditions are met[0]; basically it tells you that locally, the boundary of $\Omega$ can be represented by the graph of a certain $C^2$ function

[0]

Formal definition

We say that $\Omega$ is a $C^1$ domain if for every $x \in \partial \Omega$, there exists a coordinate system $(y_1, \dots, y_n) \equiv (\bar y', y_n)$ with origin in $x$, a sphere $B(x)$ and a function $\varphi$, defined in a neighborhood $\mathcal N \subset \mathbb R^{n-1}$ of $\bar y' = 0$, such that

$$\varphi \in C^1(\mathcal N), \varphi(0) = 0$$ $$\partial \Omega \cap B(x) = \{(\bar y', y_n): y_n = \varphi(\bar y'), \bar y' \in \mathcal N\}$$ $$\Omega \cap B(x) = \{(\bar y', y_n): y_n > \varphi(\bar y'), \bar y' \in \mathcal N\}$$

The second condition reflects the fact that $\partial \Omega$ is locally the graph of a $C^1$ function, and the third that $\Omega$ is locally on the same side with respect to $\partial \Omega$.

If you want a $C^2$ domain or a Lipschitz domain replace the above $C^1$ with the corresponding space of functions ;-)

P.S. Definition taken from "Partial differential equation" by Salsa

Answer 2

My guess would be a Hölder condition.