Does the regularity of a function say anything about the regularity of the domain?

Question

Suppose I know that $u$ is non-constant and $u \in C^{0,\alpha}(\Omega)$ and $\Omega$ is at least bounded. Can we say anything more about the regularity of the domain $\Omega$? In particular, is it possible for $\Omega$ to be like a zigzag domain? Does it mean $\Omega$ is convex?

Thank you very much.

Edit: Added $u$ is non-constant.

Answer 1

No. The restriction of any smooth function to any open set will result in a smooth function. The same goes for $C^{0,\alpha}$, etc.