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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.

  • 0
    What if $u\equiv 0$? – 2017-02-16
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    Thank you. I added a condition that $u$ is non-constant. – 2017-02-16

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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.