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I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline \Omega)$, then $inf_{\partial \Omega} u \le u(x)\le sup_{\partial \Omega}u$. Since $u$ is continuous on $\partial \Omega$, so $u( \Omega)\subset u(\partial \Omega)$.

But the Weak Maximum\Minimum Principle didn't say anything about $\Omega$ connected. Where did I go wrong?

Can anyone offer me some help? Thanks so much!:D

  • $\begingroup$ Write down your analysis on any connected component of $\Omega.$ $\endgroup$ – mfl Oct 18 '14 at 22:19
  • $\begingroup$ @mfl Could you please give me more hint. I REALLY don't understand the connected part.:D $\endgroup$ – Sherry Oct 18 '14 at 22:20
  • $\begingroup$ Only a doubt. Does your definition of domain includes connectedness? $\endgroup$ – mfl Oct 18 '14 at 22:27
  • $\begingroup$ @mfl I don't know. Should definition of domain include connectedness? $\endgroup$ – Sherry Oct 18 '14 at 22:33
  • $\begingroup$ According to wikipedia yes: en.wikipedia.org/wiki/Domain_%28mathematical_analysis%29. On the other hand, forgot my first comment. It is wrong, because the boundary of some connected component may not be a subset of the boundary of $\Omega.$ $\endgroup$ – mfl Oct 18 '14 at 22:35

You seem to infer, from continuity of $u$, that every number between $\inf_{\partial\Omega}u$ and $\sup_{\partial\Omega}u$ is in $u(\partial\Omega)$. That won't work without some connectedness assumption.

  • $\begingroup$ Why do we need the connectedness assumption? Is it tivial or need to prove that? :D $\endgroup$ – Sherry Oct 18 '14 at 22:39
  • $\begingroup$ The intermediate value theorem, which seems to be what you're tacitly using here, has a hypothesis that the domain of the function in question is connected. $\endgroup$ – Andreas Blass Oct 18 '14 at 22:58

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