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The maximum-minimum principle says that

A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant.

Here is my question:

If we restrict our attention in ${\mathbb R}^2$ or ${\mathbb R}^3$, what's the hypothesis for the domain? (bounded? closed? open?)

According to the proof of this principle, it seems that the domain is open. I could not find the context which may indicate the properties of the domain.

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3 Answers

up vote 3 down vote accepted

A domain is usually defined as an open connected set.

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If the maximum principle is stated in your book exactly as you stated, then what the book means by a domain is open, connected, bounded set. For the maximum principle is not valid for unbounded domains. But I must say usually domain means just open connected set. Often times "nonempty" is also included in the definition.

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In Folland's Introduction to Partial Differential Equations, the word domain is used to mean an open set $\Omega\subset{\mathbb R}^n$, not necessarily connected, such that $\partial \Omega=\partial({\mathbb R}^n\setminus\overline{\Omega})$. That is, all the boundary points of $\Omega$ are "accessible from the outside."

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