3
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

A domain is usually defined as an open connected set.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.