0
$\begingroup$

Can you please help me to prove that bounded harmonic function is constant?

Thanks a lot!

  • 5
    -1 No effort shown in this question at all.2012-01-28

2 Answers 2

12

If $u$ is a harmonic function then there exists a conjugate function $v$ and an analytic function $f=u+iv$. Thus $\exp(f)$ is bounded, applying the Liouville's Theorem shows that $\exp(f)$ is constant.It's easy to prove that $f$ is constant, as well as $u$.

  • 0
    @PKStyles... It seems fioreb assumes we are in dimension $2$, which the OP never said.2017-10-21
13

E. Nelson, "A Proof of Liouville's Theorem", Proc. Amer. Math. Soc. 12 (1961) 995

9 lines long. Not the shortest paper ever, but maximizes importance/length ...

http://www.jstor.org/stable/2034412

http://en.wikipedia.org/wiki/Harmonic_function#Liouville.27s_theorem

Edward Nelson's paper is freely and legally available here:

$\bullet\ $ pdf file,

$\bullet\ $ html page.

  • 0
    Very nice and short proof.2012-01-28