# compactly supported harmonic functions [closed]

Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist?

Thanks!

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## closed as off topic by Qiaochu Yuan, Chris Godsil, quid, Vidit Nanda, unknown (google) Aug 7 '12 at 3:11

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Probably this question is not well suited for this site, as it is supposed to deal with research level ones; maybe you can try math.stackexchange.com instead, where you can get better responses. By the way, there aren't any nontrivial examples of such functions (think of maximum principle). – Mateusz Wasilewski Aug 6 '12 at 20:32
Your title concerns harmonic functions and your actual question does not. This is not encouraging... – Yemon Choi Aug 6 '12 at 20:45
It might also have helped if you had given some indication that you had tried the case $n=1$, which is a natural place to start testing a question for general $n$... – Yemon Choi Aug 7 '12 at 0:21

The only such functions are $0$.
Compact support implies $$\int_{\Omega} \Delta u = 0.$$
This along with the subsolution hypothesis means that $\Delta u = 0$. Any compactly supported harmonic function is identically zero by analyticity.
What do you mean by analyticity for $n\neq 2$? – Yemon Choi Aug 6 '12 at 20:45
@Yemon: It means at each point $x$ the Taylor series of $u$ converges on a nontrivial neighbourhood of $x$. – timur Aug 6 '12 at 20:51