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


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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

1 Answer 1

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.

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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
@Yemon: Yes, what timur said. One way to show this is to use the interior gradient estimates for harmonic functions (which in turn follow from the mean value property, which holds in all dimensions). –  Connor Mooney Aug 6 '12 at 21:58
@timur, @Connor: thanks, so you did indeed mean real-analyticity. (I wasn't quite sure when quickly skimming over what you wrote) –  Yemon Choi Aug 7 '12 at 0:18

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