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Recently I was reading the theory of elliptic operators and there was a statement like this: given an elliptic partial differential operator $L$ on $C^2(\mathbb{R}^n)$ the Liouville theorem accounts to the strong maximum principle (SMP). Unfortunately I cannot find now such place (and a paper in fact) - so I am curious what was the meaning of this statement.

That's why I wonder: is SMP sufficient for the Liouville theorem to hold for an elliptic operator $L$? From my side I tried to prove Liouville theorem using only SMP - but I haven't succeed.

I assume that SMP and Liouville theorem are known, but I can provide definitions if needed.

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    You might want to improve the question by stating the excact version of Liouville's theorem you are referring to (there are several), which script or book which you are reading, and a statement of the strong maximum principle.2011-06-21
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    @Glen: I've already mentioned that I do not remember where I read it exactly.2011-06-21
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    If you are just after the classical version, then you may be satisfied with Theorem 3.5 from Gilbart and Trudinger.2011-06-21
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    @Gortaur, apologies, it wasn't clear to me that you could not find the original reference. The proof of the classical version is quite simple and does indeed follow from the maximum principle (in a sense).2011-06-21
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    @Glen: could you please explain how does it follow? Btw, I read Gilbarg and Trudinger at the current moment.2011-06-21
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    @Glen: Slightly off-topic, but in case you don't know it look at [Nelson's nine-line paper from 1961](http://dx.doi.org/10.1090/S0002-9939-1961-0259149-4). A marvellous proof of the classical Liouville theorem from the mean-value property.2011-06-21
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    @Theo Thanks for the reference. I was aware of this proof but not aware of the reference!2011-06-21
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    @Gortaur If a function is bounded above (or below) on $\mathbb{R}^n$ then it achieves its maximum (or minimum) on it also (not at infinity).2011-06-21
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    @Theo: this proof I usually saw for this theorem. The language is similar to the Nash paper on his equilibrium. Apparently, that time they didn't have to write formulas.2011-06-21
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    @Glen: $\mathrm e^x$ - bounded from below.2011-06-21
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    @Gortaur I think you need to check the conditions of Theorem 3.5 from G&T. In particular those on the elliptic operator. For that function, you need to have $c<0$, and so only reach a contradiction for functions which are *bounded above*. (As you have observed.)2011-06-21
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    @Glen: I've only commented on your statement "If a function is bounded above (or below) on $\mathbb{R}^n$ then it achieves its maximum (or minimum) on it also (not at infinity)". This statement together with 3.5 leads to Liouville theorem - but this statement does not hold in general.2011-06-21
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    @Gortaur I don't see your point: $e^x$ is bounded below and achieves a minimum at zero. It is not bounded above.2011-06-21
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    @Glen: It doesn't achieve a minimum, does it?2011-06-21
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    @Glen and @Gortaur: such conversations are probably best carried out using our awesome chat system. Using the comments for it is rather detracting from the purpose of the comments, somewhat cluttering, and rather inefficient.2011-06-21
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    BTW, I retagged. Neither elliptic functions nor harmonic analysis are appropriate tags for this question.2011-06-21
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    @Gortaur I was thinking of a different exponential map (too much differential geometry). It seems I was wrong about Theorem 3.5 from G&T as you rightly pointed out. It only gives the statement of Liouville's theorem if you instead of boundedness assume an achieved maximum or minimum, which is plainly silly.2011-06-21
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    @Willie I had no intention of such a long comment exchange. I'll take anything further to chat.2011-06-21
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    Sure, thanks anyway.2011-06-21

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In general I don't think Liouville's theorem follows from the SMP (at least I cannot work out a proof at the moment). The SMP is of the formal form that for a certain class of functions $\mathcal{F}$ defined over $\mathbb{R}^d$

If $f\in \mathcal{F}$, let $\Omega$ be bounded with (say) smooth boundary, then for any $\Omega' \Subset \Omega$ you have $\sup_{\Omega'} f \leq \sup_{\partial\Omega} f$ with equality only when $f$ is constant.

Using just this it is clearly impossible to conclude Liouville's theorem, since $f(x) = \tan^{-1}(x_1)$ satisfies the above "abstract" theorem but not Liouville, which says that

If $f\in \mathcal{F}$, and if $|f|$ is bounded, then $f$ is constant.

As far as I know, to actually obtain Liouville's theorem from the general sorts of argument you get for uniformly elliptic operators, you need to appeal to something more quantitative (in particular a way to put a bound on the growth rate). The usual arguments require a use of Harnack's inequality. Now, Harnack's inequality can also be used (as a starting point to derive Hopf-like lemmas) to establish the strong maximum principle, which maybe where you saw the connection drawn between SMP and Liouville.

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    As I understand, with $\mathcal{F}$ you denote some class of functions. That is exactly what I mean. SMP alone seems not to be sufficient - because Harnack's inequality also does not follow from SMP. As I can see from Gilbarg and Trudinger, they used mostly mean-value theorem rather then SMP. That's why I cannot understand arguments by Glen.2011-06-21