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.

  • 1
    $\begingroup$ 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. $\endgroup$ – Glen Wheeler Jun 21 '11 at 8:53
  • $\begingroup$ @Glen: I've already mentioned that I do not remember where I read it exactly. $\endgroup$ – Ilya Jun 21 '11 at 8:59
  • $\begingroup$ If you are just after the classical version, then you may be satisfied with Theorem 3.5 from Gilbart and Trudinger. $\endgroup$ – Glen Wheeler Jun 21 '11 at 9:01
  • $\begingroup$ @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). $\endgroup$ – Glen Wheeler Jun 21 '11 at 9:02
  • $\begingroup$ @Glen: could you please explain how does it follow? Btw, I read Gilbarg and Trudinger at the current moment. $\endgroup$ – Ilya Jun 21 '11 at 9:06

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.

  • $\begingroup$ 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. $\endgroup$ – Ilya Jun 21 '11 at 11:58

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