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Liouville theorem for superharmonic functions states that

Any bounded function $f:\mathbb R^n\to\mathbb R$ admitting an inequality $\Delta f\leq 0$ on $\mathbb R^n$ is a constant function.

Here $\Delta$ is a Laplacian. I wonder what are the extension of this theorem to other class of operators, i.e. what are necessary and what are sufficient conditions for this theorem to hold.

I am especially interested if there is such a theorem for a discrete Laplacian.

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I think Byron's and George's answers here fit the bill of (part of) what you're looking for. See also (strong) elliptic operators, maximum principle and Hopf maximum principle, so you have plenty of stuff to read while waiting for a serious answer. –  t.b. Aug 26 '11 at 11:42
This paper (found via google search) may be relevant. –  Matt E Aug 26 '11 at 11:53
@Theo: thank you, I've just taken a look, interesting. Matt E: thanks also, –  Ilya Aug 26 '11 at 15:37

2 Answers 2

up vote 3 down vote accepted

If you are interested in the case for the discrete Laplacian, check out the paper of Rigoli, Salvatori, and Vignati titled "Liouville properties on graphs" (DOI: 10.1112/S0025579300012031).

Among the results proven is the following:

Let $G$ be a graph and let $q$ be an arbitrary point in $G$. Let $u$ be a $p$-subharmonic function on $G$ for $p > 1$. Suppose that for all $R$ sufficiently large $$ \sup_{B_R(q)} u \lesssim \frac{(R\log R)^{(p-1)/p}}{|S_R(q)|^{1/p}} $$ and $$ |S_R(q)| \lesssim (R\log R)^{p-1} $$ where $S_R(q) = B_R(q) \setminus B_{R-1}(q) $ is the "sphere of the radius $R$", then $u$ is constant.

The requirement on the volume growth rate of balls of radius $R$ is typical: this is not just the case for graphs. Liouville theorems for non-compact, complete Riemannian manifolds are usually proven under the assumption of a lower bound on the Ricci curvature, which can be used to prove volume growth bounds on the Riemannian manifold (the simplest example being the Bishop-Gromov theorem).

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cool, it seems that now I can patiently wait for a serious answer as Theo Buehler suggested. –  Ilya Aug 26 '11 at 15:39

I thought that discrete Liouville theorem for the lattice $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206). But recently I knew from Alexander Khrabrov that there is an older article of Capoulade with almost the same result (Sur quelques proprietes des fonctions harmoniques et des fonctions preharmoniques, - Mathematica (Cluj), 6 (1932), 146-151.)

Unfortunately the last article is not available for me.

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Interesting, though I'm afraid I don't have an access to this paper either. –  Ilya Nov 22 '13 at 11:08
I've asked a question about "The origin of Discrete `Liouville's theorem'" at MathOverflow mathoverflow.net/questions/149621/… –  Alexey Ustinov Nov 22 '13 at 11:22

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