The Liouville theorem and linear operators satisfying the maximum principle.
Peer reviewed, Journal article
Published version
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Åpne
Permanent lenke
https://hdl.handle.net/11250/2678357Utgivelsesdato
2020Metadata
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- Institutt for matematiske fag [2486]
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Originalversjon
10.1016/j.matpur.2020.08.008Sammendrag
A result by Courrège says that linear translation invariant operators satisfy
the maximum principle if and only if they are of the form L = Lσ,b + Lμ
where
Lσ,b[u](x) = tr(σσTD2u(x)) + b · Du(x)
and
Lμ[u](x) =
Rd\{0}
u(x + z) − u(x) − z · Du(x)1|z|≤1
dμ(z).
This class of operators coincides with the infinitesimal generators of Lévy processes
in probability theory. In this paper we give a complete characterization of the
operators of this form that satisfy the Liouville theorem: Bounded solutions u of
L[u] = 0 in Rd are constant. The Liouville property is obtained as a consequence of
a periodicity result that completely characterizes bounded distributional solutions
of L[u] = 0 in Rd. The proofs combine arguments from PDEs and group theory.
They are simple and short.