Liouville theorems in halfspaces for parabolic hypoelliptic equations
type: Item
instance of: scholarly article
title: Liouville theorems in halfspaces for parabolic hypoelliptic equations
zbMATH Open document ID: 1121.35030
author: Elisabetta Alessia Kogoj
author: Ermanno Lanconelli
published in: Ricerche di Matematica
publication date: 2007-07-19T00:00:00Z
review text: Let \(L\) be a second order, parabolic partial differential operator of the form \(L=\sum_{j=1}^{m}X_j^2 +X_0- \partial_t\) in \(\mathbb R^{N+1}\), where the \(X_j\) are first order, linear partial differential operators on a homogeneous Lie group in \(\mathbb R^N\), with infinitely differentiable coefficients. The authors extend known results about solutions of classical parabolic equations, to solutions of \(Lu=0\). The first of the two main theorems states that, if \(u\) is a lower bounded solution to \(Lu=0\) in the half-space \(\mathbb R^N\times\left]-\infty,T\right[\) for some \(T\in \mathbb R\), then \(\lim_{s\to\infty}u(\gamma(s),T-s)=\inf u\) whenever \(\gamma:\left[0,\infty\right[\to \mathbb R^N\) is a continuous curve such that \(\limsup_{s\to\infty}s^{-1}|\gamma(s)|^2<\infty\). The second states that, if \(u\) is a nonnegative solution of \(Lu=0\) in \(\mathbb R^N\times\left]-\infty,0\right[\), and continuous onto \(\mathbb R^N\times\{0\}\) with \(u(x,0)=O(|x|^n)\) as \(|x|\to\infty\) for some real number \(n\), then \(u\) is constant. Generally the references to results on classical equations are adequate, but there is one unfortunate exception. The second of the main theorems depends crucially on Corollary 4 of the paper, and the authors fail to mention that a more general result for the heat equation (than Corollary 4) was proved over thirty years ago, with a similar proof, by the reviewer in [Proc. Lond. Math. Soc. (3) 33, 251--298 (1976; Zbl 0336.35046)].
reviewed by: Neil A. Watson
Mathematics Subject Classification ID: 35H20
zbMATH DE Number: 5173039
zbMATH Keywords: Parabolic operators
zbMATH Keywords: Liouville Theorems
zbMATH Keywords: Liouville Theorems in half-spaces
MaRDI profile type: Publication
full work available at URL: s11587-006-0015-9
OpenAlex ID: W2011893989
cites work: Liouville theorems for heat functions
cites work: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérées
cites work: Phragmén-Lindelöf-type theorems and Liouville theorems for a linear parabolic equation
cites work: Classical, viscosity and average solutions for PDE's with nonnegative characteristic form
cites work: A note on the heat equation
cites work: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations
cites work: Q4672094
cites work: Q4108682
DOI: 10.1007/S11587-006-0015-9
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description: scientific article; zbMATH DE number 5173039