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Three-spheres theorems for subelliptic quasilinear equations in Carnot groups of Heisenberg-type


Authors: Tomasz Adamowicz and Ben Warhurst
Journal: Proc. Amer. Math. Soc. 144 (2016), 4291-4302
MSC (2010): Primary 35H20; Secondary 31C15, 53C17
DOI: https://doi.org/10.1090/proc/13050
Published electronically: March 25, 2016
MathSciNet review: 3531180
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Abstract: We study the arithmetic three-spheres theorems for subsolutions of subelliptic PDEs of $ p$-harmonic type in Carnot groups of Heisenberg type for $ 1<p<\infty $. In the presentation we exhibit the special cases of sub-Laplace equations ($ p=2$) and the case $ p$ is equal to the homogeneous dimension of a Carnot group. Corollaries include asymptotic behavior of subsolutions for small and large radii and the Liouville-type theorems.


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Additional Information

Tomasz Adamowicz
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland
Email: T.Adamowicz@impan.pl

Ben Warhurst
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Email: B.Warhurst@mimuw.edu.pl

DOI: https://doi.org/10.1090/proc/13050
Keywords: Carnot group, Hadamard theorem, Heisenberg group, Lie algebra, Lie group, Liouville theorem, maximum principle, $p$-harmonic, $p$-Laplace, subelliptic equation, sub-Laplace equation, sub-Riemannian, three-circles theorem, three-spheres theorem
Received by editor(s): August 4, 2015
Received by editor(s) in revised form: November 27, 2015
Published electronically: March 25, 2016
Communicated by: Jeremy T. Tyson
Article copyright: © Copyright 2016 American Mathematical Society