Dipartimento d'Ingegneria Civile e Ambientale (DICEA)
Università di Padova Via Marzolo, 9
35131 Padova, Italy
email: nicola.garofalo@unipd.it
Nicola Garofalo
Abstract:
In this note I present some properties of sub-Laplaceans associated with a
collection of smooth vector fields satisfying Hormander's finite rank
assumption. One notable aspect of this paper is the development of the
fractional powers of sub-Laplaceans as Dirichlet-to-Neumann maps of an
extension problem inspired to the famous 2007 work of Caffarelli and Silvestre
for the standard Laplacean. A key tool is an extension problem for the
fractional heat equation for which I compute the relevant Poisson kernel. I
then use the latter to: (1) find the Poisson kernel for the time-independent
case; and (2) solve the extension problem.
Published September 15, 2018.
Math Subject Classifications: 35C15, 35K05, 35J70.
Key Words: Sub-Laplaceans; mean-value formulas; fractional powers;
extension problem.
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Nicola Garofalo Dipartimento d'Ingegneria Civile e Ambientale (DICEA) Università di Padova Via Marzolo, 9 35131 Padova, Italy email: nicola.garofalo@unipd.it |
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