Two nonlinear days in Urbino 2017.
Electron. J. Diff. Eqns., Conference 25 (2018), pp. 103131.
Some properties of subLaplaceans
Nicola Garofalo
Abstract:
In this note I present some properties of subLaplaceans 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 subLaplaceans as DirichlettoNeumann 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 timeindependent
case; and (2) solve the extension problem.
Published September 15, 2018.
Math Subject Classifications: 35C15, 35K05, 35J70.
Key Words: SubLaplaceans; meanvalue 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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