Volume 24, Number 4, October–December 2018
|Page(s)||1429 - 1451|
|Published online||26 October 2018|
Singular perturbations for a subelliptic operator★,★★
Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova,
Via Trieste 63,
2 Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Padova, Via Gradenigo 6/b, 35131, Padova, Italy
3 IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
* Corresponding author: email@example.com
Accepted: 8 September 2017
We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.
Mathematics Subject Classification: 35B25 / 49L25 / 35J70 / 35H20 / 35R03 / 35B37 / 93E20
Key words: Subelliptic equations / Heisenberg group / invariant measure / singular perturbations / viscosity solutions / degenerate elliptic equations
The first and the second authors are members of the INDAM-Gnampa and are partially supported by the research project of the University of Padova “Mean-Field Games and Nonlinear PDEs” and by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”.
© EDP Sciences, SMAI 2018
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