Gradient estimates for variable coefficient parabolic equations and singular perturbation problems

1998 ◽  
Vol 120 (2) ◽  
pp. 391-439 ◽  
Author(s):  
Luis A. Caffarelli ◽  
Carlos E. Kenig
Keyword(s):  
Singular Perturbation ◽  
Parabolic Equations ◽  
Variable Coefficient ◽  
Singular Perturbation Problems ◽  
Gradient Estimates ◽  
Perturbation Problems

scholarly journals Singular perturbations for a subelliptic operator

2018 ◽  
Vol 24 (4) ◽  
pp. 1429-1451 ◽  
Author(s):  
Paola Mannucci ◽  
Claudio Marchi ◽  
Nicoletta Tchou
Keyword(s):  
Singular Perturbation ◽  
Singular Perturbation Problems ◽  
Value Functions ◽  
Infinitesimal Operator ◽  
Lipschitz Continuous ◽  
Subelliptic Operator ◽  
Whole Space ◽  
Perturbation Problems ◽  
Ergodic Problem ◽  
Logarithmic Growth

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.

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