CPAA
The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction
Paola Mannucci
Communications on Pure & Applied Analysis 2014, 13(1): 119-133 doi: 10.3934/cpaa.2014.13.119
We prove a comparison principle for viscosity solutions of a fully nonlinear equation satisfying a condition of non-degeneracy in a fixed direction. We apply these results to prove that a continuous solution of the corresponding Dirichlet problem exists. To obtain the existence of barrier functions and well-posedness, we find suitable explicit assumptions on the domain and on the ellipticity constants of the operator.
keywords: degenerate elliptic equation Viscosity solution Pucci's operators comparison principle Dirichlet problem subelliptic equation Carnot groups.
CPAA
On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations
Martino Bardi Paola Mannucci
Communications on Pure & Applied Analysis 2006, 5(4): 709-731 doi: 10.3934/cpaa.2006.5.709
We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations that satisfy some conditions of partial non-degeneracy instead of the usual uniform ellipticity or strict monotonicity. These results are applied to the well-posedness of the Dirichlet problem under suitable conditions at the characteristic points of the boundary. The examples motivating the theory are operators of the form of sum of squares of vector fields plus a nonlinear first order Hamiltonian and the Pucci operator over the Heisenberg group.
keywords: comparison principle Pucci operators. Viscosity solution subelliptic equation Heisenberg group degenerate elliptic equation
DCDS-S
Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations
Paola Mannucci Claudio Marchi Nicoletta Tchou
Discrete & Continuous Dynamical Systems - S 2019, 12(1): 119-128 doi: 10.3934/dcdss.2019008

This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part.

We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.

keywords: Subelliptic equations Grushin vector fields invariant measure singular perturbations viscosity solutions degenerate elliptic equations

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