Periodic homogenization under a hypoellipticity condition

Mannucci, Paola; Stroffolini, Bianca

doi:10.1007/s00030-014-0296-8

Cited by 4 publications

(4 citation statements)

References 22 publications

(42 reference statements)

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“…We shall follow a pure PDE-approach. In this framework, the singular perturbation problems are strictly related to homogenization problems (see also [20]); Alvarez and Bardi [1,2] extended to singular perturbation problems with periodic fast variables the celebrated perturbed test function method by Evans (see also [3] for some cases in hypoelliptic periodic setting). Let us also recall that, the papers [5,6,13] studied singular perturbation problems of uniformly elliptic operators on the whole space.…”

Section: Introductionmentioning

confidence: 99%

Singular perturbations for a subelliptic operator

2018

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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 ǫ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and data are explicitly given in terms of the invariant measure for the ergodic operator.

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Section: Introductionmentioning

confidence: 99%

Singular perturbations for a subelliptic operator

2018

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“…The study of homogenisation in subelliptic settings started with the periodic case (see e.g. [4,7,23,24,6,29,34]). The first result for the stochastic case in this degenerate setting is [20], where the authors studied the case Hamilton-Jacobi (first order) case for Hamiltonian depending on the horizontal gradient in the case of Carnot groups.…”

Section: Introductionmentioning

confidence: 99%

Γ-convergence and homogenisation for a class of degenerate functionals

et al. 2020

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This paper is on Γ-convergence for degenerate integral functionals related to homogenisation problems in the Heisenberg group. Here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coercive nor periodic, so classic results do not apply. All the results apply to the more general case of Carnot groups.

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“…essentially in a compact setting. For homogenization in subelliptic settings in the periodic case see for example [14,16,25,26,32,36], and for homogenization with singular perturbation see [2,3].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Homogenization for Functionals with Anisotropic Rescaling and Noncoercive Hamilton--Jacobi Equations

Dirr¹,

Dragoni²,

Mannucci³

et al. 2018

SIAM J. Math. Anal.

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We study the stochastic homogenization for a Cauchy problem for a firstorder Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like H(x, σ(x)p, ω) where σ(x) is a matrix associated to a Carnot group. The rescaling considered is consistent with the underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the ε-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable deterministic Hamilton-Jacobi problem.

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Periodic homogenization under a hypoellipticity condition

Cited by 4 publications

References 22 publications

Singular perturbations for a subelliptic operator

Singular perturbations for a subelliptic operator

Γ-convergence and homogenisation for a class of degenerate functionals

Stochastic Homogenization for Functionals with Anisotropic Rescaling and Noncoercive Hamilton--Jacobi Equations

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