On the characterization of $p$-harmonic functions  on the Heisenberg group by mean value properties

Ferrari, Fausto; Liu, Qing; Manfredi, Juan J.

doi:10.3934/dcds.2014.34.2779

Cited by 26 publications

(22 citation statements)

References 10 publications

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“…The characterization of p-harmonic functions by asymptotic mean value expansions from [21], originally motivated by the tug-of-war games approach from [23] for the ∞-Laplacian and [24] for the p-Laplacian, has been extended to more general mean value expansions in [2,12] including the case of variable exponent p(x), the parabolic case in [15], and the Heisenberg group [10]. We refer to the monograph [6] for historical remarks and more references.…”

Section: Introductionmentioning

confidence: 99%

Convergence of the naturalp-means for thep-Laplacian

Manfredi

Stroffolini²

2021

ESAIM: COCV

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Section: Introductionmentioning

confidence: 99%

Convergence of the naturalp-means for thep-Laplacian

Manfredi

Stroffolini²

2021

ESAIM: COCV

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“…Firstly, there exists 0 > 0 such that for all |t − t 0 | < 0 and for all p ∈ B H (p 0 , 0 ) we have: D h ϕ(p, t) = 0, and moreover, in view of Lemma 3.1 in [9], such that for every > 0 with 2 ∈ (0, 0 ), there exist points p ,M = (x ,M , y ,M , z ,M ) ∈ ∂B H (p 0 , ) and p ,m = (x ,m , y ,m , z ,m ) ∈ ∂B H (p 0 , ) such that:…”

Section: Remarkmentioning

confidence: 98%

Approximation schemes for non-linear second order equations on the Heisenberg group

Ochoa¹

2015

Communications on Pure &Amp; Applied Analysis

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In this work, we propose and analyse approximation schemes for fully non-linear second order partial differential equations defined on the Heisenberg group. We prove that a consistent, stable and monotone scheme converges to a viscosity solution of a second order PDE on the Heisenberg group provided that comparison principles exists for the limiting equation. We also provide examples where this technique is applied.2000 Mathematics Subject Classification. Primary: 35R03; Secondary: 65N06. Key words and phrases. Partial differential equations on the Heisenberg group, finite difference methods, viscosity solutions, Heisenberg group, approximation schemes. 1841 1842 PABLO OCHOAThe general organization of the paper is as follows. In Section 2, we introduce the Heisenberg group, its differential and metric structures, and basic notation. In the next Section 3, we state the definition of solutions, in the viscosity sense, to fully non-linear second order partial equations defined on the Heisenberg group. We then proceed, in Section 4, to formulate the assumptions on the approximation schemes and we prove the main convergence results. We finally provide, in Section 5, several examples where we construct schemes giving rise to viscosity sub and super solutions of the limiting equations.2. Preliminaries. Throughout this section, we introduce the definition of the Heisenberg group H together with its differential and metric structures. The notion of subelliptic jet on H and its characterization in terms of smooth functions are also introduced.2.

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“…The coincidence of game values of tug-of-war games and functions satisfying related DPPs as well as the existence and uniqueness of these functions were shown in [LPS14]. Studies on DPPs and associated tug-of-war games are ongoing under various settings, for example in nonlocal and Heisenberg group setting, as in [CGAR09,BCF12,FLM14].…”

Section: Introductionmentioning

confidence: 99%