Asymptotic mean value Laplacian in metric measure spaces

Minne, Andreas; Tewodrose, David

doi:10.1016/j.jmaa.2020.124330

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…Then ũ(0) exists and is equal to 0 while 0 / ∈ Leb(u). Therefore, according to (12) the function u is pointwise AMV harmonic on R, while (13) yields that ∆ d µ u(0) is not defined and u is pointwise AMV harmonic on R\{0} only: this is in better agreement with the weak AMV Laplacian of u being equal to a multiple of the derivative of the Dirac distribution in 0, as mentioned in the last sentence of [MT20].…”

Section: Definitionsmentioning

confidence: 66%

“…in the weak topology of C ∞ c (R n ) ′ -which coincides with this weak Laplacian. However, they are functions which admit a distributional Laplacian without admitting a weak Laplacian, like the sign function, for instance, whose distributional Laplacian is equal to a constant times the derivative of the Dirac distribution in 0 (see the last lines of [MT20]).…”

Section: Definitionsmentioning

confidence: 99%

“…For any u ∈ C 2 (R n ) and x ∈ R n , the second-order Taylor expansion of u at x yields the identity Br (x) u(y) − u(x) dy = c n r 2 ∆u(x) + o(r 2 ) as r ↓ 0, where c n := 1/(2n + 4). Building upon this elementary observation, the authors have proposed in [MT20] a definition for the pointwise Laplacian of a locally integrable function u defined on a general metric measure space (X, d, µ) by setting…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces

Minne¹,

Tewodrose²

2022

Preprint

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The asymptotic mean value Laplacian -AMV Laplacian -extends the Laplace operator from R n to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. In this paper therefore a symmetric version of the AMV Laplacian is considered, and focus lies on when the symmetric and non-symmetric AMV operators coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces including locally Ahlfors regular spaces with vanishing metric-measure boundary. In addition, we study the context of weighted domains of R n where the two operators typically differ, and provide concrete formulae for these operators also at points where the weight vanishes.

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Section: Definitionsmentioning

confidence: 66%

Section: Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces

Minne¹,

Tewodrose²

2022

Preprint

Self Cite

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show abstract

“…For recent work on amv functions on general metric measure spaces we refer the reader to [42,43] and in particular to [2] where we focus on functions with the amv property in doubling metric measure spaces and show their local Hölder regularity and the mv property for their blow-ups. As an important special case, in [2] we consider weighted Euclidean setting and find an elliptic PDE satisfied by amv-harmonic functions.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically Mean Value Harmonic Functions in Subriemannian and RCD Settings

Adamowicz

Kijowski²,

Soultanis³

2023

J Geom Anal

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We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step 2, we prove a Blaschke–Privaloff–Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces.

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Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces

Minne,

Tewodrose

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from $\mathbb {R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of $\mathbb {R}^n$ —where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.

show abstract

Asymptotic mean value Laplacian in metric measure spaces

Cited by 4 publications

References 10 publications

Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces

Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces

Asymptotically Mean Value Harmonic Functions in Subriemannian and RCD Settings

Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces

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