A1 Journal article (refereed)
Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential (2024)


Eriksson-Bique, S., & Soultanis, E. (2024). Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential. Analysis and PDE, 17(2), 455-498. https://doi.org/10.2140/apde.2024.17.455


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Publication details

All authors or editors: Eriksson-Bique, Sylvester; Soultanis, Elefterios

Journal or series: Analysis and PDE

ISSN: 2157-5045

eISSN: 1948-206X

Publication year: 2024

Publication date: 06/03/2024

Volume: 17

Issue number: 2

Pages range: 455-498

Publisher: Mathematical Sciences Publishers

Publication country: United States

Publication language: English

DOI: https://doi.org/10.2140/apde.2024.17.455

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/93989

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2102.08097


Abstract

We represent minimal upper gradients of Newtonian functions, in the range 1≤p<∞, by maximal directional derivatives along “generic” curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of p-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along p-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules.

The arising p-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger’s structure in the presence of a Poincaré inequality. In particular, it exists whenever the space is metrically doubling. It is moreover compatible with, and gives a geometric interpretation of, Gigli’s abstract differentiable structure, whenever it exists. The p-weak charts give rise to a finite-dimensional p-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.


Keywords: functional analysis; calculus of variations

Free keywords: Sobolev; test plan; minimal upper gradient; differential structure; differential; chart


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Preliminary JUFO rating: 3


Last updated on 2024-25-03 at 08:11