Differential of metric valued Sobolev maps

Gigli, Nicola; Pasqualetto, Enrico; Soultanis, Elefterios

doi:10.1016/j.jfa.2019.108403

Cited by 16 publications

(36 citation statements)

References 17 publications

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“…with |du(Z )|. The kind of construction that we use is strongly reminiscent of-and deeply motivated by-the one proposed in [21]; the difference here is that our function u is differentiable only in the direction of Z . For this reason we won't define the differential du of u, but only its action on Z .…”

Section: The Differential Du(z)mentioning

confidence: 99%

“…This paper is part of a bigger project aiming at reproducing (1.1) in such fully synthetic setting, see also [14,21] for other contributions in this direction. The purpose of the current manuscript is to generalize part of Korevaar-Schoen's theory in [26] to the case of source spaces which are RCD.…”

Section: Introductionmentioning

confidence: 99%

“…The definition in (iv) makes use of the theory of L 0 -normed modules as tools to develop first-order calculus on metric measure spaces as proposed in [16]. More in detail, our approach for defining du(Z ) should be seen as an adaptation to the current framework of the recent construction of differential of a metric-valued Sobolev map proposed in [21].…”

Section: Introductionmentioning

confidence: 99%

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Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows

Gigli

Tyulenev

2020

Math. Z.

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We develop Korevaar–Schoen’s theory of directional energies for metric-valued Sobolev maps in the case of $${\textsf {RCD}}$$ RCD source spaces; to do so we crucially rely on Ambrosio’s concept of Regular Lagrangian Flow. Our review of Korevaar–Schoen’s spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of ‘differential of a map along a vector field’ and about the parallelogram identity for $${\textsf {CAT}}(0)$$ CAT ( 0 ) targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.

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Section: The Differential Du(z)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows

Gigli

Tyulenev

2020

Math. Z.

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“…[20]. As shown in [19,Theorem 3.3], any Sobolev map u ∈ S 2 (X; Y) can be naturally associated with an L 0 (m)-linear and continuous operator…”

Section: Introductionmentioning

confidence: 99%

“…(We refer to [19,Section 2] for a brief summary of the terminology used above.) We underline that the measure µ is not given a priori, but it rather depends on the map u itself in a nontrivial manner.…”

Section: Introductionmentioning

confidence: 99%

Infinitesimal Hilbertianity of Weighted Riemannian Manifolds

Lučić

Pasqualetto

2019

Can. Math. Bull.

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The main result of this paper is the following: any 'weighted' Riemannian manifold (M, g, µ) -i.e. endowed with a generic non-negative Radon measure µ -is 'infinitesimally Hilbertian', which means that its associated Sobolev space W 1,2 (M, g, µ) is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated to any weighted reversible Finsler manifold (M, F, µ) can be isometrically embedded into the space of all measurable sections of the tangent bundle of M that are 2-integrable with respect to µ.

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Korevaar–Schoen’s energy on strongly rectifiable spaces

Gigli

Tyulenev

2021

Calc. Var.

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We extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an $$\mathsf{RCD}$$ RCD space. In this situation it appears that no version of the ‘subpartition lemma’ holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on: the fact that such spaces are ‘strongly rectifiable’ a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim’s metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density, the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential. When the target space is $$\mathsf{CAT}(0)$$ CAT ( 0 ) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.

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Differential of metric valued Sobolev maps

Cited by 16 publications

References 17 publications

Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows

Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows

Infinitesimal Hilbertianity of Weighted Riemannian Manifolds

Korevaar–Schoen’s energy on strongly rectifiable spaces

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