Lecture Notes On Differential Calculus on $\sf {RCD}$ Spaces

Gigli, Nicola

doi:10.4171/prims/54-4-4

Cited by 45 publications

(24 citation statements)

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“…This is interesting because one can define the differential du of u, even in very abstract situations [17], by means related to Sobolev calculus on the metric measure space (Y, d Y , μ := u (|du| 2 HS m)) and tangent vector fields in this metric measure space only see directions which are tangent to the graph of u (this is rather obvious in this example, but see for instance [31] for a discussion of this phenomenon in more general cases). This means that, curiously, u cannot be computed starting from du and using Sobolev calculus in the spirit developed in [27,32], simply because u does not belong to the tangent module L 2 μ (T Y)…”

Section: Remark 411mentioning

confidence: 99%

A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

Gigli

Nobili

2021

J Geom Anal

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We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$\textsf {CAT} (\kappa )$$ CAT ( κ ) -spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-\textsf {E} (y_t)$$ y t ′ ∈ - ∂ - E ( y t ) one uses in the smooth setting and more precisely that $$y_t'$$ y t ′ selects the element of minimal norm in $$-\partial ^-\textsf {E} (y_t)$$ - ∂ - E ( y t ) . This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of $$L^2$$ L 2 and (0) valued maps: we define the Laplacian of such $$L^2$$ L 2 map as the element of minimal norm in $$-\partial ^-\textsf {E} (u)$$ - ∂ - E ( u ) , provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$ L 2 -dense. Basic properties of this Laplacian are then studied.

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Section: Remark 411mentioning

confidence: 99%

A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

Gigli

Nobili

2021

J Geom Anal

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“…To keep the presentation short we assume that the reader is familiar with the concept of Sobolev functions on a metric measure space [4,5,13,28], with that of L 0 -normed modules and differentials of real valued Sobolev maps and with second order calculus on RCD spaces [16,18].…”

Section: Sobolev Calculusmentioning

confidence: 99%

“…These properties of pullbacks have been studied in [16,18] for maps satisfying u * m X ≤ Cm Y ; the generalization to the case of L 0 -normed modules has been considered in [9,22].…”

Section: Proposition 24 (Universal Property Of the Pullback) With Thmentioning

confidence: 99%

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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“…Whenever H 1,2 (X, d, m) is a Hilbert space, we will say that (X, d, m) is infinitesimally Hilbertian. In this case, we recall from [27] that the tangent module L 2 (T X) and the corresponding gradient map ∇ : H 1,2 (X, d, m) → L 2 (T X) can be characterised as follows: L 2 (T X) is an L 2 (m)-normed L ∞ (m)-module (in the sense of [28,Definition 1.3]) that is generated by ∇f :…”

Section: Preliminaries and Notationmentioning

confidence: 99%