The abstract Lewy–Stampacchia inequality and applications

Gigli, Nicola; Mosconi, Sunra

doi:10.1016/j.matpur.2015.02.007

Cited by 29 publications

(30 citation statements)

References 39 publications

(64 reference statements)

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“…In this regard, here is a natural question towards a weak form of transitivity: is it true that, for any probability measures µ 0 , µ 1 absolutely continuous and with bounded densities with respect to m, we can find a Sobolev vector field such that, calling F its regular Lagrangian flow at a fixed time, it holds F ♯ µ 0 = µ 1 ? In subsection 3.2 below, we will see how the Lewy-Stampacchia inequality, proved in this abstract framework by Gigli-Mosconi in [GM15], allows to give an (almost) affirmative answer to this question.…”

Section: Introductionmentioning

confidence: 98%

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Bruè

Semola

2019

Comm Pure Appl Math

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We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N ) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K, N ) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting. * Scuola Normale Superiore, elia.brue@sns.it. † Scuola Normale Superiore, daniele.semola@sns.it.

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Section: Introductionmentioning

confidence: 98%

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Bruè

Semola

2019

Comm Pure Appl Math

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“…This is the generalization of the Lewy-Stampacchia inequality. An abstract proof, valid in a much more general setting, can be found in Gigli and Mosconi [9]. The assumption of our theorem, that f ∈ BV (I), implies that F ′′ = f ′ is a signed measure.…”

Section: Proof and Applications Of The Fundamental Estimatementioning

confidence: 91%

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

Overgaard¹

2018

Proceedings of the 7th International Conference on Pattern Recognition Applications and Methods

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A new proof of the equivalence of the Taut String Algorithm and the one-dimensional Rudin-Osher-Fatemi model is presented. Based on duality and the projection theorem in Hilbert space, the proof is strictly elementary. Existence and uniqueness of solutions to both denoising models follow as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. Moreover, a new and fundamental bound on the denoised signal is derived. This bound implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the insignal as the regularization parameter vanishes. The methods developed in the paper can be modified to cover other interesting applications such as isotonic regression.

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“…The proof can be obtained following verbatim the arguments given in Lemma 3.1 of [33] (inspired by [4], see also [18] for an alternative approach): there the authors are interested in cut-off functions such that χ ≡ 1 on B R (x) and supp( χ ) ⊂ B 2R (x): for this reason they fix R > 0 and then claim that for all x ∈ X and 0 < r < R there exists a cut-off function χ satisfying (i), (ii) and (2.7) with C, C ′ also depending on R. However, as far as one is concerned with cut-off functions χ where the distance between { χ = 0} and { χ = 1} is always equal to 1, the proof of [33] in the case R = 1 applies and does not affect (2.7).…”

Section: Analysis and Optimal Transport In Rcd Spacesmentioning

confidence: 99%

Benamou–Brenier and duality formulas for the entropic cost on $${\textsf {RCD}}^*(K,N)$$RCD∗(K,N) spaces

Gigli

Tamanini

2019

Probab. Theory Relat. Fields

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In this paper we prove that, within the framework of RCD * (K, N ) spaces with N < ∞, the entropic cost (i.e. the minimal value of the Schrödinger problem) admits: -a threefold dynamical variational representation, in the spirit of the Benamou-Brenier formula for the Wasserstein distance;-a Hamilton-Jacobi-Bellman dual representation, in line with Bobkov-Gentil-Ledoux and Otto-Villani results on the duality between Hamilton-Jacobi and continuity equation for optimal transport; -a Kantorovich-type duality formula, where the Hopf-Lax semigroup is replaced by a suitable 'entropic' counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem (still missing even in the Riemannian setting) as well as a perfect parallelism with the analogous formulas for the Wasserstein distance.

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The abstract Lewy–Stampacchia inequality and applications

Cited by 29 publications

References 39 publications

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

On the Taut String Interpretation of the One-dimensional Rudin–Osher–Fatemi Model

Benamou–Brenier and duality formulas for the entropic cost on $${\textsf {RCD}}^*(K,N)$$RCD∗(K,N) spaces

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