LIP manifolds: from metric to finslerian structure

Cecco, Giuseppe De; Palmieri, Giuliana

doi:10.1007/bf02571901

Cited by 32 publications

(26 citation statements)

References 6 publications

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“…If the coefficients of G are only Borel, there are examples where this property is not satisfied for x in a set of positive Lebesgue measure (see [16]). For instance in R 2 , if we choose…”

Section: U |(T) ≥ G(u(t))u(t)u(t)mentioning

confidence: 99%

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Lisini¹

2008

ESAIM: COCV

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Abstract.We study existence and approximation of non-negative solutions of partial differential equations of the typewhere A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition,, we show that u is the "gradient flow" of φ with respect to the 2-Wasserstein distance between probability measures on the space R n , endowed with the Riemannian distance induced by A −1 . In the case of uniform convexity of V , long time asymptotic behaviour and decay rate to the stationary state for solutions of equation (0.1) are studied. A contraction property in Wasserstein distance for solutions of equation (0.1) is also studied in a particular case.Mathematics Subject Classification. 35K55, 35K15, 35B40.

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“…If the coefficients of G are only Borel, there are examples where this property is not satisfied for x in a set of positive Lebesgue measure (see [16]). For instance in R 2 , if we choose…”

Section: U |(T) ≥ G(u(t))u(t)u(t)mentioning

confidence: 99%

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Lisini¹

2008

ESAIM: COCV

View full text Add to dashboard Cite

show abstract

“…In particular, when the curve γ lies in Ω (i.e. γ(I) ⊂ Ω), the following holds: [11,15], Th. 2.5), where ϕ d is the Finsler metric on Ω associated to d by derivation, given by…”

Section: Proposition 22mentioning

confidence: 99%

“…We summarize in the next proposition the main properties of ϕ d . For the proof, we refer to [13,15]. (14) is Borel-measurable.…”

Section: Proposition 22mentioning

confidence: 99%

“…For later use, we need to introduce a different way to derive a distance from an element of M. Following [15], we introduce the notion of transversality: we say that a curve γ is transversal to the negligible set E if H 1 (γ(I)∩ E) = 0. Then, for each ϕ ∈ M, we define a function d ϕ on Ω × Ω through the following formula:…”

Section: Fine Properties Of Distancesmentioning

confidence: 99%

“…The next two propositions show that the upper semicontinuity property of the length functional L ϕ plays a role in this issue. These results are essentially known [13][14][15]; they have been restated here for the reader's convenience. Proof.…”

Section: Lemma 32mentioning

confidence: 99%

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