Neumann heat flow and gradient flow for the entropy on non-convex domains

Lierl, Janna; Sturm, Karl-Theodor

doi:10.1007/s00526-017-1292-8

Cited by 8 publications

(28 citation statements)

References 18 publications

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“…Remark 5.10. (i) The above Theorem provides a far reaching extension of our previous result in [LS18] which covers the case of constant negative . Now we also admit variable and 's of arbitrary signs.…”

Section: Convexificationsupporting

confidence: 56%

“…see [LS18], Lemma 2.13 (or, more precisely, estimate (10) in the proof of it). (iv) Given any rectifiable curve (γ a ) a∈ [0,1]…”

Section: Convexificationmentioning

confidence: 97%

“…ii) In general, the sets W 1,2 (Y ) and W 1,2 (Y 0 ) do not coincide, see subsequent Example. In [LS18], Section 4.2, equality of W 1,2 (Y ) and W 1,2 (Y 0 ) was erroneously stated as a general fact. Instead, it should be added there as an extra assumption.…”

Section: Ricci Bounds For Neumann Laplaciansmentioning

confidence: 99%

“…This concept of curvature bounds for the boundary has been introduced in [LS18], restricted there to the case of constant, nonpositive . As already observed there, the two most important classes of examples are Riemannian manifolds and Alexandrov spaces.…”

mentioning

confidence: 99%

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Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

Sturm¹

2020

Geom. Funct. Anal.

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We will study metric measure spaces $$(X,\mathsf{d},{\mathfrak {m}})$$ ( X , d , m ) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds $$\mathsf{BE}_1(\kappa ,\infty )$$ BE 1 ( κ , ∞ ) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary $$\psi \in \mathrm {Lip}_b(X)$$ ψ ∈ Lip b ( X ) , and which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $$Y\subset X$$ Y ⊂ X . In the latter case, the distribution-valued Ricci bound will be given by the signed measure $$\kappa = k\,{\mathfrak {m}}_Y + \ell \,\sigma _{\partial Y}$$ κ = k m Y + ℓ σ ∂ Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and $$\ell $$ ℓ denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

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Section: Convexificationsupporting

confidence: 56%

“…see [LS18], Lemma 2.13 (or, more precisely, estimate (10) in the proof of it). (iv) Given any rectifiable curve (γ a ) a∈ [0,1]…”

Section: Convexificationmentioning

confidence: 97%

Section: Ricci Bounds For Neumann Laplaciansmentioning

confidence: 99%

mentioning

confidence: 99%

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Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

Sturm¹

2020

Geom. Funct. Anal.

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“…Next we prove the first theorem in this article. When u ∈ TestF, this result has been proved in Theorem 7.1 [32] (see also Lemma 2.1 [34], Theorem 3.3 [26]). In the following Theorem 3.13, thanks to the recent results on second order differential structure of metric measure space (cf.…”

Section: K-convexity and K-monotonicitymentioning

confidence: 94%

Characterizations of monotonicity of vector fields on metric measure spaces

Han

2018

Calc. Var.

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We characterize the convexity of functions and the monotonicity of vector fields on metric measure spaces with Riemannian Ricci curvature bounded from below. Our result offers a new approach to deal with some rigidity theorems such as "splitting theorem" and "volume cone implies metric cone theorem" in non-smooth context.In the past twenty years, the displacement convexity of functionals on Wasserstein space has been deeply studied, and it has applications in many fields such as differential equation theory, probability theory, differential and metric geometry (see [4] and [44] for an overview of related theories).One of the most interesting functionals is the Boltzmann entropy. Let (M, g, V g ) be a Riemannian manifold. The Boltzmann entropy Ent Vg (·) is defined byotherwise It is known from [45] that the convexity of Ent Vg (·) in Wasserstein space characterizes the lower Ricci curvature bound of M. It is proved by Erbar in [19] that the gradient flow of Ent Vg in Wasserstein space can be identified with the heat flow in the following sense: letH t (f V g ) be the Wasserstein gradient flow of Ent Vg starting from f V g ∈ P 2 (M), H t (f ) be the solution of the heat equation with initial datumMoreover, we have the following well-known theorem. Theorem 1.1 (Von Renesse-Sturm [45], Erbar [19]). Let (M, g, V g ) be a Riemannian manifold. Then the following characterizations are equivalent.1) The Ricci curvature of M is uniformly bounded from below by a constant K ∈ R.2) The entropy Ent Vg (·) is K-convex in Wasserstein space.3) For any probability measure µ, there exists a unique EVI K -gradient flow of Ent Vg in Wasserstein space starting from µ.4) The exponential contraction of the heat flows in Wasserstein distanceholds for any two heat flows µ i t := H t (f i )V g , i = 1, 2. 5)There exits a heat kernel ρ t (x, dz)V g (z) =H t (δ x ) for any x ∈ X, such that the exponential contraction of heat kernels in Wasserstein distanceholds for any x, y ∈ X and t > 0. 6) The gradient estimate of heat flow|DH t (f )| 2 (x) ≤ e −2Kt H t (|Df | 2 )(x), V g − a.e. x ∈ Xholds for any f ∈ W 1,2 (M).

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Curvature-dimension conditions under time change

Han

Sturm

2021

Annali di Matematica

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We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry–Émery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott–Sturm–Villani will transform under time change.

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Neumann heat flow and gradient flow for the entropy on non-convex domains

Cited by 8 publications

References 18 publications

Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

Characterizations of monotonicity of vector fields on metric measure spaces

Curvature-dimension conditions under time change

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