A logarithmic Sobolev form of the Li-Yau parabolic inequality

By D. Bakry, Michel Ledoux. 2006

169 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/f941eb26bef338aed7e6e1a9b67baa11b9ad06ac

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In Applied Mathematics

Citations in the corpus

Li-Yau inequality on graphs (2013, background): “1) on manifolds, either by the maximum principle [19] or by semigroup methods [4], it is crucial to have the chain rule in hand.”
Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces (2013, background): “Our strategy is to use the dimensional Bochner inequality (1.3) in combination with the Γ-calculus developed by Bakry-Emery [9] and Bakry-Ledoux [10] in the smooth setting.”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “The first is that of dimension, and has been well understood since the work of Bakry-Émery [10] and Bakry-Ledoux [11]: in both the geometry and analysis of spaces with lower Ricci curvature bounds, it has become clear the correct statement is not that “X has Ricci curvature bounded from below by K”,”
Cones over metric measure spaces and the maximal diameter theorem (2013, background): “→ The proof of the theorem in this form can be found in [3] (see also [12, 21]).”
The Incompressible Navier–Stokes Equations on Non-compact Manifolds (2014, background): “Proof The following proof generalizes to the vectorial Laplacian some arguments yielding log Sobolev inequalities for the Laplace–Beltrami operator on Riemannian manifolds (see, for example, [6]).”
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, methodology): “The Bakry-Émery condition BE(K,∞) can then be stated in a weak integral form (strongly inspired by [11, 15, 57]) just involving the Markov semigroup (Pt)t≥0 (see (2.”
Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces (2013, background): “ ϕ∈ D(L) ∩L∞(X,m) : Lϕ∈ L∞(X,m) . When f= gwe also set Γ2[f;ϕ] := Γ2[f,f;ϕ] = Z X 1 2 Γ f Lϕ−Γ f,Lf ϕ dm, so that Γ2[f,g;ϕ] = 1 4 Γ2[f+g;ϕ] − 1 4 Γ2[f−g;ϕ]. Γ2 provides a weak version (inspired by [9, 11]) of the Bakry-Emery condition [10, 8].´ Deﬁnition 3.1 (Bakry-Emery condition)´ We say that the”
On the Bakry–Émery Condition, the Gradient Estimates and the Local-to-Global Property of $$\mathsf{RCD}^*(K,N)$$RCD∗(K,N) Metric Measure Spaces (2016, methodology): “en f= g we also set Γ2(f;ϕ) := Γ2(f,f;ϕ) = Z X 1 2 Γ f ∆ϕ−Γ f,∆f ϕ dm, (2.6) so that Γ2(f,g;ϕ) = 1 4 Γ2(f+g;ϕ) − 1 4 Γ2(f−g;ϕ). (2.7) 5 The multilinear form Γ2 provides a weak version (inspired by [8, 10]) of the Bakry-Emery´ condition [9, 7]. In the sequel, given f: X→ R, we denote by supp(f) the ”
Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds (2005, methodology): “We refer the to Bakry and Ledoux [11] for another type of diameter upper bound estimate in ter the mean of the distance function on compact Riemannian manifolds.”
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, methodology): “In our approach, we completely avoid those tools from geometry that appear typically Riemannian, and instead base our analysis on a systematic use of some entropic inequalities for the heat semigroup that are inspired by the works [7], [10], [12], and which, as we have stressed above, in the geometr”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, methodology): “If one of them is satisfied, we obtain the following: (iv) The W2-expansion bound (1.5), (v) The Bakry–Ledoux pointwise gradient estimate BL(K,N) (1.6), (vi) The Bochner inequality BE(K,N) (1.7).”
Structure theory of metric measure spaces with lower Ricci curvature bounds (2014, background): “3], Erbar-Kuwada-Sturm proved the dimen sional Bakry-LedouxL2-gradient-Laplacian estimate [12]: if (X, d,m) is aRCD∗(K,N)-space, then for every f ∈ D(Ch) and everyt > 0, one has Γ(Ht f ) + 4Kt2 N(e2Kt − 1) |∆Ht f |(2) ≤ eHt (Γ( f )) m-a.”
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations (2014, methodology): “The method we use is adapted from [7] Theorem 5.”
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces (2015, methodology): “22) should be considered as the appropriate nonlinear version of the Bakry-Émery inequality [13] (see also [7] for the non-smooth formulation, and [15] for dimensional improvements) for solutions ̺t, φT−t to the linear Heat flow d dt 1 2 ∫”
On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons (2013, background): “On the other hand, it is a good substitute of the Ricci curvature for establishing many interesting theorems in differential geometry, for example, Myers’ theorem, eigenvalues estimates, Li-Yau Harnack inequality, Liouville theorems and the Cheeger-Gromoll splitting theorem [5, 6, 4, 14, 19, 26, 31,”