Blanchet, Adrien and Bolte, Jérôme (2018) A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions. Journal of Functional Analysis, vol. 275 (n° 7). pp. 1650-1673.

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Abstract

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Łojasiewicz inequalities. We also discuss the more general case of λ-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto-Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo-Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed.

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• A family of functional inequalities. (deposited 25 Jul 2018 07:42)
  • A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions. (deposited 09 Jul 2018 14:40) [Currently Displayed]