We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an L 1 -Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-Émery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.
The relative isoperimetric inequality inside an open, convex cone C states that, at fixed volume, B r ∩C minimizes the perimeter inside C. Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov's proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside C. Our proof follows the line of reasoning in [16], though several new ideas are needed in order to deal with the lack of translation invariance in our problem.
We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log C 1,1 functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application we obtain new bounds on the entropy.
In the optimal partial transport problem, one is asked to transport a fraction 0 < m ≤ min{||f || L 1 , ||g|| L 1 } of the mass of f = f χ Ω onto g = gχ Λ while minimizing a transportation cost. If f and g are bounded away from zero and infinity on strictly convex domains Ω and Λ, respectively, and if the cost is quadratic, then away from ∂(Ω ∩ Λ) the free boundaries of the active regions are shown to be C 1,α loc hypersurfaces up to a possible singular set. This improves and generalizes a result of Caffarelli and McCann [6] and solves a problem discussed by Figalli [7, Remark 4.15]. Moreover, a method is developed to estimate the Hausdorff dimension of the singular set: assuming Ω and Λ to be uniformly convex domains with C 1,1 boundaries, we prove that the singular set is H n−2 σ-finite in the general case and H n−2 finite if Ω and Λ are separated by a hyperplane.
We consider fully nonlinear obstacle-type problems of the formwhere Ω is an unknown open set and K > 0. In particular, structural conditions on F are presented which ensure that W 2,n (B1) solutions achieve the optimal C 1,1 (B 1/2 ) regularity when f is Hölder continuous. Moreover, if f is positive on B1, Lipschitz continuous, and {u = 0} ⊂ Ω, then we obtain local C 1 regularity of the free boundary under a uniform thickness assumption on {u = 0}. Lastly, we extend these results to the parabolic setting.
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