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Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains
Matteo Bonforte, Alessio Figalli, Juan Luis Vazquez
This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form , , wher…
Geometry of minimizers for the interaction energy with mildly repulsive potentials
J. A. Carrillo, A. Figalli, F. S. Patacchini
We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class and mildly repulsive at t…
Partial regularity for optimal transport maps
Shibing Chen, Alessio Figalli
We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class for any $p \geq…
Rigidity and stability of Caffarelli's log-concave perturbation theorem
Guido De Philippis, Alessio Figalli
In this note we establish some rigidity and stability results for Caffarelli's log-concave perturbation theorem. As an application we show that if a 1-log-concave measure has almos…
Characterization of isoperimetric sets inside almost-convex cones
Eric Baer, Alessio Figalli
In this note we characterize isoperimetric regions inside almost-convex cones. More precisely, as in the case of convex cones, we show that isoperimetric sets are given by intersec…
Free boundary regularity in the parabolic fractional obstacle problem
Begoña Barrios, Alessio Figalli, Xavier Ros-Oton
The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Lévy processes. In this paper…