Alessio Figalli Filippo Santambrogio
Discrete & Continuous Dynamical Systems - A 2014, 34(4): i-ii doi: 10.3934/dcds.2014.34.4i
Optimal mass transportation can be traced back to Gaspard Monge's paper in 1781. There, for engineering/military reasons, he was studying how to minimize the cost of transporting a given distribution of mass from one location to another, giving rise to a challenging mathematical problem. This problem, an optimization problem in a certain class of maps, had to wait for almost two centuries before seeing significant progress (starting with Leonid Kantorovich in 1942), even on the very fundamental question of the existence of an optimal map. Due to these connections with several other areas of pure and applied mathematics, optimal transportation has received much renewed attention in the last twenty years. Indeed, it has become an increasingly common and powerful tool at the interface between partial differential equations, fluid mechanics, geometry, probability theory, and functional analysis. At the same time, it has led to significant developments in applied mathematics, with applications ranging from economics, biology, meteorology, design, to image processing. Because of the success and impact that this subject is still receiving, we decided to create a special issue collecting selected papers from leading experts in the area.

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An excess-decay result for a class of degenerate elliptic equations
Maria Colombo Alessio Figalli
Discrete & Continuous Dynamical Systems - S 2014, 7(4): 631-652 doi: 10.3934/dcdss.2014.7.631
We consider a family of degenerate elliptic equations of the form div $(\nabla F(\nabla u)) = f$, where $F\in C^{1,1}$ is a convex function which is elliptic outside a ball. We prove an excess-decay estimate at points where $\nabla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $\nabla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $\nabla u$.
keywords: traffic congestion excess decay Degenerate elliptic PDEs continuity of the gradient.
Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials
Alessio Figalli Vito Mandorino
Discrete & Continuous Dynamical Systems - A 2011, 31(4): 1325-1346 doi: 10.3934/dcds.2011.31.1325
In this paper we study the properties of curves minimizing mechanical Lagrangian where the potential is Sobolev. Since a Sobolev function is only defined almost everywhere, no pointwise results can be obtained in this framework, and our point of view is shifted from single curves to measures in the space of paths. This study is motived by the goal of understanding the properties of variational solutions to the incompressible Euler equations.
keywords: Non-smooth Lagrangians action-minimizing measures Euler-Lagrange equations value function.
Characterization of isoperimetric sets inside almost-convex cones
Eric Baer Alessio Figalli
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 1-14 doi: 10.3934/dcds.2017001

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 intersecting the cone with a ball centered at the origin.

keywords: Isoperimetric inequality almost-convex cones
A global existence result for the semigeostrophic equations in three dimensional convex domains
Luigi Ambrosio Maria Colombo Guido De Philippis Alessio Figalli
Discrete & Continuous Dynamical Systems - A 2014, 34(4): 1251-1268 doi: 10.3934/dcds.2014.34.1251
Exploiting recent regularity estimates for the Monge-Ampère equation, under some suitable assumptions on the initial data we prove global-in-time existence of Eulerian distributional solutions to the semigeostrophic equations in 3-dimensional convex domains.
keywords: Optimal transportation semigeostrophic equations.
Partial regularity of Brenier solutions of the Monge-Ampère equation
Alessio Figalli Young-Heon Kim
Discrete & Continuous Dynamical Systems - A 2010, 28(2): 559-565 doi: 10.3934/dcds.2010.28.559
Given $\Omega,\Lambda \subset \R^n$ two bounded open sets, and $f$ and $g$ two probability densities concentrated on $\Omega$ and $\Lambda$ respectively, we investigate the regularity of the optimal map $\nabla \varphi$ (the optimality referring to the Euclidean quadratic cost) sending $f$ onto $g$. We show that if $f$ and $g$ are both bounded away from zero and infinity, we can find two open sets $\Omega'\subset \Omega$ and $\Lambda'\subset \Lambda$ such that $f$ and $g$ are concentrated on $\Omega'$ and $\Lambda'$ respectively, and $\nabla\varphi:\Omega' \to \Lambda'$ is a (bi-Hölder) homeomorphism. This generalizes the $2$-dimensional partial regularity result of [8].
keywords: regularity. Monge-Ampère equation Brenier solutions
Asymptotics of the $s$-perimeter as $s\searrow 0$
Serena Dipierro Alessio Figalli Giampiero Palatucci Enrico Valdinoci
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 2777-2790 doi: 10.3934/dcds.2013.33.2777
We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.
keywords: Nonlinear problems minimal surfaces. fractional Laplacian fractional Sobolev spaces nonlocal perimeter

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