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DCDS

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.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

keywords:

DCDS-S

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 $\nabla u$.

DCDS

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.

DCDS

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.

DCDS

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.

DCDS

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].

DCDS

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.

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