Continuity and injectivity of optimal maps for non-negatively cross-curved costs (2009)
Figalli, Alessio, Kim, Young-Heon, McCann, Robert J.
Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If...
A VARIATIONAL METHOD FOR CLASS OF PARABOLIC PDES (2009)
Alessio Figalli, Wilfrid Gangbo, Turkay Yolcu
Abstract. In this manuscript we extend De Giorgi’s interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying...
Almost everywhere well-posedness of continuity equations with measure initial data (2009)
Ambrosio, Luigi, Figalli, Alessio
The aim of this note is to present some new results concerning "almost everywhere" well-posedness and stability of continuity equations with measure initial data. The proofs of all such results can...
Ambrosio, Luigi, Crippa, Gianluca, Figalli, Alessio, Spinolo, Laura V.
We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of...
Generalized solutions for the Euler equations in one and two dimensions (2009)
Bernot, Marc, Figalli, Alessio, Santambrogio, Filippo
In this paper we study generalized solutions (in the Brenier's sense) for the Euler equations. We prove that uniqueness holds in dimension one whenever the pressure field is smooth, while we show...
Generalized solutions for the Euler equations in one and two dimensions (2009)
Bernot, Marc, Figalli, Alessio, Santambrogio, Filippo
In this paper we study generalized solutions (in the Brenier's sense) for the Euler equations. We prove that uniqueness holds in dimension one whenever the pressure field is smooth, while we show...
Mass Transportation on Sub-Riemannian Manifolds (2008)
Figalli, Alessio, Rifford, Ludovic
We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize...
Ambrosio, Luigi, Figalli, Alessio
In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.
Optimal transportation on non-compact manifolds (2007)
Fathi, Albert, Figalli, Alessio
In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In...
On the Hausdorff Dimension of the Mather Quotient (2007)
Fathi, Albert, Figalli, Alessio, Rifford, Ludovic
Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we...
Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index (2007)
Abbondandolo, Alberto, Figalli, Alessio
We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite...
Abbondandolo, Alberto, Figalli, Alessio
Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have quadratic...
Trasporto ottimale su varietà non compatte (2006)
We consider the optimal transportation problem on non-compact manifolds. We yield existence and uniqueness of a unique transport map in the case of cost functions induced by a $C^2$-Lagrangian,...