A global existence result for the semigeostrophic equations in three dimensional convex domains
Luigi Ambrosio Maria Colombo Guido De Philippis Alessio Figalli
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.
Variational models for incompressible Euler equations
Luigi Ambrosio
In this paper we illustrate some recent work [1], [2] on Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field.
keywords: Euler equations Optimal transportation Measure-preserving maps
Leaf superposition property for integer rectifiable currents
Luigi Ambrosio Gianluca Crippa Philippe G. Lefloch
We consider the class of integer rectifiable currents without boundary in $\R^n\times\R$ satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
keywords: Metric spaces valued $BV$ functions Multi-valued functions. Integer rectifiable currents Currents in metric spaces Cartesian currents
Preface: A beautiful walk in the way of the understanding
Luigi Ambrosio Luis A. Caffarelli A. Maugeri
The International Conference "Variational Analysis and Applications" devoted to the memory of Ennio De Giorgi and Guido Stampacchia was held at the International School of Mathematics in Erice from May 9 to May 17 of 2009. About thirty lecturers from every part of the world took part in the conference and the workshop was enriched by the award of the "Third Gold Medal G. Stampacchia" to the young researcher Camillo de Lellis. Some of the lecturers, together with their well-known students, presented a paper for this special issue in honour of Ennio De Giorgi and Guido Stampacchia.

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Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability
Luigi Ambrosio Michele Miranda jr. Diego Pallara
We discuss some recent developments of the theory of $BV$ functions and sets of finite perimeter in infinite-dimensional Gaussian spaces. In this context the concepts of Hausdorff measure, approximate continuity, rectifiability have to be properly understood. After recalling the known facts, we prove a Sobolev-rectifiability result and we list some open problems.
keywords: sets with finite perimeter rectifiability. Wiener spaces

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