Sharp upper bounds for a variational problem with singular perturbation

Let Ω be a C 2 bounded open set of and consider the functionals We prove that if , |∇ u| = 1 a.e., and ∇ u∈BV, then The new result is the Γ- lim sup inequality

Density lower bound estimates for local minimizers of the 2d Mumford-Shah energy

We prove, using direct variational arguments, an explicit energy-treshold criterion for regular points of 2-dimensional Mumford-Shah energy minimizers. From this we infer an explicit constant for the density lower bound of De Giorgi, Carriero and Leaci

Dissipative continuous Euler flows

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy

Lack of uniqueness for weak solutions of the incompressible porous media equation

In this work we consider weak solutions of the incompressible 2-D porous media equation. By using the approach of De Lellis-Sz\'ekelyhidi we prove non-uniqueness for solutions in $L^\infty$ in space and time.Comment: 23 pages, 2 fugure

Rectifiability and upper Minkowski bounds for singularities of harmonic Q-valued maps

In this article we prove that the singular set of Dirichlet-minimizing Q-valued functions is countably .m2/-rectifiable and we give upper bounds for the .m2/-dimensional Minkowski content of the set of singular points with multiplicity Q

Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper "Oscillations and concentrations in weak solutions of the incompressible fluid equations", R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.Comment: 35 pages. Final revised version. To appear in Arch. Ration. Mech. Ana

On the concentration of entropy for scalar conservation laws

We prove that the entropy for an $L^\infty$-solution to a scalar conservation laws with continuous initial data is concentrated on a countably $1$-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution

Regularity of higher codimension area minimizing integral currents

This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher codimension, due originally to F. Almgren, which is contained in a series of papers in collaboration with C. De Lellis (University of Zurich).Comment: This text will appear in "Geometric Measure Theory and Real Analysis", pp. 131--192, Proceedings of the ERC school in Pisa (2013), L. Ambrosio Ed., Edizioni SNS (CRM Series

Moving in unison after perceptual interruption

Humans interact in groups through various perception and action channels. The continuity of interaction despite a transient loss of perceptual contact often exists and contributes to goal achievement. Here, we study the dynamics of this continuity, in two experiments involving groups of participants (N= 7) synchronizing their movements in space and in time. We show that behavioural unison can be maintained after perceptual contact has been lost, for about 7s. Agent similarity and spatial configuration in the group modulated synchronization performance, differently so when perceptual interaction was present or when it was memorized. Modelling these data through a network of oscillators enabled us to clarify the double origin of this memory effect, of individual and social nature. These results shed new light into why humans continue to move in unison after perceptual interruption, and are consequential for a wide variety of applications at work, in art and in sport