[CvGmt News] CVGMT: weekly bulletin

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Subject: CVGMT weekly bulletin

Weekly bulletin of http://cvgmt.sns.it/

--- Summary ---

* Seminars by: FigalliPlakhovValdinoci
* New papers by: Petrache, Bucur, Pratelli, Velichkov, Mazzoleni
* Modified papers by: Ambrosio, Cassano, Sarrocco, Petrache, Riviere, De Philippis, Garroni, Mazzieri, Catino, Braides, Deruelle, Figalli, Barilari, Mondino, Kuwert, Ruffini, De Lellis, Sz\'ekelyhidi, Buckmaster, Schygulla , Paolini, Scala, Van Goethem

--- Seminars next week ---

* Tuesday 17 dec 2013
time: 16:00
Aula Mancini, Scuola Normale Superiore

Stability results for the semisum of sets in R^n
Alessio Figalli (UT Austin)

Abstract. Given a Borel A in R^n of positive measure, one can consider its semisum S=(A+A)/2. It is clear
that S contains A, and it is not diffi cult to prove that they have the same measure if and only if A
is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is
stable: if the measure of S is close to the one of A, is A close to his convex hull? More generally, one
may consider the semisum of two different sets A and B, in which case our question corresponds
to proving a stability result for the Brunn-Minkowski inequality. When n=1, one can approximate a
set with fi nite unions of intervals to translate the problem to the integers Z. In this discrete setting
the question becomes a well-studied problem in additive combinatorics, usually known as Freiman?s
Theorem. In this talk, which is intended for a general audience, I will review some results in the onedimensional
discrete setting and show how to answer to the problem in arbitrary dimension.

* Wednesday 18 dec 2013
time: 16:00
Aula Seminari - Department of Mathematics, University of Pisa

Besicovitch's magic method and problems of minimal resistance
Alexander Plakhov 

Abstract. We consider piecewise smooth functions $u : \bar{\Omega} \rightarrow \mathbb{R}$ defined on the closure of a bounded domain $\Omega \subset \mathbb{R}^2$ satisfying the conditions $u(x)<0$ for $x\in \Omega $ and $u(x)=0$ for $x\in\partial \Omega$ (in other words, the graph of $u$ forms a "dimple" on the plane).

We also consider a flow of particles that fall on the graph of $u$ vertically down and reflect from it in the perfectly elastic manner. It is assumed that u satisfies the so-called "single impact condition" (SIC): each particle reflected at a non-singular point of the graph, further moves freely above the graph until it leaves the dimple. This condition can be stated analytically as follows: for any regular point $x\in\Omega$ and any $t > 0$ such that $x - t \nabla u ( x ) \in  \bar{\Omega}$,
\[
\frac{u(x ? t\nabla u(x)) ? u(x)}{t} \leq \frac 1 2(1 ? |\nabla u(x)|^2). \tag{1}	
\]
The force of resistance of the dimple to the flow (more precisely, the vertical projection of this force) equals $2\rho |\Omega| R(u; \Omega)$, where $?$ is the flow density, $|\Omega|$ is the area of $\Omega$, and
\[
R(u;\Omega) = \frac{1}{|\Omega|} \int_\Omega \frac{dx}{1+|\nabla u(x)|^2}.	
\]
This formula is true provided that the SIC (1) is fulfilled. 

The problem is: minimize the value of "specific resistance" $R(u; \Omega)$. It has two versions which are eventually equivalent: 
(a) $\inf_{u,\Omega} R(u; \Omega)$ 
(b) $\inf_{u} R(u;\Omega)$ for a given $\Omega$. 
Obviously, $\sup R(u; \Omega) = 1$ and $\inf R(u; \Omega) \geq 1/2$. The main question is to find if $\inf R(u; \Omega) > 1/2$ or $\inf R(u; \Omega) = 1/2$. I will prove that the latter is true. This result is somewhat counterintuitive: one needs to provide a sequence of functions with the slope of the graph being "almost" $45^\circ$ in the most part of the region $\Omega$. That is, most part of reflected particles move "almost" horizontally and do not meet obstacles on the way.
A part of the construction is borrowed from Besicovitch?s solution of the Kakeya problem: what is the minimum area of a plane region in which a unit line segment can be rotated continuously through $360^\circ$.



time: 17:00
Aula Seminari - Department of Mathematics, University of Pisa

Regularity of nonlocal minimal surfaces in low dimension
Enrico Valdinoci ((Universit? di Roma II, Tor Vergata)

Abstract. We present a full-detail proof of the fact that the only minimal cones for the nonlocal perimeter in plane are the trivial ones (equivalently, the singular set of fractional perimeter minimizers has, at most, codimension three).

As a consequence, we show that a Bernstein type result holds in this setting up to dimension three.

The technique used is quite flexible and it may be applied to obtain monotonicity and symmetry results for variational problems with quadratic energy growth. These results were obtained in collaboration with O. Savin and A. Figalli.

--- New Papers ---

* Petrache: A singular radial connection over B? minimizing the Yang-Mills energy

* Bucur, Mazzoleni, Pratelli, Velichkov: Lipschitz regularity of the eigenfunctions on optimal domains

--- Modified Papers ---

* Petrache, Riviere: Weak closure of Singular Abelian $L^p$-bundles in $3$ dimensions

* Petrache: An integrability result for $L^p$-vectorfields in the plane 

* Ambrosio, Paolini: Partial regularity for quasi minimizers of perimeter

* Kuwert, Mondino, Schygulla : Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

* Petrache: Interior partial regularity for minimal $L^p$-vectorfields with integer fluxes

* Barilari: Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

* Ruffini: Stability theorems for GNS inequalities: a reduction principle to the radial case

* De Philippis, Figalli: Higher integrability for minimizers of the Mumford-Shah functional

* Buckmaster, De Lellis, Sz\'ekelyhidi: Transporting microstructure and dissipative Euler flows

* Braides, Cassano, Garroni, Sarrocco: Evolution of damage in composites: the one-dimensional case

* Ruffini: Optimization problems for solutions of elliptic equations and stability issues

* Scala, Van Goethem: Constraint Reaction for Continuum Dislocations

* Catino, Deruelle, Mazzieri: Uniqueness of asymptotically cylindrical gradient shrinking Ricci solitons

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