SEMINARI DI CALCOLO DELLE VARIAZIONI

Minimization of the eigenvalues of the Dirichlet-Laplacian with a diameter constraint.

Relatore: 
Ilaria Lucardesi
Abstract. In this talk I present some recent results about the minimization of $\lambda_k$ under diameter constraint. After providing existence, attained at a constant width body, and optimality conditions in any dimension, I focus my attention on the optimality of the disk in the plane, giving the precise list of 17 eigenvalues for which the disk is a local minimum. This last fact is confirmed by numerical simulations, which show non circular minimizers out of the afore mentioned 17 values of $k$. These results are obtained in collaboration with B. Bogosel (CMAP) and A.
Data Seminario: 
Wednesday, October 17, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Institut Elie Cartan de Lorraine
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

A minimization approach to the wave equation on time-dependent domains.

Relatore: 
Lucia De Luca
Abstract. We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time. Joint work with Gianni Dal Maso.
Data Seminario: 
Wednesday, October 10, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Università di Pisa
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

Analysis of Novel Domain Wall Types in Ferromagnetic Nanostructures

Abstract. Recent advances in nanofabrication allow an unprecedented degree of control of ferromagnetic materials down to the atomic scale, resulting in novel nanostructures whose properties are often dominated by material interfaces. Mathematically, these systems give rise to challenging problems in the calculus of variations that feature non-convex, vectorial, topologically constrained, multi-scale variational problems.
Data Seminario: 
Wednesday, September 26, 2018
Relatore: 
Cyrill Muratov
Ora Inizio: 
Saturday, January 1, 0000 - 17:00
Ora Fine: 
Saturday, January 1, 0000 - 18:00

When does optimal transport branch?

Relatore: 
Mircea Petrache

Abstract. Consider the problem of transporting some objects
between N distinct locations. Depending on how we package
together different objects and on how the transport cost (per
unit of distance traveled) depends on the package that we are
moving, we may cook up a minimum-cost transport strategy.
Is it always the best option to let our objects travel independently
of each other, or is it sometimes more cost-efficient to merge/split
packages along the way, following a branched, tree-like, global
network?

Data Seminario: 
Wednesday, June 20, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Pontificia Universidad Católica de Chile
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

The Kakeya needle problem for rectifiable sets

Relatore: 
Alan Chang

Abstract. We show that the classical results about rotating
a line segment in arbitrarily small area, and the existence
of a Besicovitch and a Nikodym set hold if we replace the
line segment by an arbitrary rectifiable set.
This is joint work with Marianna Csörnyei.

Data Seminario: 
Wednesday, June 20, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 19:00
Affiliazione: 
University of Chicago
Ora Inizio: 
Saturday, January 1, 0000 - 18:00

Generalized Sadowsky Theory For Ribbons From Three-Dimensional Nonlinear Elasticity

Relatore: 
Roberto Paroni

Abstract. In the 1930s Sadowsky derived an asymptotic theory for narrow ribbons. We here provide a rigorous derivation of the generalized Sadowsky theory starting from nonlinear three-dimensional elasticity by means of Γ-convergence. On a technical level, this involves capturing a contribution to the asymptotic energy functional generated by a nonlinear constraint which is satisfied only approximately. It also involves the construction of fine-scale ‘corrugations’ capable of reaching a bending energy regime that is strictly below that of the original Sadowsky functional.

Data Seminario: 
Wednesday, May 30, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 19:00
Affiliazione: 
Universita' di Pisa
Ora Inizio: 
Saturday, January 1, 0000 - 18:00

Properties of Convex sets in Wiener spaces

Relatore: 
Michele Miranda

Abstract. We show some recent results on convex sets in Wiener spaces. We characterize the essential and reduced boundary of open convex sets and investigate integration by parts formulae. Of particular interest is the investigation of trace theorems for functions of bouned variation on boundaries of subsets in Wiener spaces.

Data Seminario: 
Wednesday, May 30, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Universita' di Ferrara
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

A nonlocal isoperimetric problem with dipolar repulsion

Abstract. 

We study a functional in which perimeter and regularized dipolar repulsion compete under a volume constraint.

In contrast to previously studied similar problems, the nonlocal term contributes to the perimeter term to leading order for small regularization parameters.

Data Seminario: 
Wednesday, June 6, 2018
Relatore: 
Simon Thilo
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Aula: 
Sala Seminari (Dip. Matematica)
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

A new gluing phenomenon for metrics of prescribed Q-curvature in dimension 6

Relatore: 
Luca Martinazzi
Abstract. Contrary to the Yamabe case, metric of prescribed Q-curvature (which we will briefly discuss) in dimension 4 and higher, can blow-up both on isolated points and on higher dimensional submanifolds, as discovered by Adimurthi, F. Robert and M. Struwe. We will show, in a radially symmetric situation, that both kind of blow-up behaviour can happen at the same time. This is based on a joint work with A. Hyder.
Data Seminario: 
Wednesday, May 23, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
University of Basel
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

Purely unrectifiable metric spaces and perturbations of Lipschitz functions

Relatore: 
David Bate
Abstract. We give characterisations of purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of Lipschitz functions $f\colon X \to R^m$. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, a typical $f$ (with respect to the supremum norm) has ${\mathcal H}^n(f(S))=0$. Conversely, if $E\subset X$ is $n$-rectifiable with ${\mathcal H}^n(E)>0$, a typical $f$ has ${\mathcal H}^n(f(E))>0$.
Data Seminario: 
Wednesday, May 16, 2018
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Universita' di Helsinki
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

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