SEMINARI DI CALCOLO DELLE VARIAZIONI

Willmore Flow of Networks

Abstract. We consider networks of curves in the plane moving according to the L^2-gradient flow of a variant of the elastic energy. In this talk we will prove short time existence in the case of networks composed by three curves that are required to meet in one or two triple junctions. As a variation of the result we additionally impose that they form an angle of 120 degrees at the triple junction(s). If time allows we will give some outlook on our expectations concerning the long time behaviour based on numerical work by John Barrett, Harald Garcke and Robert Nürnberg.
Data Seminario: 
Wednesday, November 29, 2017
Relatore: 
Julia Menzel
Ora Fine: 
Saturday, January 1, 0000 - 19:00
Aula: 
Sala Seminari (Dip. Matematica)
Ora Inizio: 
Saturday, January 1, 0000 - 18:00

Minimal Elastic Networks

Abstract. In this talk we will consider planar networks of three curves minimizing a combination of the elastic energy and the length functional. We will prove existence and regularity of minimizers and we will show some properties of the minimal configurations. In addition to the presentation of the results that are obtained in collaboration with Anna Dall'Acqua and Matteo Novaga, we will give a partial review of the theory of elasticae and discuss about the onset of new phenomena passing from the problem for curves to the one for networks.
Data Seminario: 
Wednesday, November 29, 2017
Relatore: 
Alessandra Pluda
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Aula: 
Sala Seminari (Dip. Matematica)
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

Functions of fractal bounded variation

Abstract. In this talk we introduce a notion of functions of fractal bounded variation. Here, the sup-norm of test functions as used in the classical definition is replaced by the Hoelder norm with respect to some exponent. Characteristic functions of domains with fractal bounderies are particular examples that belong to this class. Among a characterization in terms of currents, we state some properties that naturally extend those of classical BV functions such as higher integrability, decomposition into Hoelder functions and push forwards.
Data Seminario: 
Wednesday, November 22, 2017
Relatore: 
Roger Züst
Ora Fine: 
Saturday, January 1, 0000 - 19:00
Aula: 
Sala Seminari (Dip. Matematica)
Ora Inizio: 
Saturday, January 1, 0000 - 18:00

Optimal partitions.

Relatore: 
Edouard Oudet

ABSTRACT. We present recent numerical approaches dedicated to the identification
of optimal partitions associated to geometrical costs. We focus our
presentation on 2D surfaces and full 3D problems.

Data Seminario: 
Wednesday, December 13, 2017
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Centro De Giorgi
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

Variational approximations and relaxations of the Steiner problem

Relatore: 
Mauro Bonafini
Abstract. In this talk we focus our attention on problems involving one dimensional sets, using as guiding example the Euclidean Steiner tree problem. We first reformulate the problem over a suitable family of rank one tensor valued measures and then we take two different perspectives. In the first case we focus on the planar setting and provide a variational approximation via Gamma convergence by means of functionals of phase transition type.
Data Seminario: 
Wednesday, December 6, 2017
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Universita' di Trento
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

Asymptotic planar N-bubble

Relatore: 
Giacomo Del Nin
Abstract. We consider in the plane a fixed finite number N of "bubbles", that is disjoint finite perimeter sets which possibly share portions of their boundaries, and look for configurations that minimize, under a volume constraint, the total weighted length of their boundaries: the interface between each bubble and the exterior is given weight 1 while the interface between any two bubbles is given weight $2-\varepsilon$. We are interested in the case when $\epsilon$ converges to 0: we prove that minimizing
Data Seminario: 
Wednesday, November 22, 2017
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Universita' di Pisa
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

Branched transportation: stability and new models.

Relatore: 
Andrea Marchese
Abstract. Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree, or the nervous and the cardiovascular systems. Given a flow that transports a given mass distribution $\mu^-$ onto a target distribution $\mu^+$, along a 1-dimensional network, the transportation cost per unit length is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. This favors grouping of the particles' trajectories and prevents diffusion.
Data Seminario: 
Wednesday, November 15, 2017
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
University of Zurich
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

A Hele-Shaw tumor growth model as a gradient flow.

Relatore: 
Simone Di Marino
Abstract. In 2014 Perthame, Quiroz and Vasquez unite two types of modeling of tumor growth into a unique framework of reaction-diffusion type where the diffusive term is $\Delta p(\rho)$ and $p(\rho)=\rho^m$. The stiff limit $m \to \infty$ is in particular a Hele-Shaw type problem: we find a gradient flow formulation of this problem, namely it is the gradient flow of the negative mass with respect to the Wasserstein-Fisher-Rao distance, discovered simultaneously by many authors in the last years.
Data Seminario: 
Wednesday, November 8, 2017
Aula: 
Sala Seminari (Dip. Matematica)
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Affiliazione: 
Scuola Normale Superiore
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

Liouville-type problems on compact surfaces: a variational approach

Abstract. A class of Liouville equations and systems on compact surfaces is considered: we focus on the Toda system which is motivated in mathematical physics by the study of models in non-abelian Chern-Simons theory and in geometry in the description of holomorphic curves. We discuss its variational aspects which yield existence results.
Data Seminario: 
Wednesday, October 25, 2017
Relatore: 
Aleks Jevnikar
Ora Fine: 
Saturday, January 1, 0000 - 18:00
Aula: 
Sala Seminari (Dip. Matematica)
Ora Inizio: 
Saturday, January 1, 0000 - 17:00

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