CVGMT Seminarshttp://cvgmt.sns.it/seminars/en-usTue, 21 Nov 2017 16:43:42 -0000Asymptotic planar N-bubblehttp://cvgmt.sns.it/seminar/603/2017-11-22: G. Del Nin.
<p>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 $\varepsilon$ converges to 0: we prove that minimizing
configurations approach in the limit a configuration of disjoint disks
which maximize the number of tangencies among them. Moreover we obtain
some information about the structure of minimizers for small $\varepsilon$.</p>
http://cvgmt.sns.it/seminar/603/Functions of fractal bounded variationhttp://cvgmt.sns.it/seminar/609/2017-11-22: R. Züst.
<p>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.</p>
http://cvgmt.sns.it/seminar/609/Stable constant-mean-curvature hypersurfaces: regularity and compactness.http://cvgmt.sns.it/seminar/608/2017-11-24: C. Bellettini.
<p>This talk describes a recent joint work of the speaker with Neshan Wickramasekera (Cambridge). The work develops a regularity theory, with an associated compactness theorem, for weakly defined hypersurfaces (codimension 1 integral varifolds) of a smooth Riemannian manifold that are stationary and stable on their regular parts for volume preserving ambient deformations. The main regularity theorem gives two structural conditions on such a hypersurface that imply that, away from a set of codimension 7 or higher, the hypersurface is locally either a single smoothly embedded disk or precisely two smoothly embedded disks intersecting tangentially. Easy examples show that neither structural hypothesis can be relaxed. An important special case is when the varifold corresponds to the boundary of a Caccioppoli set, in which case the structural conditions can be considerably weakened.</p>
http://cvgmt.sns.it/seminar/608/Minimal Elastic Networkshttp://cvgmt.sns.it/seminar/610/2017-11-28: <a href="/person/1826/">A. Pluda</a>.
<p>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.</p>
http://cvgmt.sns.it/seminar/610/Willmore Flow of Networkshttp://cvgmt.sns.it/seminar/611/2017-11-28: J. Menzel.
<p>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.
The presented result is joint work with Harald Garcke and Alessandra Pluda.</p>
http://cvgmt.sns.it/seminar/611/Variational approximations and relaxations of the Steiner problemhttp://cvgmt.sns.it/seminar/604/2017-12-06: <a href="/person/2250/">M. Bonafini</a>.
<p>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. In the second case we describe a convex framework associated to the problem which provides relevant tools extensively used from a numerical point of view to identify optimal 1d structures.
</p>
<p>This is joint work with Giandomenico Orlandi (Verona) and Edouard Oudet (Grenoble).</p>
http://cvgmt.sns.it/seminar/604/Optimal partitionshttp://cvgmt.sns.it/seminar/605/2017-12-13: <a href="/person/374/">E. Oudet</a>.
<p>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.</p>
http://cvgmt.sns.it/seminar/605/