Oberseminar Analysis Summer 2014
- Thursday, April 10, 2:15 pm, seminar room 0.011
Riccardo Adami (Politecnico di Torino)
Miniminzing NLS energy on graphs
Owing to the loss of translational symmetry, the Nonlinear Schroedinger Equation on star graphs consisting of at least two infinite edges shows the lack of ground states. In this talk it will be shown how to extend this result to a more general class of graphs with Kirchhoff's (i.e. free) conditions at vertices, and an example will be given in which, on the other hand, the ground state exists.
This is a joint work in progress with E. Serra and P. Tilli (Politecnico di Torino).
- Thursday, May 8, 2:15 pm, Lipschitz-Saal
Paolo Tilli (Politecnico di Torino)
A minimization approach to hyperbolic Cauchy problems
We will discuss some recent results obtained in
collaboration with E. Serra, concerning a minimization
approach to a wide family of hyperbolic Cauchy problems.
This general and abstract approach, that stems from a
conjecture of De Giorgi in a particular case, allows the approximation of
solutions to Cauchy problems by functions that minimize
suitable functionals in space-time. Also equations
with dissipative terms can be treated. Focus will be on
recent results and some related open questions.
- Thursday, June 26, 2:15 pm, Lipschitz-Saal
A twice-relaxed sharp-interface limit in multiple slip-plane plasticity.
We study a continuum model for plastic slips in multiple parallel slip planes embedded into an elastic crystal. This model contains two length scales, the lattice parameter and the distance of the planes.
Generalizing a result by S. Conti, A. Garroni, and S.Mueller, we obtain a sharp-interface limit energy, penalizing edge dislocations.
Depending on the scaling law of the two length scales, the limit line-tension energy may feature an iterative relaxation behavior and induce microstructure at multiple length scales.
We shall outline the basic steps in proving an upper and a lower bound.
- Thursday, July 10, 2:15 pm, seminar room 2.040
Nicholas Alikakos (University of Athens)
On the structure of phase transition maps: density estimates and applications
The scalar Allen-Cahn ( or Ginzburg- Landau ) equation is
related to Minimal Surfaces and Minimal Graphs via the level sets of its
solutions. The Vector Allen-Cahn is related to Plateau Complexes. These
are non-orientable minimal objects with a hierarchical structure. After
explaining these relationships we focus on vector extensions of the
Caffarelli-Cordoba Density Estimates (L. Caffarelli and A.Cordoba , Comm.
Pure and Applied Mathematics Volume 48, Issue 1, pages 1–12, January 1995).
In particular we establish lower co-dimension density estimates. These
are useful for studying the hierarchical structure of certain entire
vector solutions. We also give applications to minimal solutions (lower
bounds, Liouville theorems)
Joint work with: Giorgio Fusco