Hynek Kovarik, Università di Brescia, Comportamento asintotico del p-Laplaciano, Wednesday, October 16, 2013, time 14:00 o'clock, Aula seminari III piano
Abstract:Abstract:
Consideriamo il primo autovalore dell operatore dato dalla somma del p-Laplaciano e di un potenziale V in R^n. Studiamo il comportamento asintotico di questo autovalore per V che tende a zero. Mostreremo, in particolare, come lo sviluppo asintotico dipende da p e dalla dimensione dello spazio.

Carlos Escudero, Universidad Autonoma de Madrid, Existence results for a fourth order equation arising in the theory of non-equilibrium phase transitions, Wednesday, October 02, 2013, time 14:00 o'clock, aula seminari VI piano
Abstract:Abstract:
In this talk we will introduce a model that arises in the theory of non-equilibrium phase transitions, in particular in the description of self-affine surfaces. We will briefly comment on the role that this model is meant to play in this physical theory. Furthermore, we will mention some open questions of physical nature related to it. Subsequently we will start with the rigorous analysis of our model, which is a fourth order partial differential equation. We will describe our progress in building an existence theory for the full model, which is a parabolic equation, and for its stationary counterpart. For the latter case existence and multiplicity results are provided, and for the former one we will show local in time existence of the solution, that can be made global for small enough data, and cannot if these are large enough. We will show how these results fit into the physical theory, and what open questions are left for the future.

A. Takatsu, Nagoya University, Evolution equations as Wasserstein gradient flows, Friday, September 27, 2013, time 14:00 o'clock, Aula seminari III piano
Abstract:Abstract:
In this talk, I study a certain evolution equation on a weighted Riemannian manifold as the gradient flow of a generalized relative entropy with respect to the Wasserstein metric. The Wasserstein metric is a metric on the space of probability measures. For example, it is known that the porous medium equation is the Wasserstein gradient flow of the Tsallis entropy. I give a criterion for generalized relative entropies to be convex. From the convexity of generalized relative entropies, I derive appropriate variants of the Poincaré and the logarithmic Sobolev inequalities, also the contraction propety of its gradient flow. This is a joint work with Shin-ichi Ohta.

Ciprian Gal, Florida International University, On degenerate parabolic equations with dynamic boundary conditions, Friday, July 05, 2013, time 11:15 o'clock, Aula seminari III piano
Abstract:Abstract:
We wish to present some recent developments concerning the solvability of some classes of quasilinear parabolic equations with dynamic boundary conditions. We will describe issues from well-posedness to the long time asymptotic behavior as time goes to infinity (attractors, dimension estimates, etc). We will explain how to derive new conditions which reflect an exact balance between the internal and the boundary mechanisms involved, even when both the nonlinear sources contribute in opposite directions. Blow up of solutions will also be touched on.

Daniele Valtorta, Università degli Studi dell Insubria, Quantitative stratification and critical sets of harmonic functions, Thursday, June 06, 2013, time 12:00, Aula seminari III piano
Abstract:Abstract:
Given a harmonic function defined on the unit ball of R^n, we discuss techniques to obtain effective volume estimates on the tubular neighborhood of its critical sets. We use a technique recently introduced by proff. Jeff Cheeger and Aaron Naber, called quantitative stratification technique. It is based on approximate symmetries of the function u at different scales. Studying how these approximate symmetries interact with each other, we obtain the effective volume estimates. These results are described in a preprint available on arXiv. We also discuss possible improvements of the results using a refined quantitative differentiation argument and packing estimates for semi-algebraic sets.

Antoine Henrot, Institut Elie Cartan, Elastic energy of a convex body, Tuesday, Febraury 26, 2013, time 15:00, Aula seminari VI piano
Abstract:Abstract:
Following L. Euler, we define the elastic energy E(K) of a regular compact set K in the plane as 1/2 times the integral over the boundary of K of the square of the boundary curvature. We will denote by $A(K)$ the area of $K$ and $P(K)$ its perimeter. In this talk, we prove that for any convex set K the quotient A(K)E(K)/P(K) is larger than or equal to pi/2, with equality only for the disk. We deduce that the disk minimizes the elastic energy with an area constraint. We will also consider analogous tridimensional problems involving the Willmore (or the Helfrich) energy linked to the modelling of vesicles.