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.
The Entropy Power Inequality states that the Shannon differential entropy of the convolution of two probability densities in R^n with given entropies is minimum when they are both Gaussian. This functional inequality is fundamental in information theory and was conjectured by C. Shannon himself, although he only proved that Gaussian densities are critical points; a full proof was later given by A. Stam, using an interpolation argument along the heat semigroup.
Abstract. In this talk I will show how one can exactly compute the minimizer of a variant of branched transportation arising as a simplified model for pattern formation in type-I superconductors. The proof relies on the natural scaling properties of the problem. A 3D version of this model has recently been derived from the full Ginzburg-Landau model together with S. Conti, S. Serfaty and F. Otto.
Abstract. In the first part of the talk I will present a comprehensive
theory that covers in a unified way a rather large class of
(possibly) nonlocal geometric flows bearing a gradient flow structure
with respect to suitable generalized perimeters. Within this framework
one can establish new existence and uniqueness results as well as
recover several examples scattered in the literature.
In the second part I will discuss a new distributional formulation
that allows one to treat the highly "degenerate" case of crystalline
We study an optimization problem with SPDE constraints, which
has the peculiarity that the control parameter $s$ is the $s$-th power
of the diffusion operator in the state equation. Before moving to the
SPDE case, we first describe the result of Sprekels-Valdinoci for the
PDE case. Then we discuss a suitable concept of solutions of the state
equation and establish pathwise differentiability properties of the
stochastic process w.r.t. the fractional parameter $s$. Finally, we show
that under certain conditions on the noise, optimality conditions for
Abstract. In the talk I will investigate the uniqueness of solutions of scalar conservation laws with discontinuous flux.
While in the smooth setting this property follows from Kruzhkov's entropy inequalities,
in the case of discontinuous fluxes these inequalities are not enough and additional dissipation conditions must be imposed at the discontinuity set of the flux.
I will explain how any entropy solution admits traces on the discontinuity set of the flux field and use this to prove the validity of a generalized Kato inequality for any pair of solutions.
We focus our attention on shape optimization problems in which one
dimensional connected objects are involved. Very old and classical
problems in calculus of variation are of this kind: euclidean Steiner's
tree problem, optimal irrigation networks, cracks propagation, etc.
In a first part we quickly recall some previous work in collaboration
with F. Santambrogio related to the functional relaxation of the
irrigation cost. We establish a $\Gamma$-convergence of Modica and
Mortola's type and illustrate its efficiency from a numerical point of