| Liouville theorems for limit problems corresponding to cubic stationary Schr÷dinger
In the study of a priori bounds for systems of stationary Schr÷dinger systems - as
arising e.g. in the Hartree-Fock theory of a mixture of Bose-Einstein condensates - one
is led, via a rescaling procedure, to analyze the solution set of cubic elliptic systems
either in the entire space or in the half space. We will focus on existence and
nonexistence results in the case where the system is not cooperative. Here the structure
of the solution set depends in a subtle way on parameters, and standard methods like the
moving plane method cannot be used.
|A uniqueness result for the continuity equation in two dimensions
Abstract: In the simplest form, our result characterizes the bounded,
divergence-free vector fields on the plane such that the Cauchy
problem for the associated continuity (or transport) equation has
a unique bounded solution in the sense of distribution.
Unlike previous results in this directions (cf. Di Perna-Lions,
Ambrosio, etc.), the proof does not rely on regularization, but rather
on a dimension-reduction argument.
Note that the "good" vectorfields are not characterized in terms of
function spaces (i.e., by inequalities), but by a qualitative property which
is completely non-linear in character.
These results have been obtained in collaboration with Stefano
Bianchini (SISSA, Trieste) and Gianluca Crippa (University of Parma).