[CvGmt News] Two CVGMT Seminars/17:00/25 May

[Valentino Magnani]

Dear All,
I am glad to announce that 

. on Thursday, 25 May 
. in ``Sala dei Seminari'' of the Mathematics Department

there will be Two seminars:

I) At 17:00. Michal Wojciechowski (Polish Academy of Science)
will present

"Singularity of vector valued measures in terms of Fourier transform"

II) At 18:00. Massimiliano Berti (University of Naples)
will present

"Cantor families of periodic solutions of wave equations via a variational 
principle"

ABSTRACT I. We study how the singularity (in the sense of Hausdorff 
dimension) of a vector valued measure can be affected by certain 
restrictions imposed on its Fourier trans form. The restrictions, we are 
interested in, concern the direction of the (vector) values of the Fourier 
transform. The results obtained could be considered as a generalizations 
of F. and M. Riesz theorem, however a phenomenon, which have no analogy in 
the scalar case, arise in the vector valued case. As an example of 
application, we show that every measure µ = (µ1,...,µd) Î M(Rd ,Rd) 
annihilating gradients of C0(1) (Rd) embedded in the natural way into 
C0(Rd ,Rd), i.e. such that åi ò¶i fdmi = 0 for f Î C0(1) (Rd), has 
Hausdorff dimension at least one. We provide examples which show both 
completeness and incompleteness of our results.
These results are joint with M. Roginskaya.


ABSTRACT II. For finite dimensional Hamiltonian systems, existence of 
periodic solutions close to an elliptic equilibrium have been proved by 
Weinstein, Moser and Fadell-Rabinowitz: by the classical Lyapunov-Schmidt 
decomposition the problem splits into (i) the range equation, solved 
through the standard Implicit Function Theorem, and (ii) the bifurcation 
equation, solved via variational arguments. On the contrary, for infinite 
dimensional Hamiltonian PDEs (i) a ``small divisors problem'' requires the 
use of a Nash-Moser implicit function theorem to solve the range equation 
and, as a consequence, the bifurcation equation (ii) is defined just on a 
Cantor like set. We presents the first existence results of periodic 
solutions for Hamiltonian PDEs solving a variational principle defined on 
a Cantor set.