25 may 2006
On Thursday, 25 May
In ``Sala dei seminari'' of the Mathematics Department
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 $µ=(µ1,...,µd)\in M(R^d ,R^d)$ annihilating gradients of $C_0^(1) (R^d)$ embedded in the natural way into $C_0(R^d ,R^d)$, i.e. such that $\sum_i \int \partial^i fd\mu_i =0$ for $f\in C_0^(1) (R^d)$, 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.