Vincenzo Maria Tortorelli

Vincenzo M.Tortorelli, February 14th 2012
CURRICULUM VITAE ET STUDIORUM
- Born in Milano April 20th 1958.
- Degree in Mathematics, A.A. 1977/78-1982/83, at the University of Pisa; title of the
final dissertation, supervisor Ennio de Giorgi:
``Gamma-limiti ed Analisi non Standard''
- I.N.d.A.M. grant: A.A. 1983/84,
- PHD at Scuola Normale Superiore: A.A. 1984-1987
- I.N.d.A.M.grant: 1988-1990
- Consultant teacher at the University of Pisa: A.A. 1990/91 e 1991/92.
- From 25/11/1992 `ricercatore universitario' at the
Mathematical Dept. ``L.Tonelli'', University of Pisa.
RESEARCH:
1. Non Standard Analysis:
Foundations and models, applications to variational operators ($Gamma$-limits).
(Degree thesis,[1], [2]).
2. Foundations of Mathematics:
The foundational project in Mathematics led by Ennio De Giorgi got steam with
several mathematicians toward the analysis of ``handy'', consistant axiomatics.
It is desiderable to have an high degree of self-description, and a fruitful open-endedness, to engraft in these different branches of knowledge. [3], [4], [7], [10], [11], [12].
3. Calculus of Variations:
3.1: The study of lower semicontinuity and of relaxation of integral of the kind
$$ F(u)= int_{Omega}f(x,u,nabla u),dx$$
in the case of linear growth of the integrand,
with respect to the gradient variable. This analysis leads to consider
functionals defined on functions presenting singularies ([5]).
3.2:This interest also concerns the analysis of integral
functionals whose minimizers model equilibrium states that
effect characterizing singularities of various kind.
A model for this kind of analysis is to find minimizers
$(u, K)$ of:
$$ G(u,K) = II_{Omegasetminus K}f(x,u,nabla
u),dx+II_Kvarphi(x, u^+,u^-,nu),dHH quadquad
u:Omegasubsetrrnrightarrow{bf R}^k$$
([6], [8]).
4. Differential Equations
4.1: With Franco Flandoli in [9] it is studied the convergence of
a discrete approximation for Ornastein-Uhlenbeck equations
and bidimensional Navier-Stokes equations with a rather general
``white-noise'' additive perturbation.
4.2: A general problem is to find notions of approximation, on spaces of
continuous paths, allowing the extension of the elementary integration of
1-forms on regular paths.
On the subject in [14] is proven:
a) almost all paths of suitable stochastic processes, such are semimartingale
and Lyons-Zheng stuctures, define 1-currents of Sobolev class $H^{-s}, ~s>0$;
b) a $gamma$-Holder curve, $gamma > 1/2$, defines a 1-current that is the
sum of an integral rectifiable 1-corrente and of the boundary of an integral
rectifiable 2-currente. The correspondence with these addenda is continuous
with respect to the $gamma$-Holder norm of continuous funcitons.
5. Geometric Motions
In [13]a partial answer to the following problem is given:
to establish the global, in time, existence for
the evolution of a partition of the disc in three components
such that:
- on the interface curves normal velocities equal curvatures;
- each among these interface curves has one end point fixed on
the boundary of the disc, and these boundary point reaches the boundary.points are different;
- the other end point of each curve is the only point (``three-point'')
of the disc where the three components all meet together.
- Here the tangent directions to each among the three interfaces meet
pair-wise with equal angles.
- All this must hold until the three-point reaches the boundary.