[CvGmt News] seminari giovedi 28

Giuseppe Buttazzo buttazzo at dm.unipi.it
Thu Feb 14 12:49:49 CET 2002

I seminari del giovedi pomeriggio continuano con:

Prof. Bianca STROFFOLINI (Universita' di Napoli)
"Convex functions in the Heisenberg Group"
Dipartimento di Matematica - Sala dei Seminari
Giovedi' 28 Febbraio - ore 17.00

Dr. Fabricio MACIA (Univ. Complutense Madrid e Univ. di Pisa)
"Wigner measures and controllability of discrete wave equations"
Dipartimento di Matematica - Sala dei Seminari
Giovedi' 28 Febbraio - ore 18.00

Seguono degli abstracts dei due seminari. Vi ricordo che oggi
pomeriggio alle 18.00 c'e' il seminario di Adriana Garroni.

Giuseppe Buttazzo

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABSTRACTS %%%%%%%%%%%%%%%%%%%%%%



\renewcommand{\theack}{}   %for  unnumbered ack's
\newcommand{\ncite}[1]{{\bf \cite{#1}}}
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%   \parindent=0mm
\title[ Convex functions in the Heisenberg Group]
       {Convex functions in the Heisenberg Group}

\author{Bianca Stroffolini}



Regularizing properties of inf-convolutions were
described in Lasry-Lions and used by  Jensen  to prove
his celebrated comparison principle for second order fully nonlinear
elliptic equations.
Shortly after Jensen-Lions-Souganidis  suggested to look at
the time-dependent eikonal equation:
$$\frac{\partial w}{\partial t}+\frac1{2}|Dw|^{2}=0 ,$$
$$\frac{\partial w}{\partial t}-\frac1{2}|Dw|^{2}=0 ,$$
with initial data
$w(\cdot,0)=u$ or $w(\cdot,0)=v$, $u$ upper semicontinuous and $v$
lower semicontinuous. The solutions of these two equations are given
by explicit formulas
in terms of sup and inf convolution respectively (Hopf-Lax formulas):
$$u_{\varepsilon}(x)=\sup_{y} \left\{u(y)-\frac1{2\varepsilon}
|x-y|^{2} \right\}$$
$$v^{\varepsilon}(x)=\inf_{y} \left\{v(y)+\frac1{2\varepsilon}
|x-y|^{2} \right\}$$

They enjoy good   semiconvexity and semiconcavity properties
respectively and they preserve the subsolution and supersolution
of the initial data $u$ and $v$.  \par

We are following the same program in Carnot Groups. Our first goal is to
study Hamilton-Jacobi equations in the
subelliptic setting. For this purpose, we have to  consider radial
hamiltonians with a geometric distance, not
equivalent to the Euclidean distance.
We will establish in this new setting a version of the Hopf-Lax formula.
This requires a careful examination of geodesics with respect to the new
Next,we present some
definitions of convexity in the subriemannian setting.
The  notion of
{\em horizontal convexity}  has the right properties.
In fact, upper-semicontinuous H-convex functions are
Lipschitz and their symmetrized horizontal second derivatives are

In addition,
the corresponding H-semiconcavity  is preserved under
evolution via the simplest hamiltonian: the square of the distance.

Finally, we use the subelliptic Hopf-Lax formula
  to build semiconcave regularizations which   preserve
the subsolution or the supersolution character of the initial data.


\centerline{\bf Fabricio MACIA}



In this talk we shall present Wigner measures as a tool for describing
the limiting energy distribution of a $L^2$-bounded sequence of discrete
functions and show how they can be applied to find necessary and
sufficient geometric conditions for the existence of uniformly bounded
least-norm controls for discrete wave equations.


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