28 feb 2002

**Abstract.**

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

\section{Abstract}

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 character 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
metric.\par
Next,we present some
definitions of convexity in the subriemannian setting.
The notion of
*horizontal convexity* has the right properties.
In fact, upper-semicontinuous H-convex functions are
Lipschitz and their symmetrized horizontal second derivatives are
measures.

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.