| Abstract: |
| In this talk we consider second-order
differential operators of the form
$$\mathcal{L}_X = \sum_{j = 1}^mX_j^2 \qquad \text{in $\mathbb{R}^n$},$$
where
the vector fields $X_j$`s satisfy H\ormander`s condition
and enjoy suitable homogeneity properties
with respect to a family of non-isotropic dilations.
The class of these operators comprehends the sub-Laplacians on
Carnot groups, the smooth Grushin-type operators and the so-called smooth
$\Delta_\lambda$-Laplacians.
By making use of a global lifting tech\-ni\-que for homogeneous vector fields,
we prove
the validity of a Gibbons-type conjecture for the operator
$\mathcal{L}_X$. Moreover, we establish
a comparison result for the solutions of the semi-linear
equation $$\mathcal{L}_Xu+f(u) = 0$$
under suitable assumptions on the non-linearity $f$. |
|