Daniele Morbidelli

On the Poincaré inequality for vector fields

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

Kelvin transform for Grushin operators and critical semilinear equations

We study positive entire solutions u = u(x, y) of the critical equation xu+ (α + 1)2|x|2α yu = −u(Q+2)/(Q−2) in R = R × R, (1) where (x, y) ∈ Rm×Rk , α > 0, and Q = m+ k(α+1). In the first part of the article, exploiting the invariance of the equation with respect to a suitable conformal inversion, we prove a “spherical symmetry” result for solutions. In the second part, we show how to reduce the dimension of the problem using a hyperbolic symmetry argument. Given any positive solution u of (1), after a suitable scaling and a translation in the variable y, the function v(x) = u(x, 0) satisfies the equation divx(p∇xv) − qv = −pv(Q+2)/(Q−2), |x| < 1, (2) with a mixed boundary condition. Here, p and q are appropriate radial functions. In the last part, we prove that if m = k = 1, the solution of (2) is unique and that for m ≥ 3 and k = 1, problem (2) has a unique solution in the class of x-radial functions.

Regular domains in homogeneous groups

We study John, uniform and non-tangentially accessible domains in homogeneous groups of steps 2 and 3. We show that C 1,1 domains in groups of step 2 are non-tangentially accessible and we give an explicit condition which ensures the John property in groups of step 3.

Isoperimetric inequality in the Grushin plane

We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric sets.

Nonsmooth Hörmander vector fields and their control balls

We prove a ball-box theorem for nonsmooth Hörmander vector fields of step s ≥ 2.

Levi umbilical surfaces in complex space

Abstract We define a complex connection on a real hypersurface of ℂ n+1 which is naturally inherited from the ambient space. Using a system of Codazzi-type equations, we classify connected real hypersurfaces in ℂ n+1, n ≧ 2, which are Levi umbilical and have non zero constant Levi curvature. It turns out that such surfaces are contained either in a sphere or in the boundary of a complex tube domain with spherical section.

Almost Exponential Maps and Integrability Results for a Class of Horizontally Regular Vector Fields

We consider a family

A Frobenius-type theorem for singular Lipschitz distributions

{\mathcal{H}}:= \{X_1, \dots, X_m\}

Stability of isometric maps in the Heisenberg group

of C1 vector fields in ℝn and we discuss the associated

On the subRiemannian cut locus in a model of free two-step Carnot group

{\mathcal{H}}