Roberto Monti

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.

Heisenberg isoperimetric problem. The axial case

Abstract We prove Pansus conjecture about the Heisenberg isoperimetric problem in the class of axially symmetric sets. The result is based on a weighted rearrangement scheme in the half plane which is of independent interest.

Extremal Curves in Nilpotent Lie Groups

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin maximum principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.

Sobolev Inequalities for Weighted Gradients

We study symmetry, existence, and uniqueness properties of extremal functions for the weighted Sobolev inequality where x ∈ ℝ m , y ∈ ℝ k with m,k ≥ 1 and n = m + k, α > 0, and Q = m + k(α + 1).

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.

The regularity problem for sub-Riemannian geodesics

We review some recent results on the regularity problem of sub-Riemannian length minimizing curves. We also discuss a new nontrivial example of singular extremal that is not length minimizing near a point where its derivative is only Holder continuous. In the final section, we list some open problems.

Isoperimetric problem and minimal surfaces in the Heisenberg group

The 2n +1-dimensional Heisenberg group is the manifold ℍ n = ℂ n × ℝ, n ∊ ℕ, endowed with the group product

Accessible domains in the Heisenberg group

We study the problem of accessibility of boundary points for domains in the sub-Riemannian setting of the first Heisenberg group. A sufficient condition for accessibility is given. It is a Dini-type continuity condition for the horizontal gradient of the defining function. The sharpness of this condition is shown by examples.