Harnack Inequality for a Subelliptic PDE in nondivergence form

We consider subelliptic equations in non divergence form of the type $Lu = \sum a_{ij} X_jX_iu=0$, where $X_j$ are the Grushin vector fields, and the matrix coefficient is uniformly elliptic. We obtain a scale invariant Harnack's inequality on the $X_j$'s CC balls for nonnegative solutions under the only assumption that the ratio between the maximum and minimum eigenvalues of the coefficient matrix is bounded. In the paper we first prove a weighted Aleksandrov Bakelman Pucci estimate, and then we show a critical density estimate, the double ball property and the power decay property. Once this is established, Harnack's inequality follows directly from the axiomatic theory developed by Di Fazio, Gutierrez and Lanconelli in [6]

Abstract approach to non homogeneous Harnack inequality in doubling quasi metric spaces

We develop an abstract theory to obtain Harnack inequality for non homogeneous PDEs in the setting of quasi metric spaces. The main idea is to adapt the notion of double ball and critical density property given by Di Fazio, Guti\'errez, Lanconelli, taking into account the right hand side of the equation. Then we apply the abstract procedure to the case of subelliptic equations in non divergence form involving Grushin vector fields and to the case of X-elliptic operators in divergence form

Integral Formulas for a Class of Curvature PDE'S and Application to Isoperimetric Inequalities and to Symmetry Problems

We prove integral formulas for closed hypersurfaces in C^(n+1); which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the "Soap Bubble Theorem" for star- shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above

Graphs with prescribed the trace of the Levi form

We prove existence and uniqueness of a viscosity solution of the Dirichlet problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by proving that it is the uniform limit of a sequence of classical solutions of elliptic problems and by building Lipschitz continuous barriers

On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups

We show by explicit estimates that the SubRiemannian distance in a Carnot group of step two is locally semiconcave away from the diagonal if and only if the group does not contain abnormal minimizing curves. Moreover, we prove that local semiconcavity fails to hold in the step-3 Engel group, even in the weaker "horizontal" sense.Comment: Revised version. To appear on J. Math. Anal- App

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

We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators. We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Finally, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.Comment: Added Section 6. Final version, to appear on Calc. Va

On the second order derivatives of convex functions on the Heisenberg group

In the Euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous H-convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for the class of continuous H-convex functions in the Heisenberg group

Maximum and comparison principles for convex functions on the Heisenberg group

We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge-Ampere type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge-Ampere measures for convex functions in the Heisenberg group.Comment: The results in this paper and the ideas of their proofs have been presented in the following talks: Analysis Seminar, Temple U., October 2002; Fabes--Chiarenza Lectures at Siracusa, December 2002; Pan-American Conference, Santiago de Chile, January 2003; Analysis Seminar, U. of Bologna, March 2003; and Analysis Seminar, U. Texas at Austin, March 200

Nonsmooth viscosity solutions of elementary symmetric functions of the complex Hessian

In this paper we prove the existence of nonsmooth viscosity solutions for Dirichlet problems involving elementary symmetric functions of the eigenvalues of the complex Hessian

A Hadamard-type open map theorem for submersions and applications to completeness results in Control Theory

We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carath\'eodory spaces and we improve a classical completeness result by Palais.Comment: 12 pages. Revised version. Minor changes. To appear on Annali di Matematic