130 research outputs found
Harnack Inequality for a Subelliptic PDE in nondivergence form
We consider subelliptic equations in non divergence form of the type , where are the Grushin vector fields, and the
matrix coefficient is uniformly elliptic. We obtain a scale invariant Harnack's
inequality on the '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 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
- …