1,720,989 research outputs found
On the inner cone property forconvex sets in two-step Carnot groups, with applications to monotone sets
Full text linkIn the setting of two-step Carnot groups we show a "cone property" forhorizontally convex sets. Namely, we prove that, given a horizontally convex set C,a pair of points P ¬ C and Q ¬ int(C), both belonging to a horizontal line , thenan open truncated subRiemannian cone around and with vertex at P is containedin C.We apply our result to the problem of classification of horizontally monotone setsin Carnot groups. We are able to show that monotone sets in the direct product H×Rof the Heisenberg group with the real line have hyperplanes as boundaries
On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups
No full textWe 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
Sub-Riemannian cut time and cut locus in Reiter–Heisenberg groups
Full text linkWe study the sub-Riemannian cut time and cut locus of a given point in a class of step-2 Carnot groups of Reiter–Heisenberg type. Following the Hamiltonian point of view, we write and analyze extremal curves, getting the cut time of any of them, and a precise description of the set of cut points
On the subRiemannian cut locus in a model of free two-step Carnot group
Full text linkWe characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators, giving a new, independent proof of a result by Myasnichenko (J Dyn Control Syst 8(4):573-597, 2002). We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Furthermore, 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
Geometric methods in PDE’s
No full textThe analysis of PDEs is a prominent discipline in mathematics research, both in terms of its theoretical aspects and its relevance in applications. In recent years, the geometric properties of linear and nonlinear second order PDEs of elliptic and parabolic type have been extensively studied by many outstanding researchers. This book collects contributions from a selected group of leading experts who took part in the INdAM meeting "Geometric methods in PDEs", on the occasion of the 70th birthday of Ermanno Lanconelli. They describe a number of new achievements and/or the state of the art in their discipline of research, providing readers an overview of recent progress and future research trends in PDEs. In particular, the volume collects significant results for sub-elliptic equations, potential theory and diffusion equations, with an emphasis on comparing different methodologies and on their implications for theory and applications.
Liouville type theorems for non-linear differential inequalities on Carnot groups
No full textWe overview some recent results on the existence and non-existence of positive solutions for differential inequalities of the kind
in the setting of Carnot groups under the Keller-Osserman condition
A HADAMARD-TYPE OPEN MAP THEOREM FOR SUBMERSIONS AND APPLICATIONS TO COMPLETENESS RESULTS IN CONTROL THEORY
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