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RESEARCH PRODUCT
Regularity of sets with constant horizontal normal in the Engel group
Costante BellettiniEnrico Le Donnesubject
Mathematics - Differential GeometryStatistics and ProbabilityClass (set theory)Pure mathematicsStructure (category theory)Group Theory (math.GR)Analysis; Statistics and Probability; Geometry and Topology; Statistics Probability and UncertaintyMathematics - Analysis of PDEsMathematics - Metric GeometryFOS: MathematicsMathematics::Metric GeometryEngel groupMathematicsta111StatisticsCarnot groupMetric Geometry (math.MG)Function (mathematics)Lipschitz continuityEuclidean distanceDifferential Geometry (math.DG)Probability and UncertaintyGeometry and TopologyStatistics Probability and UncertaintyConstant (mathematics)Mathematics - Group TheoryAnalysisAnalysis of PDEs (math.AP)description
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic horizontal upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the intrinsic horizontal upper-graph of a function that is Lipschitz-continuous with respect to the intrinsic subRiemannian cones (and in particular locally Holder-continuous for the Euclidean distance). We further discuss a PDE characterization of the class of all sets with constant horizontal normal. Finally, we show that our rectifiability argument extends to the case of filiform groups of the first kind.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2012-01-30 | Communications in Analysis and Geometry |