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Horizontal semiconcavity for the square of Carnot-Carathéodory distance on step 2 Carnot groups and applications to Hamilton-Jacobi equations

Federica Dragoni, Qing Liu, Ye Zhang

TL;DR

This work proves that on step-2 Carnot groups, the square CC distance from the origin, d^2(·,0), is h-semiconcave everywhere, first establishing the result for ideal Carnot groups and then extending to all step-2 groups. The authors develop a three-case local analysis near the unit sphere and leverage endpoint-map geometry, the abnormal set being trivial, and BR18’s C-nearly horizontal semiconcavity to obtain global regularity. They further connect this geometric regularity to Hamilton–Jacobi theory by showing that viscosity solutions obtained via the Hopf–Lax formula inherit spatial h-semiconcavity for t>0, under suitable convex Hamiltonians. The results yield Hessian and Laplacian bounds in the horizontal directions, enable generalizations to compositions with distance-like functions, and clarify the necessity of the step-2 hypothesis through Engel-type counterexamples, with implications for sub-Riemannian PDEs and distance-function analysis.

Abstract

We show that the square of Carnot-Carathéodory distance from the origin, in step 2 Carnot groups, enjoys the horizontal semiconcavity (h-semiconcavity) everywhere in the group including the origin. We first give a proof in the case of ideal Carnot groups, based on the simple group structure as well as estimates for the Euclidean semiconcavity. Our proof of the general result involves more geometric properties of step 2 Carnot groups. We further apply our h-semiconcavity result to show h-semiconcavity of the viscosity solutions to a class of non-coercive evolutive Hamilton-Jacobi equations by using the Hopf-Lax formula associated to the Carnot-Carathéodory metric.

Horizontal semiconcavity for the square of Carnot-Carathéodory distance on step 2 Carnot groups and applications to Hamilton-Jacobi equations

TL;DR

This work proves that on step-2 Carnot groups, the square CC distance from the origin, d^2(·,0), is h-semiconcave everywhere, first establishing the result for ideal Carnot groups and then extending to all step-2 groups. The authors develop a three-case local analysis near the unit sphere and leverage endpoint-map geometry, the abnormal set being trivial, and BR18’s C-nearly horizontal semiconcavity to obtain global regularity. They further connect this geometric regularity to Hamilton–Jacobi theory by showing that viscosity solutions obtained via the Hopf–Lax formula inherit spatial h-semiconcavity for t>0, under suitable convex Hamiltonians. The results yield Hessian and Laplacian bounds in the horizontal directions, enable generalizations to compositions with distance-like functions, and clarify the necessity of the step-2 hypothesis through Engel-type counterexamples, with implications for sub-Riemannian PDEs and distance-function analysis.

Abstract

We show that the square of Carnot-Carathéodory distance from the origin, in step 2 Carnot groups, enjoys the horizontal semiconcavity (h-semiconcavity) everywhere in the group including the origin. We first give a proof in the case of ideal Carnot groups, based on the simple group structure as well as estimates for the Euclidean semiconcavity. Our proof of the general result involves more geometric properties of step 2 Carnot groups. We further apply our h-semiconcavity result to show h-semiconcavity of the viscosity solutions to a class of non-coercive evolutive Hamilton-Jacobi equations by using the Hopf-Lax formula associated to the Carnot-Carathéodory metric.
Paper Structure (13 sections, 19 theorems, 152 equations, 4 figures)

This paper contains 13 sections, 19 theorems, 152 equations, 4 figures.

Table of Contents

  1. Introduction.
  2. Preliminaries
  3. Carnot groups
  4. Endpoint map and ideal Carnot groups
  5. H-concavity and h-semiconcavity
  6. H-semiconcavity of squared CC distance in ideal Carnot groups
  7. Case 1: $p \in S = \{q\in \mathbb G \,|\, d(q,0) = 1\}$.
  8. Case 2: $p\in \mathbb G$ and $p \ne 0$.
  9. Case 3: $p = 0$.
  10. Generalization to step 2 Carnot groups
  11. Further consequences and generalization
  12. Application to Hamilton-Jacobi equations
  13. Properties of special functions and details of computation

Key Result

Theorem 1.1

Let $\mathbb G$ be a step 2 Carnot group with CC distance $d$. Then $d^2(\cdot, 0)$ is h-semiconcave in $\mathbb G$.

Figures (4)

  • Figure 1: The CC ball $B_{CC}(0, 1)$ and the Euclidean ball $B_E(q, \eta/2)$
  • Figure 2: The points $p_\pm$ and the Euclidean balls centered at $q$
  • Figure 3: The graph of $d_0^2=d^2(\cdot ,0)$ in the Heisenberg group $\mathbb H$
  • Figure 4: The graph of the function $\mu$

Theorems & Definitions (51)

  • Theorem 1.1
  • Definition 2.1: Carnot group
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5: Sub-Riemannian structure
  • Definition 2.6: Carnot-Carathéodory distance
  • Proposition 2.7: AES85, Proposition 1.1
  • Proposition 2.8: BLU07, Proposition 5.1.4
  • Definition 2.9: Endpoint map
  • ...and 41 more