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Viscosity Solutions in Martinet Spaces

Thomas Bieske, Frederic Bowen

TL;DR

This work extends the viscosity solution framework to Martinet sub-Riemannian spaces, which lack both a group law and Grushin-type vector field triangularity. By developing Martinet jets and a Twisting Lemma that connects Euclidean and Martinet jets, and by introducing a Martinet-specific maximum principle, the authors establish a robust comparison principle for strictly monotone elliptic PDEs on $\mathbb{M}$. The core result is a uniqueness theory for viscosity solutions of the horizontal infinite Laplacian $\Delta_{\infty}$ on Martinet spaces, achieved through the Iterated Maximum Principle and Jensen-type auxiliary functions, culminating in a well-posed Dirichlet problem for $\Delta_{\infty} u=0$. These contributions broaden the landscape of sub-Riemannian PDE analysis to geometries where the usual Carnot group tools are unavailable, enabling well-posedness results in new nongroup, non-Grushin settings.

Abstract

In this paper, we establish the properties of viscosity solutions in Martinet spaces, which lack both the algebraic group law of Carnot groups and the triangular vector fields of Grushin-type spaces. We then prove the uniqueness of viscosity solutions to strictly monotone elliptic PDEs and to the infinite Laplace equation.

Viscosity Solutions in Martinet Spaces

TL;DR

This work extends the viscosity solution framework to Martinet sub-Riemannian spaces, which lack both a group law and Grushin-type vector field triangularity. By developing Martinet jets and a Twisting Lemma that connects Euclidean and Martinet jets, and by introducing a Martinet-specific maximum principle, the authors establish a robust comparison principle for strictly monotone elliptic PDEs on . The core result is a uniqueness theory for viscosity solutions of the horizontal infinite Laplacian on Martinet spaces, achieved through the Iterated Maximum Principle and Jensen-type auxiliary functions, culminating in a well-posed Dirichlet problem for . These contributions broaden the landscape of sub-Riemannian PDE analysis to geometries where the usual Carnot group tools are unavailable, enabling well-posedness results in new nongroup, non-Grushin settings.

Abstract

In this paper, we establish the properties of viscosity solutions in Martinet spaces, which lack both the algebraic group law of Carnot groups and the triangular vector fields of Grushin-type spaces. We then prove the uniqueness of viscosity solutions to strictly monotone elliptic PDEs and to the infinite Laplace equation.
Paper Structure (8 sections, 13 theorems, 94 equations)

This paper contains 8 sections, 13 theorems, 94 equations.

Table of Contents

  1. introduction
  2. Martinet Space
  3. Calculus
  4. Martinet Jets and Viscosity Solutions
  5. Maximum Principle
  6. Viscosity Solutions to the Infinite Laplace Equation
  7. The Iterated Maximum Principle
  8. Infinite Harmonic Functions

Key Result

Lemma 4.1

Let $O\subset \mathbb{M}$ be open, let $u: O\to\mathbb{R}$, and let $p_0=(x_1^0,x_2^0,x_3^0) \in O$. Suppose that $(\eta, X)\in J^{2,+}_{\textmd{eucl}} u(p_0)$ with $\eta=(\eta_1,\eta_2,\eta_3)$ and $X=\{X_{ij}\}$ a $3\times 3$ symmetric matrix. Then where $A(p_0)$ is the $2\times3$ matrix of Martinet vector coefficients defined by the $3\times 3$ matrix $B(p_0)$ is given by and the $2\times2$

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4.1: Twisting Lemma
  • Remark 5.1
  • Lemma 5.2
  • Lemma 5.3
  • proof
  • Theorem 5.4
  • proof
  • ...and 12 more