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Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group

Michele Circelli, Giovanna Citti, Albert Clop

TL;DR

This work proves local Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group $\mathbb{H}^n$ with $q\ge2$, addressing a degenerate, anisotropic PDE in a sub-Riemannian setting. The authors adapt Zhong's approach via a Riemannian approximation $g_\epsilon$ and a regularized potential $f_\delta$, developing mixed Caccioppoli inequalities for horizontal and vertical derivatives and employing Moser iteration to obtain uniform gradient bounds independent of $\epsilon$ and $\delta$. They then pass to the limit to establish $\nabla_H u\in L^\infty_{\text{loc}}(\Omega)$ and, consequently, local Lipschitz continuity of weak solutions, with an explicit Lipschitz estimate in terms of horizontal-gradient averages. The results push regularity theory in sub-Riemannian PDEs for degenerate, orthotropic operators and suggest possible extensions to other step-two Carnot groups, highlighting the role of the commutation relation $[X_i,X_{i+n}]=X_{2n+1}$ in the analysis.

Abstract

We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the Hölder regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected.

Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group

TL;DR

This work proves local Lipschitz regularity for solutions to an orthotropic -Laplacian-type equation in the Heisenberg group with , addressing a degenerate, anisotropic PDE in a sub-Riemannian setting. The authors adapt Zhong's approach via a Riemannian approximation and a regularized potential , developing mixed Caccioppoli inequalities for horizontal and vertical derivatives and employing Moser iteration to obtain uniform gradient bounds independent of and . They then pass to the limit to establish and, consequently, local Lipschitz continuity of weak solutions, with an explicit Lipschitz estimate in terms of horizontal-gradient averages. The results push regularity theory in sub-Riemannian PDEs for degenerate, orthotropic operators and suggest possible extensions to other step-two Carnot groups, highlighting the role of the commutation relation in the analysis.

Abstract

We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the Hölder regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected.
Paper Structure (10 sections, 11 theorems, 144 equations)

This paper contains 10 sections, 11 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Heisenberg group $\mathbb{H}^n$
  4. Riemannian approximation of the sub-Riemannian Heisenberg group
  5. The orthotropic $q$-Laplacian-type equation
  6. Riemannian approximation scheme
  7. Local Lipschitz regularity for solutions
  8. Uniform Caccioppoli-type inequalities for first derivatives of $\epsilon$-weak solutions
  9. Uniform $L^\infty$ estimate for the Riemannian gradient of $\epsilon$-weak solutions
  10. Local Lipschitz regularity for solutions

Key Result

Theorem 2.1

Let $1\leq p<N$. For any Riemannian ball $B_\epsilon(r)\subset\mathbb{H}^n$ and any $u\in W^{1,p}_\epsilon(B_\epsilon(r))$, it holds where $c=c(p,N)$ is independent of $\epsilon$.

Theorems & Definitions (21)

  • Theorem 2.1
  • Definition 3.1: Weak Solution
  • Definition 3.2: $\epsilon$-Weak Solution
  • Lemma 4.1
  • proof
  • Lemma 4.2: Caccioppoli for $X_{2n+1}u$
  • proof
  • Lemma 4.3
  • proof
  • Corollary 4.4
  • ...and 11 more