Gradient estimates for an orthotropic nonlinear diffusion equation in the Heisenberg group

TL;DR

The paper addresses local Lipschitz regularity for weak solutions to a parabolic orthotropic $p$-Laplacian-type equation in the Heisenberg group $\mathbb{H}^n$ for $2\le p\le 4$, focusing on the equation $\partial_t u=\operatorname{div}_H(Df(\nabla_H u))$ with $f(z)=\frac{1}{p}\sum_{i=1}^{n}(z_i^2+z_{i+n}^2)^{p/2}$. The authors develop a robust regularization framework via Riemannian approximation, deriving uniform Caccioppoli-type inequalities for the horizontal derivatives and a Poincaré-type control of the vertical derivative, then apply a Moser iteration to obtain uniform $L^\infty$ bounds on the horizontal gradient. By passing to the limit, they prove that the horizontal gradient $\nabla_H u$ is locally Lipschitz in space, uniformly in time, and they establish local $L^q$ integrability for the vertical derivative $Zu$ and the time derivative $\partial_t u$ for all $q<\infty$. This work extends beyond prior stationary results by handling the parabolic, degenerate, orthotropic diffusion in a sub-Riemannian setting, offering a first parabolic regularity theory in this context. The boundedness (but not Hölder continuity) of $\nabla_H u$ reflects the degeneracy along multiple submanifolds where $\lambda_i(\nabla_H u)=0$, and the results provide a foundational step for further regularity in broader ranges of $p$.

Abstract

We prove local Lipschitz regularity for weak solutions to a parabolic orthotropic $p$-Laplacian-type equation in the Heisenberg group $\Hn$, for the range $2\leq p\leq4$.

Theorems & Definitions (24)

• Theorem 1.1
• Lemma 2.1
• proof
• Definition 2.2: $\delta,\varepsilon$-weak solution
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2