Lipschitz regularity for solutions of the parabolic $p$-Laplacian in the Heisenberg group

TL;DR

This paper establishes local Lipschitz regularity for solutions to a class of degenerate parabolic PDEs modeled on the p-Laplacian within the Heisenberg group, extending to contact sub-Riemannian manifolds.

Contribution

It proves Lipschitz regularity for weak solutions of the parabolic p-Laplacian in the Heisenberg group, a significant extension of regularity results to sub-Riemannian geometries.

Findings

Lipschitz regularity holds for p between 2 and 4.

Results extend to contact sub-Riemannian manifolds.

Regularity established for degenerate parabolic PDEs in non-Euclidean settings.

Abstract

We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $\Omega\times \R^+$, where $\Omega$ is domain in the Heisenberg group $\Hn$, and $2\le p \le 4$. The result continues to hold in the more general setting of contact sub-Riemannian manifolds.