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

In their 1996 paper, Beals, Gaveau and Greiner found the fundamental solution to a 2 2 -Laplace-type equation in a class of sub-Riemannian spaces. This solution is related to the well-known fundamental solution to the p \texttt {p} -Laplace equation in Grushin-type spaces and the Heisenberg group. We extend the 2 2 -Laplace-type equation to a p \texttt {p} -Laplace-type equation. We show that the obvious generalization does not have desired properties, but rather, our generalization preserves some natural properties.

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp even under the bounded slope condition.

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

On local Lipschitz regularity for quasilinear equations in the Heisenberg group

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the 𝑛-dimensional Heisenberg group into CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), 4, 516–534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), 3, 863–934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD ⁢ ( K , N ) \mathrm{RCD}(K,N) to CAT ⁢ ( 0 ) \mathrm{CAT}(0) spaces and related results, preprint (2022), https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner–Eells–Sampson inequality for harmonic maps from RCD ⁡ ( k , n ) \operatorname{RCD}(k,n) spaces to CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces, preprint (2022), https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

In the Heisenberg group Hn, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ∂tu−∑i=12nXiAi(Xu)=K(x,t,u,Xu), where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For 2≤p≤4, we establish the local Lipschitz regularity (u∈Cloc0,1), with the horizontal gradient satisfying Xu∈Lloc∞; (ii) For 2≤p<3, we establish the local second-order horizontal Sobolev regularity (u∈HWloc2,2), with the second-order horizontal derivative satisfying XXu∈Lloc2. These results solve an open problem proposed by Capogna et al.