Regularity for Variational Problems in the Heisenberg Group

Abstract: We examine the local interior regularity of minimizers of scalar variational integrals of $p$-growth, with the $p$-Laplace equation $ \divo(|\Xu|^{p-2}\X u) = 0$ as a model example. This is done in the setting of the Heisenberg Group $\mathbb{H}^n$, which is $\mathbb{R}^{2n+1}$ endowed with a certain sub-riemannian geometry which gives rise to left invariant vector fields satisfying the Heisenberg algebra $ [X_i,X_j] = T\delta_{j,n+i}$ and hence differential operators like horizontal gradient and divergence, sub-laplacian etc. coming from them.

A longstanding literature in this area begins since the late 60's, going back to works of H\"{o}rmander in which regularity for $ p = 2$ is well established. The equation being quasilinear for $ p\neq 2$, investigating regularity is not quite trivial due to the non commutative vector fields. It is known that weak solutions are H\"{o}lder continuous due to Capogna, Danielli and Garofalo in 1993. But there has been no complete theory for the regularity of the horizontal gradient $\Xu$ and the problem remains open for atleast two decades until now, as we believe. There have been partial results for the gradient regularity over the years; Domokos and Manfredi showed H\"{o}lder continuity by Cordes perturbation technique in 2005 when $p$ is very close to $2$, Lipschitz continuity for the weak solution has been established for $2 \leq p < 4$ by G. Mingione, J. Manfredi and for $p\geq 2$ by X. Zhong. in 2007. Other regularity results include the proof of $ Tu \in L^p$ for $ 1< p< 4$ by Domokos and Marchi in 2004, where the restriction $ p < 4$ is seemingly unavoidable.

As a joint work of myself and X. Zhong, we show the local $ C^{1,\alpha}$ regularity of weak solutions of the equation for all $1 < p < \infty $ with quantitative estimates, which we believe to be optimal at least for $ p > 2$, as is the case for the corresponding Eucledean problem. We follow the De-Giorgi's method that is implemented by P. Tolksdorff and E. Debenedetto in 1982, for the same regularity of the problem in $\R^n$, taking suitable truncations of the gradient. I would like to present our solution and discuss the use of a reversed Caccioppoli type inequality for $Tu$ that allows us to get rid of the extra items coming from the commutators, the singular integrals for $ 1 < p < 2$ estimated in small measure sets, thereafter.