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Regularity for Variational Problems in the Heisenberg Group

Shirsho Mukherjee, Department of Mathematics and Statistics, University of Jyväkylä
Shirsho Mukherjee, Department of Mathematics and Statistics, University of Jyväkylä
When Jan 12, 2016
from 04:00 PM to 05:00 PM
Where LH006
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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.

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