Juan Manfredi
University of Pittsburgh
Pittsburgh, PA

Smoothness of solutions to quasilinear elliptic equations in the Heisenberg group

ABSTRACT: Consider quasilinear elliptic equations of $ p$-Laplacian type

$\displaystyle \sum_{i=1}^{2n} X_i \left( \left(\Lambda^2+ \left\vert \mathfrak{X}(u) \right\vert^2\right)^{\frac{p}{2}-1} X_i u\right) =0$ (1)

in the $ n$-dimensional Heisenberg group $ \mathcal{H}^n$, where $ \Lambda >0$ and $ p\ge 2$. These equations approximate the $ p$-Laplacian as $ \Lambda\to0$. In the Euclidean analogue, it is well known that these equations have $ C^{\infty}_{\mathrm{loc}}$ solutions. In turn, this result is used to prove the $ C_{\mathrm{loc}}^{1,\alpha}$ regularity of $ p$-harmonic functions.

In the case of the Heisenberg group much less is known without any additional hypothesis. It was shown by Capogna that the boundedness of the horizontal gradient $ \mathfrak{X}u$ implies the smoothness of the solution $ u$. In this talk we will show that when $ p$ is an interval $ 2 \le p< c(n)$ the horizontal gradient $ \mathfrak{X}u$ is indeed locally bounded. Our proof is based on using a type of unbalanced Moser iteration. We obtain the values $ c(1)=c(2)=4$ and $ c(3)=2+14/13$.

This is joint work with Giuseppe Mingione from the University of Parma.