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

∑_{i=1}^{2n} X_{i} ((Λ^{2} + |𝔛(u)|^{2})^{\frac{p}{2} −1} X_{i}u) = 0  (1)

in the n-dimensional Heisenberg group H^{n}, where Λ > 0 and p ≥ 2. These equations approximate the p-Laplacian as Λ → 0. In the Euclidean analogue, it is well known that these equations have C_{loc}^{∞} solutions. In turn, this result is used to prove the C_{loc}^{1,α} 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 𝔛u implies the smoothness of the solution u. In this talk we will show that when p is an interval 2 ≤ p < c(n) the horizontal gradient 𝔛u 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.