Regularity theory of quasilinear elliptic and parabolic equations in the Heisenberg group

Abstract

This note provides a succinct survey of the existing literature concerning the Hölder regularity for the gradient of weak solutions of PDEs of the form $$\sum _{i=1}^{2n}{X}_i{A}_i({\nabla }_{0}u)=0\text{ and }{\partial }_{t}u=\sum _{i=1}^{2n}{X}_i{A}_i({\nabla }_{0}u)$$ modeled on the $p$-Laplacian in a domain $\Omega$ in the Heisenberg group ${\mathbb{H}}^n$, with $1\le p<\infty$, and of its parabolic counterpart. We present some open problems and outline some of the difficulties they present.

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