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

TL;DR

This paper surveys the regularity of solutions to quasilinear PDEs in the Heisenberg group, focusing on Hölder continuity of gradients for equations modeled on the p-Laplacian, and discusses open problems and challenges.

Contribution

It provides a comprehensive overview of existing results on regularity theory for quasilinear PDEs in the Heisenberg group and highlights open problems in the field.

Findings

Hölder regularity results for weak solutions' gradients

Identification of key difficulties in extending regularity theory

Open problems in the regularity of quasilinear PDEs in the Heisenberg group

Abstract

This note provides a succinct survey of the existing literature concerning the H\"older 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.