Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2

TL;DR

This paper characterizes when graphs of continuous functions in Carnot groups of step 2 are $C^1_H$-regular by linking it to solutions of a Burgers' type system in the distributional sense, revealing regularity properties and limitations in higher step groups.

Contribution

It establishes a precise equivalence between $C^1_H$-regularity and distributional solutions of Burgers' type systems in Carnot groups of step 2, and shows that such solutions have $1/2$-Hölder regularity along vertical directions.

Findings

Graphs are $C^1_H$-regular iff they satisfy a Burgers' system distributionally.

Continuous distributional solutions are broad solutions to the Burgers' system.

Solutions exhibit $1/2$-Hölder regularity along vertical directions.

Abstract

We prove that in arbitrary Carnot groups $\mathbb G$ of step 2, with a splitting $\mathbb G=\mathbb W\cdot\mathbb L$ with $\mathbb L$ one-dimensional, the graph of a continuous function $\varphi\colon U\subseteq \mathbb W\to \mathbb L$ is $C^1_{\mathrm{H}}$-regular precisely when $\varphi$ satisfies, in the distributional sense, a Burgers' type system $D^{\varphi}\varphi=\omega$, with a continuous $\omega$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $\varphi$ to a Burgers' type system $D^{\varphi}\varphi=\omega$, with $\omega$ continuous, is actually a broad solution to $D^{\varphi}\varphi=\omega$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^{\varphi}\varphi=\omega$, with $\omega$ continuous, enjoy $1/2$-little H\"older regularity along vertical directions.