A note on monotonicity and Bochner formulas in Carnot groups

TL;DR

This paper establishes two new monotonicity formulas involving the right-invariant carré du champ for solutions to certain PDEs in Carnot groups, drawing parallels with classical formulas in Euclidean settings.

Contribution

It introduces novel monotonicity formulas for PDEs in Carnot groups using right-invariant structures, extending classical Euclidean results.

Findings

Monotonicity formulas hold for right-invariant carré du champ

Formulas resemble Euclidean monotonicity results

Failure of nondecreasing Almgren-type functional in this setting

Abstract

In this note we prove two monotonicity formulas for solutions of $\Delta_H f = c$ and $\Delta_H f - \p_t f = c$ in Carnot groups. Such formulas involve the right-invariant \emph{carr\'e du champ} of a function and they are false for the left-invariant one. The main results, Theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt-Caffarelli-Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an "almost monotoniccity" formula be possible. In the last section we discuss the failure of the nondecreasing monotonicity of an Almgren type functional.