Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some counterexamples to Alt-Caffarelli-Friedman monotonicity formulas in Carnot groups

Fausto Ferrari, Davide Giovagnoli

Abstract

In this paper we continue the analysis of an Alt-Caffarelli-Friedman (ACF) monotonicity formula in Carnot groups of step $s >1$ confirming the existence of counterexamples to the monotone increasing behavior. In particular, we provide a sufficient condition that implies the existence of some counterexamples to the monotone increasing behavior of the ACF formula in Carnot groups. The main tool is based on the lack of orthogonality of harmonic polynomials in Carnot groups. This paper generalizes the results proved in \cite{ferrari2023counterexample}.

Some counterexamples to Alt-Caffarelli-Friedman monotonicity formulas in Carnot groups

Abstract

In this paper we continue the analysis of an Alt-Caffarelli-Friedman (ACF) monotonicity formula in Carnot groups of step confirming the existence of counterexamples to the monotone increasing behavior. In particular, we provide a sufficient condition that implies the existence of some counterexamples to the monotone increasing behavior of the ACF formula in Carnot groups. The main tool is based on the lack of orthogonality of harmonic polynomials in Carnot groups. This paper generalizes the results proved in \cite{ferrari2023counterexample}.
Paper Structure (8 sections, 5 theorems, 78 equations)

This paper contains 8 sections, 5 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Carnot group settings
  3. A characterization via harmonic homogeneous polynomials
  4. The general case
  5. Nonexistence of an ACF formula in some Carnot groups
  6. Final remarks
  7. Data Availability
  8. Declaration

Key Result

Theorem 1.1

For any Carnot group $\mathbb{G}$ of step $s$, with $s >1$, there exists an intrinsic harmonic function $u$ such that c1 fails to be monotone increasing in a right neighborhood of $0$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1: in folland1982hardy is Proposition 1.26
  • proof
  • Remark 4.2
  • Theorem 5.1
  • proof
  • ...and 3 more