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A counterexample to the monotone increasing behavior of an Alt-Caffarelli-Friedman formula in the Heisenberg group

Fausto Ferrari, Nicolò Forcillo

Abstract

In this paper we provide a counterexample about the existence of an increasing monotonicity behavior of a function introduced in \cite{FeFo}, companion of the celebrated Alt-Caffarelli-Friedman monotonicity formula, in the noncommutative framework.

A counterexample to the monotone increasing behavior of an Alt-Caffarelli-Friedman formula in the Heisenberg group

Abstract

In this paper we provide a counterexample about the existence of an increasing monotonicity behavior of a function introduced in \cite{FeFo}, companion of the celebrated Alt-Caffarelli-Friedman monotonicity formula, in the noncommutative framework.
Paper Structure (7 sections, 15 theorems, 155 equations)

This paper contains 7 sections, 15 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminary facts: the Euclidean case
  3. The Heisenberg setting
  4. Some properties of the orthogonal polynomials in the Heisenberg group
  5. Explicit computation of the counterexample
  6. Nonexistence of an Alt-Caffarelli-Friedman type monotonicity formula in $\mathbb{H}^1$
  7. A further generalization with application

Key Result

Theorem 1.1

Let $u=x-3yt-2x^3.$ Then $\Delta_{\mathbb{H}^1}u=0$ and $J^{\mathbb{H}^1}_u(r)$ is strictly monotone decreasing in a right neighborhood of $r=0.$

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • proof
  • ...and 20 more