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

In this paper we continue the analysis of an Alt–Caffarelli–Friedman (ACF) monotonicity formula in Carnot groups of step $$s >1$$ s > 1 confirming the existence of counterexamples to the monotone increasing behavior. In particular, we provide a sufficient condition that implies the existence o...

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Bibliographic Details
Published in:Annali di matematica pura ed applicata 2024-07
Main Authors: Ferrari, Fausto, Giovagnoli, Davide
Format: Article
Language:English
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Summary:In this paper we continue the analysis of an Alt–Caffarelli–Friedman (ACF) monotonicity formula in Carnot groups of step $$s >1$$ 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 Ferrari and Forcillo (Atti Accad Naz Lincei Rend Lincei Mat Appl 34(2):295–306, 2023).
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-024-01490-8