On a fractional Alt-Caffarelli-Friedman-type monotonicity formula
Analysis of PDEs
2025-10-13 v2
Abstract
In this note, by exploiting mean value properties of -harmonic functions, we introduce some monotonicity formulas in the nonlocal setting. We take into account intrinsically nonlocal functionals mimicking those introduced by Alt, Caffarelli and Friedman in the seminal work [Alt-Caffarelli-Friedman, Trans. Amer. Math. Soc. (1984)]. Our approach is purely nonlocal and does not rely on the extension technique. As a byproduct we also established interior nonlocal gradient estimates and a nonlocal analogue of the Bochner identity.
Keywords
Cite
@article{arxiv.2509.25891,
title = {On a fractional Alt-Caffarelli-Friedman-type monotonicity formula},
author = {Fausto Ferrari and Davide Giovagnoli and Enzo Maria Merlino},
journal= {arXiv preprint arXiv:2509.25891},
year = {2025}
}