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arXiv · 2509.25891

On a fractional Alt-Caffarelli-Friedman-type monotonicity formula

Fausto Ferrari·Davide Giovagnoli·Enzo Maria Merlino
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Abstract

In this note, by exploiting mean value properties of $s$-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.

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Fausto Ferrari, Davide Giovagnoli, Enzo Maria Merlino. 2025-09-30. On a fractional Alt-Caffarelli-Friedman-type monotonicity formula. https://arxiv.org/abs/2509.25891

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