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.
Ferrari et al. (Tue,) studied this question.