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

2025/09/30 by Fausto Ferrari, Ferrari, Fausto, Davide Giovagnoli +3
Mathematics · #Mathematical Inequalities and Applications #Fractional Differential Equations Solutions #Fuzzy Systems and Optimization

paper · pdf · doi:10.48550/arxiv.2509.25891

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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Related

• A self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula
• On the mean value property of fractional harmonic functions
• A new glance to the Alt-Caffarelli-Friedman monotonicity formula
• Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings
• Existence and monotonicity of nonlocal boundary value problems: the one-dimensional case