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

Fausto Ferrari, Davide Giovagnoli, Enzo Maria Merlino

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.

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

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.

Paper Structure

This paper contains 9 sections, 23 theorems, 147 equations.

Table of Contents

  1. Notations and preliminary results
  2. Some nonlocal quantities
  3. Fractional mean value properties.
  4. Main results
  5. The distributional Riesz fractional gradient
  6. The operators $\nabla^s$ and $\mathop{\mathrm{div}}\nolimits^s$
  7. A Nonlocal Functional Based on the Fractional Gradient
  8. A nonlocal Bochner-type formula and some applications of the monotonicity formula
  9. Appendix

Key Result

Theorem 1

Let $s \in (0,1)$ and $\varepsilon, \, \delta >0$ be fixed. Assume that $u \in C_{loc}^{s+\varepsilon}(\mathop{\mathrm{\mathbb{R}}}\nolimits^n) \cap L_{s}^2(\mathop{\mathrm{\mathbb{R}}}\nolimits^n)$ and Then the map is monotone increasing for $R$ sufficiently small.

Theorems & Definitions (40)

  • Theorem 1: Monotonicity formula for $G_u$
  • Theorem 2: Stability of $J_{ACF}^{s}(u,R)$ as $s \rightarrow 1^{-}$
  • Theorem 3: Interior nonlocal gradient estimates
  • Theorem 4: Monotonicity formula for $|\nabla^s u|^2$
  • Theorem 5: Stability of $\mathfrak{J}_{ACF}^{s}(u,R)$ as $s \rightarrow 1^{-}$
  • Theorem 6
  • Lemma 1.1
  • proof
  • Remark 1.2
  • Proposition 1.3: $s$-mean value property
  • ...and 30 more