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Nonlocal parabolic De Giorgi classes

Simone Ciani, Kenta Nakamura

TL;DR

This work develops a comprehensive regularity theory for the nonlocal parabolic De Giorgi class, proving local boundedness under optimal tail conditions, two distinct weak Harnack inequalities, and a full parabolic Harnack principle, leading to local Hölder continuity and a Liouville-type rigidity result. The authors avoid parabolic covering arguments, instead leveraging De Giorgi–Nash–Moser measure-theoretic techniques with careful tail control, enabling sharp regularity results for nonlocal parabolic equations and an application to nonlocal Trudinger-type equations. The methodology extends to nonlinear and doubly nonlinear nonlocal settings, providing robust tools for long-range interaction problems and contributing a state-of-the-art nonlocal Harnack framework. The results have implications for fractional diffusion models and nonlocal variational problems, with potential extensions to nonlocal Trudinger-type dynamics and related PDE systems.

Abstract

We study the local behavior of the elements of a specific energy class of functions, called the nonlocal parabolic ($p$-homogenous) De Giorgi class. First we carry on an analysis of their local boundedness under optimal tail conditions, and then prove several weak Harnack inequalities, measure theoretical propagation lemmas, and a parabolic Harnack inequality. We show a full proof of the local Hölder continuity, eventually establishing a Liouville-type rigidity property. Finally, as an application of our method, we prove a state-of-the-art nonlocal Harnack inequality for nonnegative solutions of the nonlocal Trudinger equation.

Nonlocal parabolic De Giorgi classes

TL;DR

This work develops a comprehensive regularity theory for the nonlocal parabolic De Giorgi class, proving local boundedness under optimal tail conditions, two distinct weak Harnack inequalities, and a full parabolic Harnack principle, leading to local Hölder continuity and a Liouville-type rigidity result. The authors avoid parabolic covering arguments, instead leveraging De Giorgi–Nash–Moser measure-theoretic techniques with careful tail control, enabling sharp regularity results for nonlocal parabolic equations and an application to nonlocal Trudinger-type equations. The methodology extends to nonlinear and doubly nonlinear nonlocal settings, providing robust tools for long-range interaction problems and contributing a state-of-the-art nonlocal Harnack framework. The results have implications for fractional diffusion models and nonlocal variational problems, with potential extensions to nonlocal Trudinger-type dynamics and related PDE systems.

Abstract

We study the local behavior of the elements of a specific energy class of functions, called the nonlocal parabolic (-homogenous) De Giorgi class. First we carry on an analysis of their local boundedness under optimal tail conditions, and then prove several weak Harnack inequalities, measure theoretical propagation lemmas, and a parabolic Harnack inequality. We show a full proof of the local Hölder continuity, eventually establishing a Liouville-type rigidity property. Finally, as an application of our method, we prove a state-of-the-art nonlocal Harnack inequality for nonnegative solutions of the nonlocal Trudinger equation.

Paper Structure

This paper contains 13 sections, 8 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Origins and framing of the topic
  3. Main results and novelties
  4. The Method
  5. Novelty and Significance
  6. Organization of the paper
  7. Notation, fractional tools and notion of De Giorgi classes
  8. Notation
  9. Fractional tools
  10. Notion of De Giorgi class
  11. Proof of Theorem \ref{['Thm:boundedness']}
  12. Several expansions of positivity
  13. De Giorgi-type Lemma

Key Result

Theorem 1.1

Let $u$ belong to $\mathbf{PDG}_\pm^{s,p}(\Omega_T,\gamma_{\mathrm{DG}}, \varepsilon)$ in the sense of Definition def of DG. Suppose that $Q_{\varrho, \theta}(z_o) \subseteq \Omega_T$ with $z_o=(x_o,t_o) \in \Omega_T$, and let $\sigma \in (0,1)$ and $\nu \in (0,p]$. There exist positive constants $ holds, where

Theorems & Definitions (18)

  • Theorem 1.1: $L^\infty$--$L^\nu$ Local boundedness
  • Remark 1.2
  • Theorem 1.3: Nonlocal weak Harnack inequality I
  • Theorem 1.4: Nonlocal weak Harnack inequality II
  • Remark 1.5
  • Theorem 1.6: Nonlocal Harnack inequality
  • Remark 1.7: Time-Gap Phenomenon
  • Theorem 1.8: Hölder modulus of continuity
  • Corollary 1.9: Liouville-type rigidity theorem
  • Definition 2.1: Nonlocal parabolic De Giorgi class
  • ...and 8 more