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Partial Hölder regularity for fully nonlinear nonlocal parabolic equations with integrable kernels

Minhyun Kim, Luke Schleef, Russell W. Schwab

TL;DR

The paper develops partial Hölder regularity for solutions of fully nonlinear parabolic integro-differential equations with integrable interaction kernels, focusing on kernels truncated at scale $\rho$. By proving a scale-adapted weak Harnack inequality for supersolutions, the authors obtain Hölder estimates up to the truncation scale with constants that are uniform in $\rho$ and remain valid in the limit $\rho\to0^+$. The framework unifies and extends prior Krylov–Safonov type results for nonlocal equations (including the case $\rho=0$) to the nonlinear, parabolic regime with integrable kernels, and it encompasses linear and fully nonlinear operators with drift for $s\ge 1/2$. In addition, the work discusses approximation by truncated kernels and presents elliptic counterparts, including a Liouville-type result, highlighting the robustness and breadth of the partial regularity theory in nonlocal parabolic contexts.

Abstract

In this work, we consider solutions to (fully nonlinear) parabolic integro-differential equations with integrable interaction kernels. A typical equation would be that obtained by starting with, for $s\in(0,1)$, the $s$-fractional heat equation, but replacing the interaction kernel in the integro-differential term with one which has been truncated, for $ρ>0$, at the value $ρ^{-d-2s}$, hence integrable. We show that solutions to these equations have a partial regularity estimate which captures differences of the solution up to the scale at which the kernel has a truncation in its singularity. The estimates we provide are robust with respect to the truncation parameter, and they include the existing results for the original operators without truncation. There are some earlier results for linear and elliptic cases of this situation of integrable interaction kernels, and so our work is a generalization of those to the nonlinear and parabolic setting.

Partial Hölder regularity for fully nonlinear nonlocal parabolic equations with integrable kernels

TL;DR

The paper develops partial Hölder regularity for solutions of fully nonlinear parabolic integro-differential equations with integrable interaction kernels, focusing on kernels truncated at scale . By proving a scale-adapted weak Harnack inequality for supersolutions, the authors obtain Hölder estimates up to the truncation scale with constants that are uniform in and remain valid in the limit . The framework unifies and extends prior Krylov–Safonov type results for nonlocal equations (including the case ) to the nonlinear, parabolic regime with integrable kernels, and it encompasses linear and fully nonlinear operators with drift for . In addition, the work discusses approximation by truncated kernels and presents elliptic counterparts, including a Liouville-type result, highlighting the robustness and breadth of the partial regularity theory in nonlocal parabolic contexts.

Abstract

In this work, we consider solutions to (fully nonlinear) parabolic integro-differential equations with integrable interaction kernels. A typical equation would be that obtained by starting with, for , the -fractional heat equation, but replacing the interaction kernel in the integro-differential term with one which has been truncated, for , at the value , hence integrable. We show that solutions to these equations have a partial regularity estimate which captures differences of the solution up to the scale at which the kernel has a truncation in its singularity. The estimates we provide are robust with respect to the truncation parameter, and they include the existing results for the original operators without truncation. There are some earlier results for linear and elliptic cases of this situation of integrable interaction kernels, and so our work is a generalization of those to the nonlinear and parabolic setting.
Paper Structure (13 sections, 17 theorems, 174 equations)

This paper contains 13 sections, 17 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Background and related results
  3. Preliminaries
  4. Sets and functions in $\mathbb R^d\times(-\infty,0]$
  5. Assumptions on the operators
  6. Some properties of $K$, $L_K$, and $\mathcal{M}_\rho^\pm$
  7. Definition of Viscosity solutions and some properties
  8. Weak Harnack inequality
  9. Partial Hölder regularity
  10. Related results: approximation via integrable kernels and elliptic problems
  11. Approximation of solutions via truncated kernels
  12. The elliptic case
  13. Additional Proofs

Key Result

Theorem 1.1

Let $\rho>0$ and assume that $K$ satisfies, for some $0<\lambda<\Lambda$, with also $K(x,y,t)=K(x,-y,t)$ if $s=1/2$. Let $R>\rho$ and $f\in L^\infty(B_R\times I_R)$, where $I_R = (-R^{2s},0]$. Assume that $u\in L^\infty({\mathbb R}^d\times I_R)$, $Lu$ is as in eqIntro:Operator, and $u$ is a viscosity solution of Then there exist universal parameters $\alpha\in(0,2s)$ and $C>0$ such that

Theorems & Definitions (41)

  • Theorem 1.1: Simplest Setting
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 1.6
  • Definition 1.7
  • Lemma 1.8
  • Theorem 2.1
  • Theorem 2.2
  • ...and 31 more