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Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term

Yuzhou Fang, Juha Kinnunen, Chao Zhang

TL;DR

The article develops a comprehensive regularity theory for degenerate fully nonlinear nonlocal equations with Hamiltonian terms. By leveraging the Ishii–Lions viscosity framework, perturbation techniques, and compactness arguments, it establishes Lipschitz and gradient Hölder regularity, with uniform control as the fractional parameter $\sigma$ approaches 2. It further extends the regularity to $C^1$ under minimal degeneracy hypotheses via a non-collapsing modulus framework and affine approximation schemes. Together, these results advance the understanding of interior regularity for a broad class of degenerate nonlocal Hamilton–Jacobi problems and connect near-local behavior to fully nonlocal dynamics as $\sigma\to2$.

Abstract

We investigate a class of degenerate fully nonlinear nonlocal elliptic equations with Hamiltonian terms. By precisely characterizing the interaction between the degeneracy law of equations and the growth behavior of the Hamiltonian terms, we establish the Lipschitz regularity of viscosity solutions by the Ishii-Lions method, and further show the gradient Hölder continuity for solutions via utilizing perturbation techniques. Additionally, under minimal assumptions on the degeneracy pattern, the $C^1$-differentiability property of solutions is explored as well.

Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term

TL;DR

The article develops a comprehensive regularity theory for degenerate fully nonlinear nonlocal equations with Hamiltonian terms. By leveraging the Ishii–Lions viscosity framework, perturbation techniques, and compactness arguments, it establishes Lipschitz and gradient Hölder regularity, with uniform control as the fractional parameter approaches 2. It further extends the regularity to under minimal degeneracy hypotheses via a non-collapsing modulus framework and affine approximation schemes. Together, these results advance the understanding of interior regularity for a broad class of degenerate nonlocal Hamilton–Jacobi problems and connect near-local behavior to fully nonlocal dynamics as .

Abstract

We investigate a class of degenerate fully nonlinear nonlocal elliptic equations with Hamiltonian terms. By precisely characterizing the interaction between the degeneracy law of equations and the growth behavior of the Hamiltonian terms, we establish the Lipschitz regularity of viscosity solutions by the Ishii-Lions method, and further show the gradient Hölder continuity for solutions via utilizing perturbation techniques. Additionally, under minimal assumptions on the degeneracy pattern, the -differentiability property of solutions is explored as well.

Paper Structure

This paper contains 5 sections, 15 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz continuity of solutions
  4. Gradient Hölder regularity of solutions
  5. $C^1$-regularity of solutions

Key Result

Theorem 1.1

Let $u\in C(\overline{B}_1)$ be a viscosity solution to main, and assume that the conditions $(A_1)$--$(A_4)$ in Section sec2 hold with $0\le q\le p+1$. Then there exists $\sigma_0\in (1, 2)$, close enough to $2$, such that if $\sigma\in(\sigma_0, 2)$ then $u$ is locally $C^{1,\alpha}(B_1)$-regular Here $0<\alpha<\min\bigl\{\overline{\alpha},\frac{\sigma-1}{1+p}\bigr\}$ and $\overline{\alpha}$ is

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1: viscosity solutions
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • ...and 25 more