Nonlocal Liouville theorems with gradient nonlinearity

TL;DR

The paper develops a unified Liouville theory for nonlinear nonlocal equations with gradient nonlinearity, addressing equations of the form $-\,\mathcal{I} u + H(\nabla u)=0$ in $\mathbb{R}^n$, where $\mathcal{I}$ is a fractional Pucci-type operator of order $2s$ and $H$ captures gradient nonlinearity in three models. The authors adapt the Ishii–Lions doubling-variables method to the nonlocal setting, introducing a penalization with small Hölder seminorm and establishing a suite of technical lemmas to control nonlocal terms and localization errors. They prove Liouville-type rigidity under subcritical growth conditions for various ranges of $s$ and gradient-nonlinearity structures, including a symmetric-kernel case and a coercive Hamiltonian model, and they answer an open problem from prior work. These Liouville results are then leveraged to obtain interior regularity conclusions, such as $C^{\gamma}$ regularity for critical diffusion ($s=1/2$) and related regularity for nonlinear nonlocal equations, illustrating the practical impact on the regularity theory of nonlocal equations. The work thus provides a versatile, unified framework linking nonlocal Liouville properties to subsequent regularity consequences in a broad class of Hamilton–Jacobi–Bellman-type equations with gradient nonlinearity.

Abstract

In this article we consider a large family of nonlinear nonlocal equations involving gradient nonlinearity and provide a unified approach, based on the Ishii-Lions type technique, to establish Liouville properties of the solutions. We also answer an open problem raised by [24]. Some applications to regularity issues are also studied.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.3
• Theorem 1.6
• Corollary 1.7