Regularity theory for mixed local-nonlocal problem involving general stable operators

TL;DR

The paper addresses regularity for a linear elliptic equation with a mixed local-nonlocal operator $\mathcal{E}=L-\operatorname{div}(a(x)\nabla\cdot)$, where $L$ is the generator of a symmetric stable Lévy process and $a(x)$ is a positive Hölder weight. The authors develop a maximum principle and a Liouville-type theorem in the whole space, then combine blow-up analysis and compactness to derive interior regularity, followed by boundary regularity via $W^{2,p}$ estimates and a boundary maximum principle. Key contributions include interior Hölder and higher-regularity results in two parameter regimes depending on $s$ and $\alpha$, and a thorough treatment of regularity up to the boundary for constant $a$ and $C^{1,1}$ domains. The results extend the regularity theory for general symmetric stable operators to mixed local-nonlocal settings, with implications for models that couple diffusion and jump processes in physics, ecology, and finance.

Abstract

In this paper, we study the regularity of solutions to a linear elliptic equation involving a mixed local-nonlocal operator of the form $$Lu - \operatorname{div}\big(a(x)\nabla u(x)\big)= f, \quad \text{in } Ω\subset \mathbb{R}^n,$$ where $L$ is a general stable Lévy type operator and $a(\cdot)$ is a positive Hölder continuous weight. By establishing a maximum principle and a Liouville-type result in the entire space, we are able to derive the interior regularity and the regularity up to the boundary of the solutions under suitable assumptions on $f(x)$ and $a(x)$ .

Theorems & Definitions (39)

• Definition 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of Theorem \ref{['maxprin']}
• Proposition 2.1
• proof
• Theorem 2.2
• proof