次二次分数阶p-拉普拉斯方程的比较原理与对称性

中文摘要 AI 辅助

本文针对次二次分数阶p-拉普拉斯算子（即1 < p < 2），在极小正则性假设下建立了新的比较原理框架，该算子因具有奇异性而长期是重大挑战问题。我们的结果为该情形下的移动平面法提供了必要的分析工具。首先，我们在测度足够小的有界区域中，对(-Δ)_p^s u = f(u)证明了弱比较原理，仅要求弱解有界。更重要的是，我们在参数范围s ∈ (0, 1/2)且1/(1-s) < p < 2时，对连续弱解建立了强比较原理。我们的证明引入了局部障碍函数，既不要求弱解具有任何赫尔德正则性，也不要求区域具有任何光滑性，这与此前严重依赖赫尔德甚至C^{1,1}正则性的研究形成显著对比。作为直接应用，我们利用这些比较原理证明了(-Δ)_p^s u = f(u)的弱解在温和假设下的对称性，这极大拓展了非局部拟线性方程的现有对称性理论。

英文摘要

This paper establishes a new comparison principle framework for the subquadratic fractional $p$-Laplacian, i.e.~$1 < p < 2$ under minimal regularity assumptions, that has remained a significant challenging issue due to the singularity of the operator. Our results provide the essential analytical tools required for the moving plane method in this setting. We prove first a weak comparison principle for $(-Δ)_p^s u = f(u)$ in bounded domains with sufficiently small measure, where only the boundedness of the weak solution is required. More importantly, we establish a strong comparison principle for continuous weak solutions in the parameter range $s \in (0, \frac{1}{2})$ and $\frac{1}{1-s} < p < 2$. Our proof introduces a localized barrier function and does not require any Hölder regularity of the weak solution, nor any smoothness of the domain. This presents a substantial contrast over previous study, which relied heavily on Hölder or even $C^{1,1}$ regularity. As a direct application, we employ these comparison principles to prove the symmetry of weak solutions to $(-Δ)_p^s u = f(u)$ under mild assumptions, which significantly extend existing symmetry theories for nonlocal quasilinear equations.

发表机构

• East China Normal University（华东师范大学）
• Zhejiang Normal University（浙江师范大学）

机构由 AI 辅助整理，请以论文原文为准。