海森堡群中的混合局部-非局部特征值问题：谱理论与奇异重数

中文摘要 AI 辅助

本文研究由一族结合两种不同扩散机制的算子所控制的非线性特征值问题：一种经典的局部扩散，用于描述短程相互作用；以及一种分数阶非局部扩散，用于捕捉长程效应。分析在海森堡群中进行，这是一种非欧几里得几何背景，其中运动被限制在一组特定的水平方向上。作为所得谱理论的一个应用，我们在纯非局部和混合局部-非局部两种情形下，建立了受扰奇异问题的重数结果。对于混合情形，一个梯度收敛定理在推导这些重数结果中起着关键作用。我们分析的一个关键特征是处理可变奇异指数，允许奇异性在整个定义域内变化。据我们所知，这是首次对海森堡群中混合局部-非局部算子的非线性特征值问题和具有可变奇异指数的奇异问题进行系统研究。因此，这些结果为亚椭圆算子的谱理论开辟了新的方向，并为从反常扩散和非局部相变到非完整系统上的图像分析和控制理论等应用提供了严格的数学基础。

英文摘要

This paper investigates nonlinear eigenvalue problems governed by a family of operators that combine two distinct diffusion mechanisms: a classical local diffusion, which accounts for short-range interactions, and a fractional nonlocal diffusion, which captures long-range effects. The analysis is carried out in the Heisenberg group, a non-Euclidean geometric setting in which motion is constrained to a distinguished set of horizontal directions. As an application of the obtained spectral theory, we establish multiplicity results for perturbed singular problems in both the purely nonlocal and mixed local--nonlocal settings. For the mixed case, a gradient convergence theorem plays a crucial role in deriving these multiplicity results. A key feature of our analysis is the treatment of variable singular exponents, allowing the singularity to vary throughout the domain. To the best of our knowledge, this is the first systematic study of nonlinear eigenvalue problems and singular problems with variable singularity exponents for mixed local-nonlocal operators in the Heisenberg group. The results therefore open a new direction in the spectral theory of subelliptic operators and provide a rigorous mathematical foundation for applications ranging from anomalous diffusion and nonlocal phase transitions to image analysis and control theory on nonholonomic systems.

发表机构

• Indian Institute of Science Education and Research Berhampur（印度科学教育研究学院伯赫拉普尔分校）
• AGH University of Kraków（克拉科夫AGH科技大学）
• Brno University of Technology（布尔诺理工大学）

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